text:
# Analysis of Cardiac Motion and Deformation

## Paper List

Acrobat .pdf Format Gzipped Postscript Format.
See note below for access restrictions.

## Papers

### The Active Elastic Model

Xenophon Papademetris, E. Turan Onat, Albert J. Sinusas, Donald P. Dione, R. Todd Constable and James S. Duncan.
To appear in the proceedings of Information Processing in Medical Imaging (IPMI), 2001.

#### Abstract

Continuum mechanical models have been used to regularize ill-posed problems in many applications in medical imaging analysis such as image registration and left ventricular motion estimation. In this work, we present a significant extension to the common elastic model which we call the \emph{active elastic model}. The active elastic model is designed to reduce bias in deformation estimation and to allow the imposition of proper priors on deformation estimation problems that contain information regarding both the expected magnitude and the expected variability of the deformation to be estimated. We test this model on the problem of left ventricular deformation estimation, and present ideas for its application in image registration and brain deformation during neurosurgery.

#### BibTeX Entry

@INPROCEEDINGS{xPapademetris2001,
author = "X.  Papademetris and  E. T. Onat and A.  J.  Sinusas  and
D.  P.  Dione and R. T. Constable, J. S.  Duncan",
title = "The Active Elastic Model",
booktitle = "Information Processing in Medical Imaging (IPMI)",
year = "2001",
month = "June"}

### Estimation of 3D Left Ventricular Deformation from Echocardiography

Xenophon Papademetris, Albert J. Sinusas, Donald P. Dione and James S. Duncan.
Medical Image Analysis. In-press (March 2001)

#### Abstract

The quantitative estimation of regional cardiac deformation from 3D image sequences has important clinical implications for the assessment of viability in the heart wall. Such estimates have so far been obtained almost exclusively from Magnetic Resonance (MR) images, specifically MR tagging. In this paper we describe a methodology for estimating cardiac deformations from 3D echocardiography (3DE). The images are segmented interactively and then initial correspondence is established using a shape-tracking approach. A dense motion field is then estimated using an anisotropic linear elastic model, which accounts for the fiber directions in the left ventricle. The dense motion field is in turn used to calculate the deformation of the heart wall in terms of strain in cardiac specific directions. The strains obtained using this approach in open-chest dogs before and after coronary occlusion, show good agreement with previously published results in the literature. They also exhibit a high correlation with strains produced in the same animals using implanted sonomicrometers. This proposed method provides quantitative regional 3D estimates of heart deformation from ultrasound images.

#### BibTeX Entry

@article{xPapademetris2001b,
author ="X.  Papademetris and  A.  J.  Sinusas  and D.  P.  Dione and J. S.  Duncan",
title ="Estimation  of {3D} Left Ventricular Deformation from Echocardiography",
journal="Medical Image Analysis",
year ="in-press"}
The accompanying quicktime movies are also available.

### Estimating 3D Strain from 4D Cine-MRI and Echocardiography: In-Vivo Validation

Xenophon Papademetris, Albert J. Sinusas, Donald P. Dione, R. Todd Constable and James S. Duncan.
In the proceedings of MICCAI' 2000 to-appear

#### Abstract

The quantitative estimation of regional cardiac deformation from 3D image sequences has important clinical implications for the assessment of myocardial viability. The validation of such image-derived estimates, however, is a non-trivial problem as it is very difficult to obtain ground truth. In this work we present an approach to validating strain estimates derived from 3D cine-Magnetic Resonance (MR) and 3D Echocardiography (3DE) images using our previously-developed shape-based tracking algorithm. The images are segmented interactively and then initial correspondence is established using a shape-tracking approach. A dense motion field is then estimated using a transversely linear elastic model, which accounts for the fiber directions in the left ventricle. The dense motion field is in turn used to calculate the deformation of the heart wall in terms of strains. The strains obtained using our algorithm are compared to strains estimated using implanted markers and sonomicrometers, which are used as the gold standards. These preliminary studies show encouraging results.

#### BibTeX Entry

@INPROCEEDINGS{xPapademetris99b,
author = "X.  Papademetris and  A.  J.  Sinusas  and
D.  P.  Dione and R. T. Constable, J. S.  Duncan",
title = "Estimating 3D Strain from 4D Cine-MRI and Echocardiography: In-Vivo Validation",
booktitle = "Medical Image Computing and Computer Aided Intervention (MICCAI)",
year = "2000",
month = "October"}

### Cardiac Image Analysis: Motion and Deformation

Xenophon Papademetris and James .S. Duncan
In SPIE Handbook on Medical Imaging - Volume III: Medical Image Processing and Analysis. June 2000.

#### Abstract

In this chapter, we describe research in the area of estimation of cardiac motion and deformation from medical images. We focus primarily on the use of 3D mag-netic resonance image sequences, but we will also discuss the application of some methods to ultrafast CT and 3D echo.

#### BibTeX Entry

@incollection(xPapademetris2000,
author = {X. Papademetris and J. S. Duncan},
title =  {Cardiac Image Analysis: Motion and Deformation},
editor = {J. M. Fitzpatrick and M. Sonka},
booktitle =  {SPIE Handbook on Medical Imaging - Volume III: Medical Image Processing and
Analysis},
publisher = {SPIE},
year =  {2000})

### 3D Cardiac Deformation from Ultrasound Images

Xenophon Papademetris, Albert J. Sinusas, Donald P. Dione and James S. Duncan.
In the proceedings of the Second International Conference on Medical Image Computing and Computer-Assisted Intervention, Cambridge England, September 1999.

#### Abstract

The quantitative estimation of regional cardiac deformation from 3D image sequences has important clinical implications for the assessment of viability in the heart wall. Such estimates have so far been obtained almost exclusively from Magnetic Resonance (MR) images, specifically MR tagging. In this paper we describe a methodology for estimating cardiac deformations from 3D ultrasound images. The images are segmented interactively and then initial correspondence is established using a shape-tracking approach. A dense motion field is then estimated using an anisotropic linear elastic model, which accounts for the fiber directions in the left-ventricle. The dense motion field is in turn used to calculate the deformation of the heart wall in terms of strain in cardiac specific directions. The strains obtained using this approach in open-chest dogs before and after coronary occlusion related to changes in blood flow, show good agreement with previously published results in the literature. This proposed method provides quantitative regional 3D estimates of heart deformation from ultrasound images.

#### BibTeX Entry

@INPROCEEDINGS{xPapademetris99b,
author = "X.  Papademetris and  A.  J.  Sinusas  and
D.  P.  Dione and J. S.  Duncan",
title = "{3D} Cardiac Deformation from Ultrasound Images",
booktitle = "Medical Image Computing and Computer Aided Intervention (MICCAI)",
year = "1999",
pages =  "420-429",
month = "September"}

### Volumetric Deformation Analysis Using Mechanics--Based Data Fusion: Applications in Cardiac Motion Recovery

Pengcheng Shi, Albert J. Sinusas, R. Todd Constable and James S. Duncan.
International Journal of Computer Vision, November 1999.

#### Abstract

Non--rigid motion estimation from image sequences is essential in analyzing and understanding the dynamic behaviors of physical objects. One important example is the dense field motion analysis of the cardiac wall, which could potentially help to better understand the physiological processes associated with heart disease and to provide improvement in patient diagnosis and treatment. In this paper, we present a new method of estimating volumetric deformation by integrating intrinsic instantaneous velocity data with geometrical token displacement information, based upon continuum mechanics principles. This object--dependent approach allows the incorporation of physically meaningful constraints into the ill--posed motion recovery problem, and the integration of the two disparate but complementary data sources overcomes some of the limitations of the single image source based motion estimation approaches.

#### BibTeX Entry

@article(xShi99,
author = "P. Shi and A.J. Sinusas and R. T.  Constable and J.S. Duncan",
title =  "Volumetric Deformation Analysis Using
Mechanics--Based Data Fusion: Applications in Cardiac Motion
Recovery.",
journal = "International Journal of Computer Vision",
volume = "35",
number = "1",
pages = "65-85",
month = "November",
year = "1999")

### Recovery of Soft Tissue Object Deformation using Biomechanical Models

Xenophon Papademetris, Pengcheng Shi, Donald P. Dione, Albert J. Sinusas, R. Todd Constable, and James S. Duncan.
In the proceedings of Information Processing in Medical Imaging (IPMI)", Visegrad Hungary, 1999.

#### Abstract

The estimation of soft tissue deformation from 3D image sequences is an important problem in a number of fields such as diagnosis of heart disease and image guided surgery. In this paper we describe a methodology for using biomechanical material models, within a Bayesian framework which allows for proper modeling of image noise, in order to estimate these deformations. The resulting partial differential equations are discretized and solved using the finite element method. We demonstrate the application of this method to estimating strains from sequences of in-vivo left ventricular MR images, where we incorporate information about the fibrous structure of the ventricle. The deformation estimates obtained exhibit similar patterns with measurements obtained from more invasive techniques, used as a gold standard.

#### BibTeX Entry

@INPROCEEDINGS{xPapademetris99a,
author = "X.  Papademetris and  P.  Shi and D.  P.  Dione
and A.  J.  Sinusas  and J.  S.  Duncan",
title = "Recovery of Soft Tissue Object Deformation using Biomechanical Models. ",
booktitle = "Information Processing in Medical Imaging",
year = "1999",
pages =  "352-357",
month = "June"}

### Point-Tracked Quantitative Analysis of Left Ventricular Motion from 3D Image Sequences

Pengcheng Shi, Albert J. Sinusas, R. Todd Constable Eric Ritman and James .S. Duncan
IEEE TMI January 2000

#### Abstract

We propose and validate the hypothesis that we can use differential shape properties of the myocardial surfaces to recover dense field motion from standard three--dimensional image data (MRI and CT). Quantitative measures of left ventricular regional function can be further inferred from the point correspondence maps. The non-invasive, algorithm--derived results are validated in two levels: the motion trajectories are compared to those of implanted imaging--opaque markers of a canine model in two imaging modalities, where sub--pixel accuracy is achieved, and the validity of using motion parameters (path length and thickness changes) in detecting myocardial injury is tested by comparison to post--mortem TTC staining of myocardial tissue, where the Pearson product--moment correlation value is 0.968.

#### BibTeX Entry

@ARTICLE{xship2000,
author = "P.  Shi and A.  J.  Sinusas and R.  T.  Constable
and E.  Ritman and J.  S.  Duncan",
title = "Point-Tracked Quantitative Analysis of
Left Ventricular Motion from {3D} Image Sequences",
journal = "IEEE Transactions on Medical Imaging",
year = "2000",
month="January",
volume="19",
number="1",
pages="36-50" }

### Physical and geometrical modeling for image-based recovery of left ventricular deformation

J.S. Duncan, P.Shi, R.T. Constable, and A.J. Sinusas.
Progress in Biophysics and Molecular Biology, 69(2-3):333--351, 1998.

#### Abstract

Information about left ventricular (LV) mechanical performance is of critical importance in understanding the etiology of ischemic heart disease. Regional measurements derived from noninvasive imaging to assist in assessing this performance have been in use for decades, and certain parameters derived from these measurements often are useful clinically, as they correlate to some extent with gross physiological hypotheses. However, relatively little work has been done to date to carefully understand the relationship of regional myocardial injury to the local mechanical performance of the heart as derived from image data acquired non- invasively for a particular patient in 3 spatial dimensions over time. This paper describes efforts to take advantage of recent developments in 3D non-invasive imaging and biome- chanical modeling to design an integrated computational platform capable of assembling a variety of displacement and velocity data derived from each image frame to deform a vol- umetric model representation of a portion of the myocardium. A brief description of both the reasoning behind this strategy is put forth, as well as an overview of the approach and some initial results are described.

#### BibTeX Entry

@article(xDuncan98,
author = "J. S.  Duncan and P.  Shi and R. T.  Constable and A.  Sinusas",
title =  "Physical and Geometrical Modeling for Image-Based Recovery
of Left Ventricular Deformation",
journal = "Progress in Biophysics and Molecular Biology",
volume = "69",
number = "2-3",
pages = "333-351",
year = "1998")

### Visually Interactive Cine-3D Segmentation of Cardiac MR Images

Xenophon Papademetris, James Rambo, Donald P. Dione, Albert J. Sinusas and James S. Duncan.
Supplement to the Journal of the American College of Cardiology Volume 31, Number 2 (Supplement A) February 1998.

A Handout that was given during the poster session is available on-line.

#### BibTeX Entry

@ARTICLE{xPapademetris98,
author = "X.  Papademetris and  J.  V.  Rambo  and D.  P.  Dione and
A.  J.  Sinusas  and  J.  S.  Duncan",
title = "Visually Interactive Cine-{3D} Segmentation of Cardiac {MR} Images",
journal = "Suppl.  to the J.  Am.  Coll.  of Cardiology",
volume = "31",
number = "2.  Suppl.  A",
month = "February",
year = "1998"}

### Tracking Myocardial Deformation Using Phase Contrast MR Velocity Fields: A Stochastic Approach

Meyer F.G., Constable R.T., Sinusas A.J., Duncan J.S.
IEEE Transactions on Medical Imaging, Vol. 15, No. 4, August 1996, pages 453-465.

#### Abstract

In this paper, we propose a new approach for tracking the deformation of the Left Ventricular (LV) myocardium from two-dimensional Magnetic Resonance (MR) phase contrast velocity fields. The use of phase contrast MR velocity data in cardiac motion problems has been introduced by others [1] and shown to be po tentially useful for tracking discrete tissue elements, and therefore characteri zing LV motion. However, we show here that these velocity data i.) are extremely noisy near the LV borders and ii.) cannot alone be used to estimate the motion and the deformation of the entire myocardium due to noise in the velocity fields . In this new approach, we use the natural spatial constraints of the endocardia l and epicardial contours, detected semi-automatically in each image frame, to h elp remove noisy velocity vectors at the LV contours. The information from both the boundaries and the phase contrast velocity data is then integrated into a de forming mesh that is placed over the myocardium at one time frame and then track ed over the entire cardiac cycle. The deformation is guided by a Kalman filter t hat provides a compromise between a.) believing the dense field velocity and the contour data when it is crisp and coherent in a local spatial and temporal sens e and b.) employing a temporally smooth cyclic model of cardiac motion when cont our and velocity data are not trustworthy. The Kalman filter is particularly wel l suited to this task as it produces an optimal estimate of the LV's kinematics (in the sense that the error is statistically minimized) given incomplete and no ise corrupted data, and given a basic dynamical model of the LV. The method has been evaluated with simulated data ; the average error between tracked nodes and theoretical position was 1:8% of the total path length. The algorithm has also been evaluated with phantom data ; the average error was 4:4% of the total path length. We show that in our initial tests with phantoms that the new approach sh ows small, but concrete improvements over previous techniques that used primaril y phase contrast velocity data alone. We feel that these improvements will be am plified greatly as we move to direct comparisons in in vivo and three-dimensional datasets.

#### BibTeX Entry

@ARTICLE{xMeyer96,
AUTHOR = "F.  G.  Meyer and R.  T.  Constable and A. J.
Sinusas and J.  S.  Duncan",
TITLE = "Tracking Myocardial Deformation Using Phase Contrast {MR}
Velocity Fields: A Stochastic Approach",
JOURNAL ="IEEE Transactions on Medical Imaging",
VOLUME = "15",
NUMBER = "4",
MONTH = "August",
YEAR = 1996}

### A Model-Based Integrated Approach to Track Myocardial Deformation Using Displacement and Velocity Constraints

Pengcheng Shi, Glynn Robinson, R. Todd Constable, Albert J. Sinusas and James S. Duncan.
Proceedings of the International Conference on Computer Vision (ICCV) 1995.

#### Abstract

Accurate estimation of heart wall dense field motion and deformation could help to better understand the physiological processes associated with ischemic heart diseases, and to provide significant improvement in patient treatment. We present a new method of estimating left ventricular deformation which integrates instantaneous velocity information obtained within the mid-wall region with shape information found on the boundaries of the left ventricle. Velocity information is obtained from phase contrast magnetic resonance images, and boundary information is obtained from shape-based motion tracking of the endo- and epi-cardial boundaries. The integration takes place within a continuum biomechanical heart model which is embedded in a finite element framework. We also employ a feedback mechanism to improve tracking accuracy. The integration of the two disparate but complementary sources overcomes some of the limitations of previous work in the field which concentrates on motion estimation from a single image-derived source.

#### BibTeX Entry

@inproceedings{xShi_ICCV95,
author = "P. Shi and G. Robinson and R. T. Constable and A. Sinusas and J. Duncan",
title = "A Model-Based Integrated Approach to Track Myocardial Deformation
Using Displacement and Velocity Constraints",
booktitle =  "Fifth International Conference on Computer Vision",
year =  "1995",
pages =  "687-692"}

### A Unified Framework to Assess Myocardial Function from 4D Images

Pengcheng Shi, Glynn P. Robinson, Amit Chakraborty, Lawrence H. Staib, R. Todd Constable, Albert J. Sinusas and James S. Duncan.
Proceedings of the Computer Vision, Virtual Reality and Robotics in Medicine (CVRMed), Nice, France April 1995.

#### Abstract

This paper describes efforts aimed at developing a unified framework to more accurately quantify the local, regional and global function of the left ventricle (LV) of the heart, under both normal and ischemic conditions, using four--dimensional (4D) imaging data over the entire cardiac cycle. The approach incorporates motion information derived from the shape properties of the endocardial and epicardial surfaces of the LV, as well as mid--wall 3D instantaneous velocity information from phase contrast MR images, and/or mid--wall displacement information from tagged MR images. The integration of the disparate but complementary sources of information overcomes the limitations of previous work which concentrates on motion estimation from a single image--derived source.

#### BibTeX Entry

@inproceedings{xShi_CVRMed95,
author = "P. Shi and G. Robinson and A. Chakraborty and L. Staib and R. T. Constable
and A. Sinusas and J. Duncan",
title = "A Unified Framework to Assess Myocardial Function from {4D} Images",
booktitle =  "Lecture Notes in Computer Science: First
International Conference on Computer Vision, Virtual Reality,
and Robotics in Medicine",
year =  "1995",
pages =  "327-337"}

### A Recursive Filter for Phase Velocity Assisted Shape-based Tracking of Cardiac Non-rigid Motion

J. McEachen, F. Meyer,R. Constable, A. Nehorai and J.S. Duncan
Proceedings of the International Conference on Computer Vision (ICCV) 1995.

### Myocardial Motion and Function Assessment Using 4D Images

P. Shi, G. Robinson, and J. Duncan.
In the proceedings of the IEEE conference on Visualization in Biomedical Computing, Rochester MN, Oct 1994.

#### Abstract

This paper describes efforts aimed at more objectively and accurately quantifying the local, regional and global function of the left ventricle (LV) of the heart from four-dimensional (4D) image data. Using our shape-based image analysis methods, point-wise myocardial motion vector fields between successive image frames through the entire cardiac cycle will be computed. Quantitative LV motion, thickening, and strain measurements will then be established from the point correspondence maps. In the paper, we will also briefly describe an in vivo experimental model which uses implanted imaging-opaque markers to validate the results of our image analysis methods. Finally, initial experimental results using image sequences from two different modalities will be presented.

#### BibTeX Entry

@inproceedings{xShi_VBC94,
author = "P. Shi and G. Robinson and J. Duncan",
title = "Myocardial Motion and Function Assessment Using {4D} Images",
booktitle =  "Visualization in Biomedical Computing",
year =  "1994",
pages =  "148-159"}

### Analysis of Cardiac Motion with Recursive Comb Filtering

J. McEachen, A. Nehorai and J.S. Duncan

#### BibTeX Entry

@inproceedings{xMcEachen,
author = "J. C. McEachen II and A. Nehorai and J. S. Duncan",
title = "A recursive filter for temporal analysis of cardiac motion",
booktitle =  "Proceedings of the IEEE Workshop on Biomedical Image Analysis",
year =  "1994",
pages =  "124-133"}

## Sonomicrometer Papers

This is related work on obtaining cardiac strains from implanted sonomicrometers.

### Three-dimensional regional left ventricular deformation from digital sonomicrometry

D.P. Dione, P.Shi, W.Smith, P.De Man, J.Soares, J.S. Duncan, and A.J. Sinusas.
In 19th Annual International Conference of the IEEE Engineering in Medicine and Biology Society}, pages 848--851, Chigago, IL, March 1997.

#### Abstract

Understanding how the left ventricle deforms in 3D and how this deformation is altered with coronary occlusion may lead to the development of non-invasive imaging techniques to determine the extent of permanent injury. To determine regional 3D strains in the left ventricle of the heart we employed digital sonomicrometry, with high temporal and spatial resolution. Two cubic arrays of 8 omni-directional transceiver crystals were implanted in two regions of the left ventricle in an open chest canine preparation ($n=6$). Additional crystals were used to define a fixed external reference space and the long axis of the ventricle. Using ultrasound transit time the distances between all the crystals were recorded. A multidimensional scaling technique was then applied to transform the distances to 3D crystal coordinates. A least squares fit of the displacement field was applied to calculate homogeneous strains for each cube. Cardiac specific directions were determined and strains rotated into the local coordinate space. This technique was applied pre- and post- coronary occlusion. Alterations in strain patterns were evident in the ischemic region and subtle temporal changes in the control region. Thus, digital sonomicrometry, with high temporal and spatial resolution, enhances our ability to analyze regional left ventricular 3D strain patterns.

#### BibTeX Entry

@INPROCEEDINGS{xDione97,
author="D.  P.  Dione and P.  Shi and W.  Smith and P.  De Man and J.  Soares
and J.  S.  Duncan and A.  J.  Sinusas",
title ="Three-Dimensional Regional Left Ventricular Deformation
from Digital Sonomicrometry",
booktitle = "19th Annual International Conference of the IEEE Engineering
in Medicine and Biology Society",
pages = "848-851",
year = "1997",
month="March"}

### Three dimensional digital sonomicrometry: Comparison with biplane radiography

D. Meoli, R. Mazhari, D. P. Dione, J. Omens, A. McCulloch, and A. J. Sinusas.
In Proceedings of IEEE 24th Annual Northeast Bioengineering Conference, pages 64--67, 1998.

#### Abstract

This paper describes a three-dimensional (3D) digital sonomicrometry approach for locating and tracking 3D objects. A commercial digital sonomicrometry system was employed to measure scalar distances between omni-directional sonomicrometers. 3D coordinates were then derived using the statistical technique of multidimensional scaling (MDS). 3D digital sonomicrometry was directly compared with biplane radiography of the ultrasound crystals for estimation of 3D distances in static phantoms and in vivo using an experimental canine preparation. An excellent correlation (r=0.992) was seen when comparing inter-crystal distances derived from biplane radiography and sonomicrometry 3D coordinate data in the gel phantom. A Bland-Altman analysis shows that the average difference in coordinate determined distance between these two different methodologies was only 0.63$\pm$0.46 mm, over a range of inter-crystal distances of 3.14 to 17.28 mm. In the in vivo canine preparation, the correlation between the sonomicrometry derived and biplane derived distances was also excellent (r=0.992) with a slope of 1.05 and an intercept of 0.06. The Bland-Altman analysis shows that the average difference in coordinate determined distance between these two different methodologies was only 0.78$\pm$0.74 mm, over a range of inter-crystal distances of 2.90 to 27.66 mm. We have demonstrated the feasibility of accurately measuring scalar distances using 3D digital sonomicrometry. Digital sonomicrometry combines high spatial and temporal resolution with availability and portability to accurately measure distances in a closely packed array of implanted piezoelectric crystals.

#### BibTeX Entry

@INPROCEEDINGS{xMeoli98,
author = "D.  Meoli and R.  Mazhari and D.  P.  Dione and J.  Omens and
A.  McCulloch and A.  J.  Sinusas",
title = "Three Dimensional Digital Sonomicrometry: Comparison with
booktitle = "Proceedings of IEEE 24th Annual Northeast Bioengineering Conference",
year = "1998",
pages = "64-67" }

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text:
How many iterations are needed for the bisection algorithm?

<TL;DR>

• The bisection algorithm is a very simple algorithm for finding the root of a 1-D function.
• Working out the number of iterations of the algorithm required to determine the root location within a specified tolerance can be determined from a very simple little hack, which I explain here.
• Things get more interesting when we consider variants of the bisection algorithm, where we cut an interval into unequal portions.

</TL;DR>

A little while ago a colleague mentioned that they were repeatedly using an off-the-shelf bisection algorithm to find the root of a function. The algorithm required the user to specify the number of iterations to run the bisection for. Since my colleague was running the algorithm repeatedly they wanted to set the number of iterations efficiently and also to achieve a guaranteed level of accuracy, but they didn’t know how to do this.

I mentioned that it was very simple to do this and it was a couple of lines of arithmetic in a little hack that I’d used many times. Then I realised that the hack was obvious and known to me because I was old – I’d been doing this sort of thing for years. My colleague hadn’t. So I thought the hack would be a good subject for a short blog post.

The idea behind a bisection algorithm is simple and illustrated in Figure 1 below.

At each iteration we determine whether the root is to the right of the current mid-point, in the right-hand interval, or to the left of the current mid-point, in the left-hand interval. In either case, the range within which we locate the root halves. We have gone from knowing it was in the interval $[x_{lower}, x_{upper}]$, which has width $x_{upper}-x_{lower}$, to knowing it is in an interval of width $\frac{1}{2}(x_{upper}-x_{lower})$. So with every iteration we reduce our uncertainty of where the root is located by half. After $N$ iterations we have reduced our initial uncertainty by $(1/2)^{N}$. Given our initial uncertainty is determined by the initial bracketing of the root, i.e.  an interval of width $(x_{upper}^{(initial)}-x_{lower}^{(initial)})$, we can now work out that after $N$ iterations we have narrowed down the root to an interval of width ${\rm initial\;width} \times \left ( \frac{1}{2}\right ) ^{N}$. Now if we want to locate the root to within a tolerance ${\rm tol}$, we just have to keep iterating until the uncertainty reaches ${\rm tol}$. That is, we run for $N$ iterations where $N$ satisfies,

$\displaystyle N\;=\; -\frac{\ln({\rm initial\;width/tol})}{\ln\left (\frac{1}{2} \right )}$

Strictly speaking we need to run for $\lceil N \rceil$ iterations. Usually I will add on a few extra iterations, e.g. 3 to 5, as an engineering safety factor.

As a means of easily and quickly determining the number of iterations to run a bisection algorithm the calculation above is simple, easy to understand and a great little hack to remember.

Is bisection optimal?

The bisection algorithm works by dividing into two our current estimate of the interval in which the root lies. Dividing the interval in two is efficient. It is like we are playing the childhood game “Guess Who”, where we ask questions about the characters’ features in order to eliminate them.

Asking about a feature that approximately half the remaining characters possess is the most efficient – it has a reasonable probability of applying to the target character and eliminates half of the remaining characters. If we have single question, with a binary outcome and a probability $p$ of one of those outcomes, then the question that has $p = \frac{1}{2}$ maximizes the expected information (the entropy), $p\ln (p)\;+\; (1-p)\ln(1-p)$.

Dividing the interval unequally

When we first played “Guess Who” as kids we learnt that asking questions with a much lower probability $p$ of being correct didn’t win the game. Is the same true for our root finding algorithm? If instead we divide each interval into unequal portions is the root finding less efficient than when we bisect the interval?

Let’s repeat the derivation but with a different cut-point e.g. 25% along the current interval bracketing the root. In general we can test whether the root is to the left of right of a point that is a proportion $\phi$ along the current interval, meaning the cut-point is $x_{lower} + \phi (x_{upper}-x_{lower})$. At each iteration we don’t know in advance which side of the cut-point the root lies until we test for it, so in trying to determine in advance the number of iterations we need to run, we have to assume the worst case scenario and assume that the root is still in the larger of the two intervals. The reduction in uncertainty is then, ${\rm max}\{\phi, 1-\phi\}$. Repeating the derivation we find that we have to run at least,

$\displaystyle N_{Worst\;Case}\;=\;\ -\frac{\ln({\rm initial\;width/tol})}{\ln\left ({\rm max}\{\phi, 1 - \phi \right \})}$

iterations to be guaranteed that we have located the root to within $tol$.

Now to determine the cut-point $\phi$ that minimizes the upper bound on number of iterations required, we simply differentiate the expression above with respect to $\phi$. Doing so we find,

$\displaystyle \frac{\partial N_{Worst\;Case}}{\partial \phi} \;=\; -\frac{\ln({\rm initial\;width/tol})}{ (1-\phi) \left ( \ln (1 - \phi) \right )^{2}} \;\;,\;\; \phi < \frac{1}{2}$

and

$\displaystyle \frac{\partial N_{Worst\;Case}}{\partial \phi} \;=\; \frac{\ln({\rm initial\;width/tol})}{\phi \left ( \ln (\phi) \right)^{2}} \;\;,\;\; \phi > \frac{1}{2}$

The minimum of $N_{Worst\;Case}$ is at $\phi =\frac{1}{2}$, although $\phi=\frac{1}{2}$ is not a stationary point of the upper bound $N_{Worst\;Case}$, as $N_{Worst\;Case}$ has a discontinuous gradient there.

That is the behaviour of the worst-case scenario. A similar analysis can be applied to the best-case scenario – we simply replace $max$ with $min$ in all the above formula. That is, in the best-case scenario the number of iterations required is given by,

$\displaystyle N_{Best\;Case}\;=\;-\frac{\ln({\rm initial\;width/tol})}{\ln\left ({\rm min}\{\phi, 1 - \phi \right \})}$

Here, the maximum of the best-case number of iterations occurs when $\phi = \frac{1}{2}$.

That’s the worst-case and best-case scenarios, but how many iterations do we expect to use on average? Let’s look at the expected reduction in uncertainty in the root location after $N$ iterations. In a single iteration a root that is randomly located within our interval will lie, with probability $\phi$, in segement to the left of our cut-point and leads to a reduction in the uncertainty by a factor of $\phi$. Similarly, we get a reduction in uncertainty of $1-\phi$ with probability $1-\phi$ if our randomly located root is to the right of the cut-point. So after $N$ iterations the expected reduction in uncertainty is,

$\displaystyle {\rm Expected\;reduction}\;=\;\left ( \phi^{2}\;+\;(1-\phi)^{2}\right )^{N}$

Using this as an approximation to determine the typical number of iterations, we get,

$\displaystyle N_{Expected\;Reduction}\;=\;-\frac{\ln({\rm initial\;width/tol})}{\ln\left ( \phi^{2} + (1-\phi)^{2} \right )}$

This still isn’t the expected number of iterations, but to see how it compares Figure 2 belows shows simulation estimates of $\mathbb{E}\left ( N \right )$ plotted against $\phi$ when the root is random and uniformly distributed within the original interval.

For Figure 2 we have set $w = ({\rm initial\;width/tol}) = 0.01$. Also plotted in Figure 2 are our three theoretical estimates, $\lceil N_{Worst\;Case}\rceil, \lceil N_{Best\;Case}\rceil, \lceil N_{Expected\;Reduction}\rceil$. The stepped structure in these 3 integer quantities is clearly apparent, as is how many more iterations are required under the worst case method when $\phi \neq \frac{1}{2}$.

The expected number of iterations required, $\mathbb{E}( N )$, actually shows a rich structure that isn’t clear unless you zoom in. Some aspects of that structure were unexpected, but requires some more involved mathematics to understand. I may save that for a follow-up post at a later date.

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text:
# 0!=1

Why is $0! = 1$ ?

Note by Srijan Singh
11 months, 3 weeks ago

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Note: $\displaystyle n! = \Gamma(n+1) = \int_{0}^{\infty}t^n e^{-t}dt$

\begin{aligned} 0! = \Gamma(1) = \int_{0}^{\infty}t^0 e^{-t}dt &= \int_{0}^{\infty}e^{-t}dt \\ &= -e^{-t}\bigg|^{\infty}_{0} \\ &= -e^{-\infty} -(-e^0) = 0 + 1 = \boxed{1} \end{aligned}

- 9 months, 3 weeks ago

Thanks for explanation.

- 9 months, 3 weeks ago

- 11 months, 3 weeks ago

0! = 1 :

1! = 1

2! = 1!*2

2! = 2

3! = 2!*3

3! = 6

4! = 3!*4

4! = 24

turn this around:

4! = 24

3! = 4!/4

3! = 6

2! = 3!/3

2! = 2

1! = 2!/2

1! = 1

0! = 1!/1

0! = 1

- 11 months, 3 weeks ago

Yeah, same reason!

- 11 months, 3 weeks ago

Its the only proof I know for 0! @Mahdi Raza, do you know any others?

- 11 months, 3 weeks ago

There was one related to $n \choose 0$. The ways of choosing 0 objects from a group of $n$ objects is $\boxed{1}$. Hence:

\begin{aligned} n \choose 0 &= \dfrac{n!}{(n-0)!0!} \\ \\ 1 &= \dfrac{n!}{n!0!} \\ \\ 0! &= \dfrac{\cancel{n!}}{\cancel{n!}} \\ \\ 0! &= \boxed{1} \end{aligned}

- 11 months, 3 weeks ago

Cool @Mahdi Raza, Thanks :)

- 11 months, 3 weeks ago

I was about to comment the proof-related to combinations after I saw Percy's proof, but then I saw you writing this (sad noises) LOL. I will write the proof-related to permutations then.

- 11 months, 2 weeks ago

@Percy Jackson nice

- 11 months, 3 weeks ago

Thanks :)

- 11 months, 3 weeks ago

The formula for no. of permutations is ${ _{ n }{ P }_{ k } }=P(n,k)=\frac { n! }{ (n-k)! }$ or basically if we take any $k$ objects from $n$ objects at a time, then how many ways can the $k$ objects be arranged is the number of permutations.

So, if we take $n$ objects from $n$ objects and ask for the number of ways we can arrange them, basically we are asking how many ways can all the $n$ objects be arranged which by a primary definition of factorials is n! So using permutations we can write as,

$n!=\frac { n! }{ (n-n)! }$

Thus, $n!=\frac { n! }{ 0! }$

By simplifying, we get 0!=1.

- 11 months, 2 weeks ago

A common-sensical(Yeah, I just invented that word, deal with it) approach would be that 0! is 0 multiplied 0 times because 1 is greater than 0. 0 multiplied 0 times is $0^{0}$, aka 1 :) @SRIJAN Singh @Siddharth Chakravarty

- 11 months, 2 weeks ago

What do you even mean? I would suggest the common-sensical way, Lol that 0! Means how many ways can we arrange 0 objects which should be 1 i.e there is no way or to do nothing.

- 11 months, 2 weeks ago

$n! = n(n-1)(n-2) \ldots 1$ or $n!$ is product of all numbers less than n and greater than 1. There are 0 ways to reach 1 from 0 like that $0! = 0^{0} = 1$. THis is in a vague sense @Siddharth Chakravarty

- 11 months, 2 weeks ago

Did you just defy yourself because you said product of all number less than n and greater than 1, and 0 has no number less than it which is greater than 1? Basically the factorial notation was brought into due to permutations and combinations as I said in the comment.

- 11 months, 2 weeks ago

For a (possibly) more intuitive explanation, recall that $n!$ may be interpreted as the number of ways to arrange $n$ distinct objects in a line.

For example, $2!=2$ and we can arrange $\square$ and $\blacksquare$ in 2 ways: $\square\blacksquare$ and $\blacksquare\square$.

$1!=1$, and there is one way to arrange the single item $\blacksquare$, like this: $\blacksquare$.

So $0!$ is the number of ways to arrange nothing. There is one way to arrange nothing in a line, like this:

So $0!=1$, though I admit that the interpretation of arranging "nothing" is a little philosophical.

- 11 months, 2 weeks ago

What's with the notifications? I got like 10 of them in this note, leading nowhere, are you guys sending and deleting comments @Siddharth Chakravarty and @SRIJAN Singh ????

- 11 months, 2 weeks ago

i got 34 comments a day

- 11 months, 2 weeks ago

did it ever go to 100+? I left brilliant for one day, I had 100+ notifications :)

- 11 months, 2 weeks ago

yah i also if i hover to notification button and then click ,if i get bore scrolling comments than CLICK DISMISS ALL LOL USED IF many times

- 11 months, 2 weeks ago

- 11 months, 2 weeks ago

@Mahdi Raza. do this

- 11 months, 2 weeks ago

- 11 months, 2 weeks ago

To trouble you. LOL :) Actually, I am not deleting @SRIJAN Singh did, he had the main comment.

- 11 months, 2 weeks ago

$\LARGE \textsf{Unsubscribe :)}$

- 11 months, 2 weeks ago

$\text{\large I think to do the same, but I am not doing, incase somebody comments something useful, notifications don't worry much}$

- 11 months, 2 weeks ago

$\Large \textsf{Why am I still getting notifications?}$

$\Large \textsf{I already Unsubbed this note.........}$

- 11 months, 2 weeks ago

because this note loves you

- 11 months, 2 weeks ago

# ....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

- 11 months, 2 weeks ago

@Percy Jackson why are you making drawings of fullstop lol

- 11 months, 2 weeks ago

- 11 months, 2 weeks ago

- 11 months, 2 weeks ago

i asked some of the questions like how was your naest exam and ask him to do my latest problems

- 11 months, 2 weeks ago

@Percy Jackson,@Siddharth Chakravarty.do this

- 11 months, 2 weeks ago

$n!$ is the number of ways you can order a set with $n$ elements, and the only way to order an empty set (with $0$ elements) is the empty set {} itself.

- 9 months, 3 weeks ago

...hi @SRIJAN Singh, you actually look pissed

What is going on with you both (referring to NEED TO NOTIFY THE STAFF? COMMENT BELOW)

If you feel this comment is offensive, I will delete this comment, k?

I just wanna know what's happening between you and Andrew (yes, that is his name...)

- 9 months ago

nothin much he's tryin to become big daddy and treats me like i m his target idk why?

- 9 months ago


text:
[<< wikibooks] Physics Study Guide/Style Guide
==== Equations ====
For uniformity in the book a template should be used for embeding equations.  View the template at Template:PSG/eq use it as follows

{{PSG/eq|$ax^2+bx+c=0$}}

Which gives

==== Units ====
Units should be given using the /mathrm{} command and small spaces \, between the units.  Vectors should be bold and italicized if they are variables with the /boldsymbol{} command.

{{PSG/eq|$\sum\Delta\boldsymbol\left\{p\right\}=0~\mathrm\left\{kg\,m/s\right\}$}}

Barry N. Taylor (2004), Guide for the Use of the International System of Units (SI) (version 2.2). Available[1]. National Institute of Standards and Technology, Gaithersburg, MD.

== Global Style Guidelines ==
Wikibooks Manual of Style


text:
Weak analogue of Hartogs' Theorem: On Noetherian schemes, the restriction of a coherent sheaf to an open set with complement of codimension 2 in the sheaf's support, is coherent.

See and see [IV, Proposition 7.2.2, EGA] for a special case.

Proposition 51.8.7 (Kollár). Let $j : U \to X$ be an open immersion of locally Noetherian schemes with complement $Z$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module. The following are equivalent

1. $j_*\mathcal{F}$ is coherent,

2. for $x \in \text{Ass}(\mathcal{F})$ and $z \in Z \cap \overline{\{ x\} }$ and any associated prime $\mathfrak p$ of the completion $\mathcal{O}_{\overline{\{ x\} }, z}^\wedge$ we have $\dim (\mathcal{O}_{\overline{\{ x\} }, z}^\wedge /\mathfrak p) \geq 2$.

Proof. If (2) holds we get (1) by a combination of Lemmas 51.8.1, Remark 51.8.5, and Lemma 51.8.4. If (2) does not hold, then $j_*i_{x, *}\mathcal{O}_{W_ x}$ is not finite for some $x \in \text{Ass}(\mathcal{F})$ by the discussion in Remark 51.8.6 (and Remark 51.8.5). Thus $j_*\mathcal{F}$ is not coherent by Lemma 51.8.1. $\square$

Comment #2598 by Rogier Brussee on

Suggested slogan: (weak analogue of Hartogh's theorem) On Noetherian schemes, the restriction of a coherent sheaf to an open set with complement of codimension 2 in the sheaf's support, is coherent.

Comment #3409 by Kestutis Cesnavicius on

One could include a reference to a special case: EGAIV2, 7.2.2.

Comment #3468 by on

Yes, you are right. I have added the reference. Strangely the proof of the result in EGA shows something more general than what they state (this almost never happens in EGA). See changes here.

Comment #4932 by Kollar on

This is now Thm 2 in

@ARTICLE{k-coherent, AUTHOR = {Koll{\'a}r, J{\'a}nos}, TITLE = {Coherence of local and global hulls}, JOURNAL = {Methods Appl. Anal.}, FJOURNAL = {Methods and Applications of Analysis}, VOLUME = {24}, YEAR = {2017}, NUMBER = {1}, PAGES = {63--70}, ISSN = {1073-2772}, MRCLASS = {14F05 (13B22 13E05)}, MRNUMBER = {3694300}, URL = {https://doi.org/10.4310/MAA.2017.v24.n1.a5}, }

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).


text:
# Using LaTeX for writing research papers

Many researchers are using Microsoft Word  for writing research papers. However, Microsoft Word has several problems or limitations.  In this blog post, I will discuss the use of LaTeX as an alternative to Microsoft Word for writing research papers.

What is LaTeX?

LaTeX is a document preparation system, proposed in the 1980s. It is used to create documents such as research papers, books, or even slides for presentations.

The key difference between LaTeX and software like Microsoft Word is that Microsoft Word let you directly edit your document and immediately see the result, while using LaTeX is a bit like programming. To write a research paper using LaTeX, you have to write a text file with the .tex extension using a formatting language to roughly indicate how your paper should look like. Then, you can run the LaTeX engine to generate a PDF file of your research paper. The following figure illustrate this process:

In the above example, I have created a very simple LaTeX document (Example.tex) and then I have generated the corresponding PDF for visualization (Example.pdf).

Why using LaTeX?

There are several reasons why many researchers prefer LaTeX to Microsoft Word for writing research papers. I will explain some of them, and then I will discuss also some problems about using LaTeX.

Reason 1: LaTeX papers generally look better

LaTeX papers often look better than papers written using Microsoft Word. This is especially true for fields like computer science, mathematics and engineering where mathematical equations are used.  To illustrate this point, I will show you some screenshots of a paper that I have written for the ADMA 2012 conference a few years ago. For this paper, I had made two versions: one using the Springer LNCS LaTeX template and the other one using the Springer LNCS Microsoft Word template.

This is the first page of the paper.

The first page is quite similar. The main difference is the font being used, which is different using LaTeX. Personally, I prefer the default LaTeX font. Now let’s compare how the mathematical equations appears in Latex and Word.

Here, we can see that mathematical symbols are more beautiful using LaTeX. For example, the set union  and the subset inclusion operators are in my opinion quite ugly in Microsoft Word. The set union operator of Word looks too much like the letter “U”. In this example, the mathematical equations are quite simple. But LaTeX really shines when displaying more complex mathematical equations, for example using matrices.

Now let’s look at another paragraph of text from the paper to further compare the appearance of Word and LaTeX papers:

In the above picture,  it can be argued that both LaTeX and Word papers look quite similar. For me, the big difference is again in the font being used. In the Springer Word  template, the Times New Roman font, while LaTeX has its own default font.  I prefer the LaTeX font. Also, I think that the URLs look better in LaTeX using the url package.

Reason 2: LaTeX is available for all platforms

The LaTeX system is free and available for most operating systems, and documents will look the same on all operating systems.

To install LaTeX on your computer you need to install a LaTeX distribution such as MikTeK  ( https://miktex.org/ ). After installing LaTeX, you can start working on LaTeX documents using a text editor such as Notepad. However, it is more convenient to also install an editor such as TexWorks or WinShell. Personally, I use TexWorks.  This is a screenshot of my working environment using TexWorks:

I will open my LaTeX document on the left window. Then, the right window will display the PDF generated by LaTeX. Thus, I can work on the LaTeX code of my documents on the left and see the result on the right.

If you want to try LaTeX without installing it on your computer, you can use an online LaTeX editor such as ShareLatex (http://www.sharelatex.org ) or OverLeaf.  Using these editors, it is not necessary to install LaTeX on your computer. I personally sometimes use ShareLatex as it also has some function for collaboration (history, chat, etc.), which is very useful when working on a research paper with other people.

Reason 3: LaTeX offers many packages

Besides the basic functionalities of LaTeX, you can install hundreds of packages to add more features to LaTeX. If you use MikTek for example, there is a tool called the “MikTek package manager” that let you choose and install packages.  There are packages for about everything from packages to display algorithms to packages for displaying chessboards. For example, here is some algorithm pseudocode that I have written in one of my recent paper using a LaTeX package called algorithm2e:

As you can see the presentation of the algorithm is quite nice. Doing the same using Word would be very difficult. For example, it would be quite difficult to add a vertical line for the “for” loop using Microsoft Word.

Reason 4: You don’t need to worry about how your document will look like

When writing a LaTeX document, you don’t need to worry about how your final document will look like. For example, you don’t need to worry about where the figures and tables will appear in your document or where the page breaks will be.  All of this is handled by the LaTeX engine during the compilation of your document. When writing document, you only need to use some basic formatting instructions such as indicating when a new section starts in your document. This let you focus on writing.

Reason 5: LaTeX can generate and update your bibliography automatically

Another reason for using LaTeX is that it can generate the bibliography of a document automatically. There are different ways of writing a bibliography using LaTeX. One of the most common way is to use a .bib file. A .bib file provide  a list of references that can be used in your  document.  Then, you can use these references in your .tex document using the \cite{} command and the bibliography will be automatically generated.

I will illustrate this with an example:

A), I have created a Latex document (a .tex file) where I cite a paper called “efim” using the LaTeX command \cite{efim}.

B) I have created a corresponding LaTeX bib file that provides bibliographical information about the “efim” paper.

C) I have generated  the PDF file using the  .tex file and the .bib file.  As you can see, the \cite{} command has been replaced by 25, and the corresponding entry 25 has been automatically generated in the correct format for this paper and added to the bibliography.

The function for generating a bibliography using LaTeX can save a lot of time to researchers especially for documents containing many references such as thesis, books, and journal papers.

Moreover, once you have created a .bib file, you can reuse it in many different papers.  And  it is also very easy to change the style of your bibliography. For example, if you want to change from the APA style to the IEEE style, it can be done almost automatically, which saves lot of time.

In Microsoft Word, there is some basic tool for generating a bibliography but it provides much less features than LaTeX.

Reason 6: LaTeX works very well for large documents

LaTeX also provides many features that are useful for large documents such as Ph.D thesis and books. These features include generating tables of contents, tables of figures, and dividing a document into several files. Some of these features are also provided in Microsoft Word but are not as flexible as in LaTeX. I have personally written both my M.Sc. and Ph.D. thesis using LaTeX and I have saved a lot of time by doing this. I have simply downloaded the LaTeX style file from my university and then used it in my LaTeX document, and after that all my thesis was properly formatted according to the university style, without too much effort.

Problems of LaTeX

Now, let’s talk about the disadvantage or problems faced using LaTeX. The first problem is that there is a somewhat steep learning curve. LaTeX is actually not so difficult to learn but it is more difficult than using Word. It is necessary to learn various commands for preparing LaTeX documents. Moreover, some errors are not so easy to debug. However, the good news is that there exist some good places to ask questions  and obtain answers when encountering problems with LaTeX such as Tex.StackExchange ( http://tex.stackexchange.com/ ).  There also exist some free books such as the Not So Short Introduction To LaTeX that are quite good for learning LaTeX, and that I use as reference.  Actually, although, there is a steep learning curve, I think that it is an excellent investment to learn to use LaTeX for researchers. Moreover, some journals in academia actually only accept LaTeX papers.

The second problem with LaTeX is that it is actually not necessary to use LaTeX for writing simple documents.  LaTeX is best used for large documents or documents with complex layouts or for special needs such as displaying mathematical equations and algorithms.  I personally use LaTeX only for writing research papers. For other things, I use Microsoft Word. Some people also use LaTeX for preparing slides using packages such as beamer, instead of using Powerpoint. This can be useful for preparing a presentation with lot of mathematical equations.

Conclusion

In this blog post, I have discussed the use of LaTeX for writing research papers. I hope that you have enjoyed this blog post.

Philippe Fournier-Viger is a professor of Computer Science and also the founder of the open-source data mining software SPMF, offering more than 120 data mining algorithms.

(Visited 15,691 times, 23 visits today)

### Related posts:

#### Using LaTeX for writing research papers — 3 Comments

1. Hi Philippe,

Thanks for a brief introduction of Latex. I am not using Latex code directly but using Lyx, a GUI processor of Latex. When I just moved from Word to Latex for preparing my papers, I felt Latex was so inconvenient since I was very familiar with the WYSIWYG feature in MS Word. However, day by day, I was more comfortable with Latex and recognized it has many advantages over Word like you said.

Just want to point out a minor disadvantage of Latex (maybe I don’t know the way to fix it). Latex attempts to arrange texts to fill the page; so there are sometimes many big gaps appearing between paragraphs.

Cheers,
Dang

• Hi Dang,

Yes, I understand. At first, I also did not feel very comfortable with Latex. I first did my Master degree thesis in LaTeX. But I still used Word for writing papers, as I felt it was more simple. Then it really took me a few years before I really start to like LaTeX. It was when I started to work on data mining papers. Since there is more math symbols in math papers, I was convinced to really switch to LaTeX for all my papers. And now, I do all my papers in LaTeX.

Yes, I agree, the layout generated by LaTeX is sometimes strange. I did not find a solution to this issue eithe. After a while, I just accepted it. I know that there are some commands to say that some figures can be floating in your documents or be placed in a specific position. But I did not do much more than that about trying to fix layout problems.

Best regards,

Philippe


text:
# ICHEP2018 SEOUL

Jul 4 – 11, 2018
COEX, SEOUL
Asia/Seoul timezone

## Small-Strip Thin Gap Chambers for the Muon Spectrometer Upgrade of the ATLAS Experiment

Jul 5, 2018, 2:48 PM
12m
202 (COEX, Seoul)

### 202

#### COEX, Seoul

Parallel Detector: R&D for Present and Future Facilities

### Speaker

Jesse Alan Heilman (Carleton University (CA))

### Description

The instantaneous luminosity of the Large Hadron Collider at CERN will be increased up to a factor of five with respect to the design value by undergoing an extensive upgrade program over the coming decade. Such increase will allow for precise measurements of Higgs boson properties and extend the search for new physics phenomena beyond the Standard Model. The largest phase-1 upgrade project for the ATLAS Muon System is the replacement of the present first station in the forward regions with the so-called New Small Wheels (NSWs) during the long-LHC shutdown in 2019/20. Along with Micromegas, the NSWs will be equipped with eight layers of small-strip thin gap chambers (sTGC) arranged in multilayers of two quadruplets, for a total active surface of more than 2500 m$^2$. All quadruplets have trapezoidal shapes with surface areas up to 2 m$^2$. To retain the good precision tracking and trigger capabilities in the high background environment of the high luminosity LHC, each sTGC plane must achieve a spatial resolution better than 100 μm to allow the Level-1 trigger track segments to be reconstructed with an angular resolution of approximately 1mrad. The basic sTGC structure consists of a grid of gold-plated tungsten wires sandwiched between two resistive cathode planes at a small distance from the wire plane. The precision cathode plane has strips with a 3.2mm pitch for precision readout and the cathode plane on the other side has pads for triggering. The position of each strip must be known with an accuracy of 30 µm along the precision coordinate and 80 µm along the beam. The mechanical precision is a key point and must be controlled and monitored all along the process of construction and integration. The sTGC detectors are currently being produced and tested in five countries and assembled into wedges at CERN for integration into ATLAS. The sTGC design, performance, construction and integration status will be discussed, along with results from tests of the chambers with nearly final electronics with beams and cosmic rays.

### Primary author

Jesse Alan Heilman (Carleton University (CA))


text:
## Reaching for that Number Theory book (Part I)

I now turn to number theory, because this field seems like a solid foundation for mathematics and physics. I start with Chapter 1 of the esteemed book, Theory and Application of Infinite Series [1]. The title of Chapter 1 is “Principles of the theory of real numbers.” Section 1 is called “The system of rational numbers and its gaps.” I have wondered about numbers and how they relate to the number line encountered in math classes. Hopefully this section clears up some of my confusion, but I don’t want to get my hopes up too high… I do not […]


text:
# Model Vision Sensor Detections

This example shows how to model and simulate the output of an automotive vision sensor for different driving scenarios. Generating synthetic vision detections is important for testing and validating tracking and sensor fusion algorithms in corner cases or when sensor hardware is unavailable. This example analyzes the differences between vision measurements and vehicle ground truth position and velocity for a forward collision warning (FCW) scenario, a passing vehicle scenario, and a hill descent scenario.

In this example, you generate vision detections programmatically. You can also generate detections by using the Driving Scenario Designer app. For an example, see Create Driving Scenario Interactively and Generate Synthetic Sensor Data.

### Introduction

Vehicles that contain advanced driver assistance system (ADAS) features or are designed to be fully autonomous typically rely on multiple types of sensors. These sensors include sonar, radar, lidar, and vision. A robust solution includes a sensor fusion algorithm to combine the strengths across the various types of sensors included in the system. For more information about sensor fusion of synthetic detections from a multisensor ADAS system, see Sensor Fusion Using Synthetic Radar and Vision Data.

When using synthetic detections for testing and validating tracking and sensor fusion algorithms, it is important to understand how the generated detections model the sensor's unique performance characteristics. Each kind of automotive sensor provides a specific set of strengths and weaknesses which contribute to the fused solution. This example presents some important performance characteristics of automotive vision sensors and shows how the sensor performance is modeled by using synthetic detections.

### Vision Sensor Model

This example uses visionDetectionGenerator to generate synthetic vision sensor detections. visionDetectionGenerator models the following performance characteristics of automotive vision sensors:

Strengths

• Good lateral position and velocity accuracy

• One detection reported per target

Weaknesses

• Poor longitudinal position and velocity accuracy

• Inability to detect occluded targets

• Longitudinal biases for elevated targets

FCW Driving Scenario

Create a forward collision warning (FCW) test scenario, which is used to illustrate how to measure a target's position with an automotive vision sensor. The scenario consists of a moving ego vehicle and a stationary target vehicle placed 75 meters down the road. The ego vehicle has an initial speed of 50 kph before applying its brakes to achieve a constant deceleration of 3 m/s^2. The vehicle then comes to a complete stop 1 meter before the target vehicle's rear bumper.

rng default; initialDist = 75; % m finalDist = 1; % m initialSpeed = 50; % kph brakeAccel = 3; % m/s^2 [scenario, egoCar] = helperCreateSensorDemoScenario('FCW', initialDist, initialSpeed, brakeAccel, finalDist);

Forward-Facing Vision Sensor

Create a forward-facing vision sensor mounted on the ego vehicle's front windshield, 1.1 m above the ground. The sensor is pitched down 1 degree toward the road and generates measurements every 0.1 second. The sensor's camera has a 480-by-640 pixel imaging array and a focal length of 800 pixels. The sensor can locate objects within a single image with an accuracy of 5 pixels and has a maximum detection range of 150 m. The ActorProfiles property specifies the physical dimensions of the vehicles seen by the vision sensor in the simulation.

visionSensor = visionDetectionGenerator(... 'SensorIndex', 1, ... 'UpdateInterval', 0.1, ... 'SensorLocation', [0.75*egoCar.Wheelbase 0], ... 'Height', 1.1, ... 'Pitch', 1, ... 'Intrinsics', cameraIntrinsics(800, [320 240], [480 640]), ... 'BoundingBoxAccuracy', 5, ... 'MaxRange', 150, ... 'ActorProfiles', actorProfiles(scenario))
visionSensor = visionDetectionGenerator with properties: SensorIndex: 1 UpdateInterval: 0.1000 SensorLocation: [2.1000 0] Height: 1.1000 Yaw: 0 Pitch: 1 Roll: 0 Intrinsics: [1x1 cameraIntrinsics] DetectorOutput: 'Objects only' FieldOfView: [43.6028 33.3985] MaxRange: 150 MaxSpeed: 100 MaxAllowedOcclusion: 0.5000 MinObjectImageSize: [15 15] DetectionProbability: 0.9000 FalsePositivesPerImage: 0.1000 Use get to show all properties

Simulation of Vision Detections

Simulate the vision sensor measuring the position of the target vehicle by advancing the simulation time of the scenario. The vision sensor generates detections from the true target pose (position, velocity, and orientation) expressed in the ego vehicle's coordinate frame.

The vision sensor is configured to generate detections at 0.1-second intervals, which is consistent with the update rate of typical automotive vision sensors. However, to accurately model the motion of the vehicles, the scenario simulation advances every 0.01 seconds. The sensor returns a logical flag, isValidTime, that is true when the vision sensor reaches its required update interval, indicating that this simulation time step will generate detections.

% Create display for FCW scenario [bep, figScene] = helperCreateSensorDemoDisplay(scenario, egoCar, visionSensor); metrics = struct; % Initialize struct to collect scenario metrics while advance(scenario) % Update vehicle positions gTruth = targetPoses(egoCar); % Get target positions in ego vehicle coordinates % Generate time-stamped vision detections time = scenario.SimulationTime; [dets, ~, isValidTime] = visionSensor(gTruth, time); if isValidTime % Update Bird's-Eye Plot with detections and road boundaries helperUpdateSensorDemoDisplay(bep, egoCar, visionSensor, dets); % Collect vision detections and ground truth for offline analysis metrics = helperCollectScenarioMetrics(metrics, gTruth, dets); end % Take a snapshot for the published example helperPublishSnapshot(figScene, time>=6); end

### Position Measurements

Over the duration of the FCW test, the target vehicle's distance from the ego vehicle spans a wide range of values. By comparing the vision sensor's measured longitudinal and lateral positions of the target vehicle to the target vehicle's ground truth position, you can observe the accuracy of the sensor's measured positions.

Use helperPlotSensorDemoDetections to plot the longitudinal and lateral position errors as the difference between the measured position reported by the vision sensor and the target vehicle's ground truth. The ground truth reference for the target vehicle is the point on the ground directly below the center of the target vehicle's rear axle, which is 1 meter in front of the car's bumper.

helperPlotSensorDemoDetections(metrics, 'position', 'reverse range', [-6 6]); % Show rear overhang of target vehicle tgtCar = scenario.Actors(2); rearOverhang = tgtCar.RearOverhang; subplot(1,2,1); hold on; plot(-rearOverhang*[1 1], ylim, 'k'); hold off; legend('Error', '2\sigma noise', 'Rear overhang');

The vision sensor converts the target's position in the camera's image to longitudinal and lateral positions in the ego vehicle's coordinate system. The sensor does this conversion by assuming that the detected points in the image lie on a flat road that is at the same elevation as the ego vehicle.

Longitudinal Position Measurements

For a forward-facing vision sensor configuration, longitudinal position measurements are derived primarily from the target's vertical placement in the camera's image.

An object's vertical position in the image is strongly correlated to the object's height above the road, but it is weakly correlated to the object's distance from the camera. This weak correlation causes a monocular vision sensor's longitudinal position errors to become large as an object moves away from the sensor. The longitudinal position errors in the preceding plot on the left show how the sensor's longitudinal errors quickly increase when the target vehicle is far from the sensor. The sensor's longitudinal measurement noise is less than 1 meter when the ground truth range to the target vehicle is less than 30 meters, but grows to more than 5 meters at ranges beyond 70 meters from the ego vehicle.

The longitudinal position errors also show a -1 meter bias between the longitude measured by the vision sensor and the target's ground truth position. The -1 meter bias indicates that the sensor consistently measures the target to be closer to the ego vehicle than the target vehicle's ground truth position. Instead of approximating the target as a single point in space, the vision sensor models the physical dimensions of the vehicle's body. For the FCW scenario, the vision sensor views the target vehicle's rear side. The -1 meter bias in the detections generated from this side corresponds to the vehicle's rear overhang. A vehicle's rear overhang defines the distance between the vehicle's rear side and its rear axle, which is where the ground truth reference is located.

Lateral Position Measurements

For a forward-facing vision sensor configuration, lateral position is derived from the target's horizontal location in the camera's image.

Unlike longitudinal position, an object's lateral position is strongly correlated to its horizontal position in the vision sensor's image. This strong correlation produces accurate lateral position measurements that do not degrade quickly with an object's distance from the sensor. The lateral position errors in the preceding plot on the right grow slowly with range. The measurement noise reported by the sensor remains below 0.2 meters out to a ground truth range of 70 meters.

### Velocity Measurements and Target Occlusion

Create a driving scenario with two target vehicles (a lead car and a passing car) to illustrate the accuracy of a vision sensor's longitudinal and lateral velocity measurements. The lead car is placed 40 meters in front of the ego vehicle and is traveling with the same speed. The passing car starts in the left lane alongside the ego vehicle, passes the ego vehicle, and merges into the right lane just behind the lead car. This merging maneuver generates longitudinal and lateral velocity components, enabling you to compare the sensor's accuracy along these two dimensions.

Because the lead car is directly in front of the sensor, it has a purely longitudinal velocity component. The passing car has a velocity profile with both longitudinal and lateral velocity components. These components change as the car passes the ego vehicle and moves into the right lane behind the lead car. Comparing the sensor's measured longitudinal and lateral velocities of the target vehicles to their ground truth velocities illustrates the vision sensor's ability to observe both of these velocity components.

% Create passing scenario leadDist = 40; % m speed = 50; % kph passSpeed = 70; % kph mergeFract = 0.55; % Merge 55% into right lane [scenario, egoCar] = helperCreateSensorDemoScenario('Passing', leadDist, speed, passSpeed, mergeFract);

Configuration of Vision Sensor Velocity Measurements

The vision sensor cannot determine an object's velocity from a single image. To estimate velocity, the vision sensor compares the object's movement between multiple images. The extracted target positions from multiple images are processed by using a smoothing filter. In addition to estimating velocity, this filter produces a smoothed position estimate. To adjust the amount of smoothing that the filter applies, you can set the sensor's process noise intensity. The sensor's process noise should be set to be of the order of the maximum acceleration magnitude expected from a target that must be detected by the sensor.

Take the vision sensor used in the previous section, and configure it to generate position and velocity estimates from a smoothing filter with a process noise intensity of 5 m/s^2.

% Configure vision sensor's noise intensity used by smoothing filter release(visionSensor); visionSensor.ProcessNoiseIntensity = 5; % m/s^2 % Use actor profiles for the passing car scenario visionSensor.ActorProfiles = actorProfiles(scenario);

Use helperRunSensorDemoScenario to simulate the motion of the ego and target vehicles. This function also collects the simulated metrics, as was previously done for the FCW driving scenario.

snapTime = 5.9; % Simulation time to take snapshot for publishing metrics = helperRunSensorDemoScenario(scenario, egoCar, visionSensor, snapTime);

Use helperPlotSensorDemoDetections to plot the vision sensor's longitudinal and lateral velocity errors as the difference between the measured velocity reported by the sensor and the target vehicle's ground truth.

helperPlotSensorDemoDetections(metrics, 'velocity', 'time', [-25 25]); subplot(1,2,1); legend('Lead car error', 'Lead car 2\sigma noise', 'Pass car error', 'Pass car 2\sigma noise');

Longitudinal Velocity Measurements

Forward-facing vision sensors measure longitudinal velocity by comparing how the sensor's longitudinal position measurements change between sensor update intervals. Because the sensor's longitudinal position errors grow with range, the longitudinal velocity errors will also grow with increasing target range.

The longitudinal velocity errors from the passing vehicle scenario are shown in the preceding plot on the left. Because the lead car maintains a constant distance from the vision sensor, its errors (displayed as red points) show the same measurement noise throughout the scenario. However, the passing car's distance from the sensor is not constant, but this distance increases as the car passes the sensor and moves toward the lead car. The passing car's longitudinal velocity errors (displayed as yellow points) are small when it first enters the sensor's field of view at 2 seconds. The passing car is close to the vision sensor at this point in the scenario. From 2 seconds to 6 seconds, the passing car is moving away from the ego vehicle and approaching the lead car. Its longitudinal velocity errors grow as its distance from the sensor increases. Once the passing car merges into the right lane behind the lead car, it maintains a constant distance from the sensor, and its measurement noise remains constant.

Lateral Velocity Measurements

Forward-facing vision sensors measure lateral velocity by comparing how the sensor's lateral position measurements change between sensor update intervals. Because the sensor's lateral position errors are not strongly correlated with the target's range from the sensor, the lateral velocity errors will also show little dependence on target range.

The lateral velocity errors from the passing vehicle scenario are shown in the preceding plot on the right. The errors from the lead car (red points) and the passing car (yellow points) have nearly the same measurement noise for the entire scenario. The passing car's reported lateral velocity errors show little change as it moves away from the sensor.

Detection of Targets with Partial Occlusion

In the preceding velocity error plots, the lead car (red points) is reliably detected during the first 6 seconds of the scenario. The passing car (yellow points) is detected at 2 seconds when it first enters the camera's field of view. Detections are then generated on both of the target vehicles until 6 seconds. At 6 seconds, the passing car merges into the right lane and moves between the ego vehicle and the lead car. For the remainder of the scenario, the passing car partially occludes the vision sensor's view of the lead car. 55% of the lead car's rear side is occluded, leaving only 45% visible to the sensor for detection. This occluded view of the lead car prevents the sensor from finding the car in the camera's image and generating detections.

A vision sensor's ability to provide reliable detections is strongly dependent on an unobstructed view of the object it is detecting. In dense traffic, the visibility of vehicles in the scenario can change rapidly as distances between vehicles change and vehicles move in and out of lanes. This inability to maintain detection on obstructed targets poses a challenge to tracking algorithms processing the vision sensor's detections.

Rerun the passing vehicle scenario with a vision sensor that can detect targets with as much as 60% of the target's viewable area occluded.

% Configure vision sensor to support maximum occlusion of 60% release(visionSensor); visionSensor.MaxAllowedOcclusion = 0.6; % Run simulation and collect detections and ground truth for offline analysis metrics = helperRunSensorDemoScenario(scenario, egoCar, visionSensor); % Plot longitudinal and lateral velocity errors helperPlotSensorDemoDetections(metrics, 'velocity', 'time', [-25 25]); subplot(1,2,1); legend('Lead car error', 'Lead car 2\sigma noise', 'Pass car error', 'Pass car 2\sigma noise');

In the preceding plot, velocity errors are shown for both the lead car (red points) and the passing car (yellow points). The same error performance is observed as before, but now detections on the lead car are maintained after the passing car has merged behind it at 6 seconds. By adjusting the maximum allowed occlusion, you can model a vision sensor's sensitivity to target occlusion.

### Longitudinal Position Bias from Target Elevation

An object's vertical location in the camera's image is strongly correlated to its height above the road. Because a monocular vision sensor generates longitudinal position measurements from the vertical location of objects in its camera's image, large errors can arise for targets at different elevations from the ego vehicle. When an object changes elevation, the sensor incorrectly interprets the vertical displacement in the camera's image as a change in the object's longitudinal position.

Run the FCW scenario again with a stationary target vehicle placed at a location 2 meters lower than the initial position of the ego vehicle. The ego vehicle descends a small hill as it approaches the target vehicle. As the ego vehicle descends the hill, the target vehicle's vertical location in the camera's image changes, introducing a bias in the sensor's measured longitudinal position.

% Create FCW hill descent scenario initialDist = 75; % m finalDist = 1; % m initialSpeed = 50; % kph brakeAccel = 3; % m/s^2 [scenario, egoCar] = helperCreateSensorDemoScenario('FCW', initialDist, initialSpeed, brakeAccel, finalDist, false, 2); % Use actor profiles for the FCW hill descent scenario release(visionSensor); visionSensor.ActorProfiles = actorProfiles(scenario); % Run simulation and collect detections and ground truth for offline analysis snapTime = 3; % Simulation time to take snapshot for publishing metrics = helperRunSensorDemoScenario(scenario, egoCar, visionSensor, snapTime, true);

Plot the position errors generated for the target vehicle as the ego vehicle descends the small hill.

helperPlotSensorDemoDetections(metrics, 'position', 'reverse range', [-6 6;0 80]); subplot(1,2,1); xlim([-10 60]); ylim([0 80]); legend('Error', '2\sigma noise');

The preceding plots show the longitudinal position errors (on the left) and the lateral position errors (on the right) for the hill descent scenario. Note that in the longitudinal position error plot, the limits of the error axis have been increased to accommodate the large bias induced by the target's elevation relative to the camera sensor as the ego vehicle descends the hill.

The ego vehicle begins its descent when it is 75 meters away from the target vehicle. Because the ego vehicle is pitched down as it descends the hill, the target appears at an elevated position near the top of the camera's image. As the ego vehicle descends the hill, the target vehicle's location in the camera's image moves from the top of the image and crosses the horizon line. For monocular vision sensors, targets located near the horizon line in the camera's image are mapped to positions that are very far from the sensor. (By definition, points on the horizon line are located at infinity.) The vision sensor does not generate detections for objects appearing above the horizon in the camera's image, because these points do not map to locations on the road's surface.

Large changes in the vehicle's longitudinal position as its location in the image moves away from the horizon also cause the sensor's smoothing filter to generate large longitudinal velocity estimates. The sensor rejects detections with speeds exceeding its MaxSpeed property. These large longitudinal velocities produced by the target's elevation also prevent the sensor from generating detections when the target vehicle is near the camera's horizon.

When the ego vehicle is approximately 40 meters from the target vehicle, the target vehicle's image location has crossed the horizon line and the sensor's velocity estimates satisfy its max speed constraint. At this distance, the vision sensor begins to generate detections from the target vehicle. The mapping of target locations near the camera's horizon to points on the road far from the sensor explains the large longitudinal errors modeled by the monocular vision sensor when it begins detecting the target vehicle. The longitudinal bias continues to decrease as the ego vehicle approaches the bottom of the hill and the target's location moves away from the horizon line in the camera's image. At the end of the ego vehicle's descent, the target is at the same elevation as the ego vehicle. Only the -1 meter bias corresponding to the target vehicle's rear overhang is present. The sensor's lateral position errors show no bias, because the pitch of the ego vehicle as it descends the hill does not change the target's horizontal location in the camera's image.

### Pedestrian and Vehicle Detection

A vision sensor's ability to detect an object in its camera's image depends on the number of pixels the object occupies in the image. When an object's size in the image is large (hundreds of pixels), the sensor can easily identify the object and generate a detection. However when an object's size in the image is small (tens of pixels) the sensor might not find it and will not generate a detection. An object's projected size on the camera's imaging array is a function of both the object's physical size and its distance from the camera. Therefore, when a vehicle is positioned farther from the camera than a pedestrian, both the vehicle and the pedestrian might have similar sizes in the camera's image. This means that a vision sensor will detect large objects (vehicles) at longer ranges than smaller objects (pedestrians).

Run the FCW scenario again with both a stationary car and pedestrian 75 meters in front of the sensor. This scenario illustrates the difference in the sensor's detection range for these two objects. The ego vehicle, stationary car, and pedestrian are all placed at the same elevation.

% Create FCW test scenario initialDist = 75; % m finalDist = 1; % m initialSpeed = 50; % kph brakeAccel = 3; % m/s^2 [scenario, egoCar] = helperCreateSensorDemoScenario('FCW', initialDist, initialSpeed, brakeAccel, finalDist, true); % Use actor profiles for the FCW hill descent scenario release(visionSensor); visionSensor.ActorProfiles = actorProfiles(scenario); % Run simulation and collect detections and ground truth for offline analysis snapTime = 5; % Simulation time to take snapshot for publishing metrics = helperRunSensorDemoScenario(scenario, egoCar, visionSensor, snapTime);

Plot the position errors generated for the target vehicle and pedestrian from the FCW scenario.

helperPlotSensorDemoDetections(metrics, 'position', 'reverse range', [-6 6]); legend('Car error','Car 2\sigma noise', 'Pedestrian error', 'Pedestrian 2\sigma noise');

The preceding plots show the longitudinal position errors (on the left) and lateral position errors (on the right) generated by the vision sensor's detections of the target vehicle and pedestrian. Errors from detections for the target vehicle (shown in red) are generated out to the farthest range included in this test (75 m), but errors for the pedestrian (shown in yellow) do not appear until the ego vehicle has reached a distance of approximately 30 m. This difference in the detection ranges is due to the difference in the sizes of these two objects.

The sensor stops generating detections from the pedestrian at ranges less than 12 m. At this range, the offset of the pedestrian from the camera's optical axis moves the pedestrian outside of the camera's horizontal field of view. Because the target vehicle is directly in front of the camera, it remains centered within the camera's image for the entire FCW test.

Some vision sensors can detect objects with smaller image sizes, enabling the sensors to detect objects at longer ranges. In the previous scenario, the sensor's detection of the pedestrian is limited by the pedestrian's width (0.45 m), which is much narrower than the width of the car (1.8 m). To increase the sensor's detection range for pedestrians to 40 m, compute the width of the pedestrian in the camera's image when it is at 40 m.

Find physical width modeled for a pedestrian

profiles = actorProfiles(scenario); pedWidth = profiles(3).Width % Compute width of pedestrian in camera's image in pixels at 40 meters from ego vehicle cameraRange = 40-visionSensor.SensorLocation(1); focalLength = visionSensor.Intrinsics.FocalLength(1); pedImageWidth = focalLength*pedWidth/cameraRange
pedWidth = 0.4500 pedImageWidth = 9.4987

At 40 m, the pedestrian has a width of 9.5 pixels in the camera's image. Set the vision sensor's minimum object width to match the pedestrian's width at 40 m.

% Configure sensor to detect pedestrians out to a range of 40 m. release(visionSensor); visionSensor.MinObjectImageSize(2) = pedImageWidth
visionSensor = visionDetectionGenerator with properties: SensorIndex: 1 UpdateInterval: 0.1000 SensorLocation: [2.1000 0] Height: 1.1000 Yaw: 0 Pitch: 1 Roll: 0 Intrinsics: [1x1 cameraIntrinsics] DetectorOutput: 'Objects only' FieldOfView: [43.6028 33.3985] MaxRange: 150 MaxSpeed: 100 MaxAllowedOcclusion: 0.6000 MinObjectImageSize: [15 9.4987] DetectionProbability: 0.9000 FalsePositivesPerImage: 0.1000 Use get to show all properties

Run the scenario again and plot the position errors to show the revised detection ranges for the vehicle and pedestrian.

% Run simulation and collect detections and ground truth for offline % analysis. metrics = helperRunSensorDemoScenario(scenario, egoCar, visionSensor); % Plot position errors for the target vehicle and pedestrian. helperPlotSensorDemoDetections(metrics, 'position', 'reverse range', [-6 6]); legend('Car error','Car 2\sigma noise', 'Pedestrian error', 'Pedestrian 2\sigma noise');

The preceding plots show the longitudinal position errors (on the left) and lateral position errors (on the right) for a vision sensor configured to support pedestrian detection out to a range of 40 m. The vehicle (shown in red) is still detected out to the farthest test range, but now detections on the pedestrian (shown in yellow) are generated out to 40 m from the sensor.

### Lane Boundary Measurements and Lane Occlusion

The vision detection generator can also be configured to detect lanes. Recreate the two-lane driving scenario with the lead car and passing car to illustrate the accuracy of a vision sensor's lane boundary measurements. This same merging maneuver is used to occlusion of the lane markings.

% Create passing scenario leadDist = 40; % m speed = 50; % kph passSpeed = 70; % kph mergeFract = 0.55; % Merge 55% into right lane [scenario, egoCar] = helperCreateSensorDemoScenario('Passing', leadDist, speed, passSpeed, mergeFract);

Configuration of Vision Sensor Lane Boundary Measurements

Configure the vision sensor used in the previous section, and configure it to generate position and velocity estimates from a smoothing filter with a process noise intensity of 5 m/s^2.

% Configure vision sensor to detect both lanes and objects release(visionSensor); visionSensor.DetectorOutput = 'lanes and objects'; % Use actor profiles for the passing car scenario visionSensor.ActorProfiles = actorProfiles(scenario);

Use helperRunSensorDemoScenario to simulate the motion of the ego and target vehicles.

snapTime = 5.9; % Simulation time to take snapshot for publishing helperRunSensorDemoScenario(scenario, egoCar, visionSensor, snapTime);

As can be seen above, the default detector can see lane boundaries out to 45 meters or so when presented with an unoccluded view. You can change the intrinsics of the detector to observe its effect.

% show camera intrinsics. visionSensor.Intrinsics
ans = cameraIntrinsics with properties: FocalLength: [800 800] PrincipalPoint: [320 240] ImageSize: [480 640] RadialDistortion: [0 0] TangentialDistortion: [0 0] Skew: 0 K: [3x3 double]
% increase the focal length and observe its effect. release(visionSensor); visionSensor.Intrinsics = cameraIntrinsics([1200 1200],[320 240],[480 640]) helperRunSensorDemoScenario(scenario, egoCar, visionSensor, snapTime);
visionSensor = visionDetectionGenerator with properties: SensorIndex: 1 UpdateInterval: 0.1000 SensorLocation: [2.1000 0] Height: 1.1000 Yaw: 0 Pitch: 1 Roll: 0 Intrinsics: [1x1 cameraIntrinsics] DetectorOutput: 'Lanes and objects' FieldOfView: [29.8628 22.6199] MaxRange: 150 MaxSpeed: 100 MaxAllowedOcclusion: 0.6000 MinObjectImageSize: [15 9.4987] MinLaneImageSize: [20 3] DetectionProbability: 0.9000 FalsePositivesPerImage: 0.1000 Use get to show all properties

Changing the focal length from 800 pixels to 1200 in both x- and y-directions zooms the camera, enabling it to detect out to further ranges.

### Summary

This example demonstrated how to model the output of automotive vision sensors using synthetic detections. In particular, it presented how the visionDetectionGenerator model:

• Provides accurate lateral position and velocity measurements over long ranges, but has limited longitudinal accuracy at long ranges

• Limits detection according to a target's physical dimensions and a target's occlusion by other objects in the scenario

• Includes longitudinal biases for targets located at different elevations than the ego vehicle

• Adjusts object and lane detections due to monocular camera intrinsics.


text:
# The Duplicators, Part I: Eutocius’ Collection of Cube Duplications - Eratosthenes

Author(s):
Colin B. P. McKinney (Wabash College)

Biographical note: Eratosthenes of Cyrene (ca. 276 BCE – ca. 194 BCE). Eratosthenes is known for a variety of mathematical topics, including the so-called sieve (related to prime number theory) and a relatively accurate estimate for the circumference of the Earth. He was a polymath but not “best” in any field, and supposedly was nicknamed “Beta” for this reason. Read more about Eratosthenes at MacTutor.

(Heiberg 84.13) Let there be two unequal straight lines, ΑΕ and ΔΘ, between which it is required to find two means in continuous proportion. Let ΑΕ be placed at right angles to some straight line ΕΘ, and on ΕΘ, let three parallelograms be set up, ΑΖ, ΖΙ, and ΙΘ, and let the diameters ΑΖ, ΛΗ, and ΙΘ be drawn in each: hence these diameters will be parallel.

Above: Figure 13.1: Eratosthenes’ First Diagram. This diagram is static. The point F is unlabeled in the manuscripts. In the next diagram, it is movable, and I label it here for continuity.

When the middle parallelogram ΖΙ remains fixed, let the parallelogram ΑΖ be slid along the top surface of the middle one, and ΙΘ along the bottom surface of the middle one, as in the second diagram, until the points Α, Β, Γ, and Δ are in a straight line. Let the line through the points Α, Β, Γ, and Δ be drawn, and let it intersect ΕΘ (having been extended) at Κ.

Above: Eratosthenes’ Second Diagram. The objective with this diagram is to have the points Β and Γ on the red line ΑΔ. This is accomplished by moving the doors (by moving points Α and F). The point Δ is also moveable; adjusting it changes the ratio ΑΕ : ΔΘ. By default, ΑΕ is twice ΔΘ. I have added a few labels to this diagram. First, the point F, which is unlabelled in the manuscripts. Second, the manuscripts only label the points Λ and Ι (and their counterparts, Λ2 and Ι2) in the first diagram. For completeness, I include them here; when the doors are moved to their starting positions, the Λ points coincide and the Ι points coincide. Similarly for the points Ζ and Η: while they are also labeled in the second diagram, their counterparts Z1 and Η1 are not. (Note that Eratosthenes’ First Diagram corresponds to a static version of this diagram.)

Indeed, in the parallels ΑΕ and ΖΒ, we have by Elements VI.4, $$\tag{97} \text{ ΑΚ : ΚΒ = ΕΚ : ΚΖ;}$$and in the parallels ΑΖ and ΒΗ,  $$\tag{98} \text{ΑΚ : ΚΒ = ΖΚ : ΚΗ. }$$Therefore [by Elements V.11]  $$\tag{99} \text{ ΑΚ : ΚΒ = ΕΚ : ΚΖ = ΚΖ : ΚΗ.}$$Again, in the parallels ΒΖ and ΓΗ, $$\tag{100} \text{ ΒΚ : ΚΓ = ZK : ΚΗ;}$$and in the parallels ΒΗ and ΓΘ, $$\tag{101} \text{ ΒΚ : ΚΓ = ΗΚ : ΚΘ.}$$Therefore $$\tag{102} \text{ΒΚ : ΚΓ = ΖΚ : ΚΗ = ΗΚ : ΚΘ. }$$But $$\tag{103} \text{ΖΚ : ΚΗ = ΕΚ : ΚΖ; }$$and therefore $$\tag{104} \text{ ΕΚ : ΚΖ = ΖΚ : ΗΚ = ΗΚ : ΚΘ.}$$

But $$\tag{105} \text{ΕΚ : ΚΖ = ΑΕ : ΒΖ,}$$and $$\tag{106} \text{ ΖΚ : ΚΗ = ΒΖ : ΓΗ,}$$and $$\tag{107} \text{ ΗΚ : ΚΘ = ΓΗ : ΔΘ;}$$therefore $$\tag{108} \text{ ΑΕ : ΒΖ = ΒΖ : ΓΗ = ΓΗ : ΔΘ.}$$Therefore both ΒΖ and ΓΗ have been found as two mean proportionals between ΑΕ and ΔΘ.

Colin B. P. McKinney (Wabash College), "The Duplicators, Part I: Eutocius’ Collection of Cube Duplications - Eratosthenes," Convergence (May 2016)


text:
# zbMATH — the first resource for mathematics

On subspaces of quotients of ($$\sum G_n$$)$$l_p$$ and ($$\sum G_n$$)$$c_0$$. Proc. internat. Sympos. partial diff. Equ. Geometry normed lin. Spaces II. (English) Zbl 0252.46025

##### MSC:
 46A45 Sequence spaces (including Köthe sequence spaces) 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces