53184402
A Convex Geodesic Selective Model for Image Segmentation
Selective segmentation is an important application of image processing. In contrast to global segmentation in which all objects are segmented, selective segmentation is used to isolate specific objects in an image and is of particular interest in medical imaging—permitting segmentation and review of a single organ. An important consideration is to minimise the amount of user input to obtain the segmentation; this differs from interactive segmentation in which more user input is allowed than selective segmentation. To achieve selection, we propose a selective segmentation model which uses the edge-weighted geodesic distance from a marker set as a penalty term. It is demonstrated that this edge-weighted geodesic penalty term improves on previous selective penalty terms. A convex formulation of the model is also presented, allowing arbitrary initialisation. It is shown that the proposed model is less parameter dependent and requires less user input than previous models. Further modifications are made to the edge-weighted geodesic distance term to ensure segmentation robustness to noise and blur. We can show that the overall Euler–Lagrange equation admits a unique viscosity solution. Numerical results show that the result is robust to user input and permits selective segmentations that are not possible with other models.
11754708
Seeded Region Growing
We present here a new algorithm for segmentation of intensity images which is robust, rapid, and free of tuning parameters. The method, however, requires the input of a number of seeds, either individual pixels or regions, which will control the formation of regions into which the image will be segmented. In this correspondence, we present the algorithm, discuss briefly its properties, and suggest two ways in which it can be employed, namely, by using manual seed selection or by automated procedures. >
15436061
Watersheds in Digital Spaces: An Efficient Algorithm Based on Immersion Simulations
A fast and flexible algorithm for computing watersheds in digital gray-scale images is introduced. A review of watersheds and related motion is first presented, and the major methods to determine watersheds are discussed. The algorithm is based on an immersion process analogy, in which the flooding of the water in the picture is efficiently simulated using of queue of pixel. It is described in detail provided in a pseudo C language. The accuracy of this algorithm is proven to be superior to that of the existing implementations, and it is shown that its adaptation to any kind of digital grid and its generalization to n-dimensional images (and even to graphs) are straightforward. The algorithm is reported to be faster than any other watershed algorithm. Applications of this algorithm with regard to picture segmentation are presented for magnetic resonance (MR) imagery and for digital elevation models. An example of 3-D watershed is also provided. >
7602622
Active contours without edges
We propose a new model for active contours to detect objects in a given image, based on techniques of curve evolution, Mumford-Shah (1989) functional for segmentation and level sets. Our model can detect objects whose boundaries are not necessarily defined by the gradient. We minimize an energy which can be seen as a particular case of the minimal partition problem. In the level set formulation, the problem becomes a "mean-curvature flow"-like evolving the active contour, which will stop on the desired boundary. However, the stopping term does not depend on the gradient of the image, as in the classical active contour models, but is instead related to a particular segmentation of the image. We give a numerical algorithm using finite differences. Finally, we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable. Also, the initial curve can be anywhere in the image, and interior contours are automatically detected.
13128194
Geodesic active contours
A novel scheme for the detection of object boundaries is presented. The technique is based on active contours deforming according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric as defined by the image content. This geodesic approach for object segmentation allows to connect classical "snakes" based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved as showed by a number of examples. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well.<<ETX>>
12849354
Snakes: Active contour models
A snake is an energy-minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges. Snakes are active contour models: they lock onto nearby edges, localizing them accurately. Scale-space continuation can be used to enlarge the capture region surrounding a feature. Snakes provide a unified account of a number of visual problems, including detection of edges, lines, and subjective contours; motion tracking; and stereo matching. We have used snakes successfully for interactive interpretation, in which user-imposed constraint forces guide the snake near features of interest.
1522654
Algorithms for Finding Global Minimizers of Image Segmentation and Denoising Models
We show how certain nonconvex optimization problems that arise in image processing and computer vision can be restated as convex minimization problems. This allows, in particular, the finding of global minimizers via standard convex minimization schemes.
28456729
Image selective segmentation under geometrical constraints using an active contour approach
In this paper we propose a new model for segmentation of an image under some geometrical constraints in order to detect special regions of interest. Our work is based on the recent work by Gout et al. [Numer. Algorithms, 39 (2005), pp. 155-173 and 48 (2008), pp. 105-133] using geodesic active contours models, by combining it with the idea of a piecewise constant Mumford-Shah model as with the non-selective Chan-Vese segmentation. Numerical tests show that our method is more robust than the previous works. AMS subject classifications: 62H35, 65N22, 65N55, 74G65, 74G75
123837563
Coefficient of Variation Based Image Selective Segmentation Model Using Active Contours
Most image segmentation techniques efficiently segment images with prominent edges, but are less efficient for some images with low frequencies and overlapping regions of homogeneous intensities. A recently proposed selective segmentation model often works well, but not for such challenging images. In this paper, we introduce a new model using the coefficient of variation as a fidelity term, and our test results show it performs much better in these challenging cases.
625818
Segmentation under geometrical conditions using geodesic active contours and interpolation using level set methods
Abstract
Let I :Ω→ℜ be a given bounded image function, where Ω is an open and bounded domain which belongs to ℜn. Let us consider n=2 for the purpose of illustration. Also, let S={xi}i∈Ω be a finite set of given points. We would like to find a contour Γ⊂Ω, such that Γ is an object boundary interpolating the points from S. We combine the ideas of the geodesic active contour (cf. Caselles et al. [7,8]) and of interpolation of points (cf. Zhao et al. [40]) in a level set approach developed by Osher and Sethian [33]. We present modelling of the proposed method, both theoretical results (viscosity solution) and numerical results are given.

16471303
Robust Interactive Image Segmentation Using Convex Active Contours
The state-of-the-art interactive image segmentation algorithms are sensitive to the user inputs and often unable to produce an accurate boundary with a small amount of user interaction. They frequently rely on laborious user editing to refine the segmentation boundary. In this paper, we propose a robust and accurate interactive method based on the recently developed continuous-domain convex active contour model. The proposed method exhibits many desirable properties of an effective interactive image segmentation algorithm, including robustness to user inputs and different initializations, the ability to produce a smooth and accurate boundary contour, and the ability to handle topology changes. Experimental results on a benchmark data set show that the proposed tool is highly effective and outperforms the state-of-the-art interactive image segmentation algorithms.
121968158
A New Variational Model with Dual Level Set Functions for Selective Segmentation
In this paper we present a selective segmentation model using a dual level set variational formulation. Our variational model aims to segment all objects with one level set function (global) and the selected object, which is the closest to the geo- metric constraints (markers), with another level set (local). It is a combination of edge detection, markers distance function and active contour without edges. Experimental results show that our model is more robust than previous work. AMS subject classifications: 62H35, 65N22, 65N55, 74G65, 74G75
120449398
Improved Selective Segmentation Model Using One Level-Set
Variational segmentation models have proven to be extremely efficient for segmenting and tracking boundaries and in most cases all of the boundaries in an image. Such models are for global segmentation. For a large class of image segmentation tasks where only one object is required to be extracted automatically, global models cannot deliver the solution and we need selective segmentation techniques. In this paper, we propose a novel, variational and single level-set function for the selective segmentation task. The model is much faster to implement than the previously dual level set model by Rada-Chen [29] by having the same efficiency and reliability. In comparing with interactive image segmentation algorithm of Nguyen-Cai-Zhang-Zheng method [2], our model shows some improvement in some cases. Several new ideas are incorporated in this new work: i) the distance function is only needed optionally and its inclusion does not affect the result; ii) an adaptive parameter is introduced in the edge detection function; iii) a new area-based fitting term is added to enhance the model's reliability (different from the idea of minimizing the area of the object). We develop an additive operator splitting method for solving the resulting Euler-Lagrange equation. Test results show that the new model finds the desired local boundaries successfully in various challenging cases and indeed it is not much dependent on the prior information of markers or the distance function based on them. More importantly, the new model gives an overall improvement over the previous models and can be recommended for selective segmentation.
96456091
A CONVEX AND SELECTIVE VARIATIONAL MODEL FOR IMAGE SEGMENTATION
Selective image segmentation is the task of extracting one object of interest from an image, based on minimal user input. Recent level set based variational models have shown to be effective and reliable, although they can be sensitive to initialization due to the minimization problems being nonconvex. This sometimes means that successful segmentation relies too heavily on user input or a solution found is only a local minimizer, i.e. not the correct solution. The same principle applies to variational models that extract all objects in an image (global segmentation); however, in recent years, some have been successfully reformulated as convex optimization problems, allowing global minimizers to be found. There are, however, problems associated with extending the convex formulation to the current selective models, which provides the motivation for the proposal of a new selective model. In this paper we propose a new selective segmentation model, combining ideas from global segmentation, that can be reformulated in a convex way such that a global minimizer can be found independently of initialization. Numerical results are given that demonstrate its reliability in terms of removing the sensitivity to initialization present in previous models, and its robustness to user input.
5548370
Geodesic Matting: A Framework for Fast Interactive Image and Video Segmentation and Matting
An interactive framework for soft segmentation and matting of natural images and videos is presented in this paper. The proposed technique is based on the optimal, linear time, computation of weighted geodesic distances to user-provided scribbles, from which the whole data is automatically segmented. The weights are based on spatial and/or temporal gradients, considering the statistics of the pixels scribbled by the user, without explicit optical flow or any advanced and often computationally expensive feature detectors. These could be naturally added to the proposed framework as well if desired, in the form of weights in the geodesic distances. An automatic localized refinement step follows this fast segmentation in order to further improve the results and accurately compute the corresponding matte function. Additional constraints into the distance definition permit to efficiently handle occlusions such as people or objects crossing each other in a video sequence. The presentation of the framework is complemented with numerous and diverse examples, including extraction of moving foreground from dynamic background in video, natural and 3D medical images, and comparisons with the recent literature.
13120590
Interactive Image Segmentation via Adaptive Weighted Distances
An interactive algorithm for soft segmentation of natural images is presented in this paper. The user first roughly scribbles different regions of interest, and from them, the whole image is automatically segmented. This soft segmentation is obtained via fast, linear complexity computation of weighted distances to the user-provided scribbles. The adaptive weights are obtained from a series of Gabor filters, and are automatically computed according to the ability of each single filter to discriminate between the selected regions of interest. We present the underlying framework and examples showing the capability of the algorithm to segment diverse images
7619283
Geodesic active contour under geometrical conditions: theory and 3D applications
In this paper, we propose a new scheme for both detection of boundaries and fitting of geometrical data based on a geometrical partial differential equation, which allows a rigorous mathematical analysis. The model is a geodesic-active-contour-based model, in which we are trying to determine a curve that best approaches the given geometrical conditions (for instance a set of points or curves to approach) while detecting the object under consideration. Formal results concerning existence, uniqueness (viscosity solution) and stability are presented as well. We give the discretization of the method using an additive operator splitting scheme which is very efficient for this kind of problem. We also give 2D and 3D numerical examples on real data sets.
3393428
Automatic Brain Tumor Segmentation Using Cascaded Anisotropic Convolutional Neural Networks
A cascade of fully convolutional neural networks is proposed to segment multi-modal Magnetic Resonance (MR) images with brain tumor into background and three hierarchical regions: whole tumor, tumor core and enhancing tumor core. The cascade is designed to decompose the multi-class segmentation problem into a sequence of three binary segmentation problems according to the subregion hierarchy. The whole tumor is segmented in the first step and the bounding box of the result is used for the tumor core segmentation in the second step. The enhancing tumor core is then segmented based on the bounding box of the tumor core segmentation result. Our networks consist of multiple layers of anisotropic and dilated convolution filters, and they are combined with multi-view fusion to reduce false positives. Residual connections and multi-scale predictions are employed in these networks to boost the segmentation performance. Experiments with BraTS 2017 validation set show that the proposed method achieved average Dice scores of 0.7859, 0.9050, 0.8378 for enhancing tumor core, whole tumor and tumor core, respectively. The corresponding values for BraTS 2017 testing set were 0.7831, 0.8739, and 0.7748, respectively.
214883166
Interactive Medical Image Segmentation Using Deep Learning With Image-Specific Fine Tuning
Convolutional neural networks (CNNs) have achieved state-of-the-art performance for automatic medical image segmentation. However, they have not demonstrated sufficiently accurate and robust results for clinical use. In addition, they are limited by the lack of image-specific adaptation and the lack of generalizability to previously unseen object classes (a.k.a. zero-shot learning). To address these problems, we propose a novel deep learning-based interactive segmentation framework by incorporating CNNs into a bounding box and scribble-based segmentation pipeline. We propose image-specific fine tuning to make a CNN model adaptive to a specific test image, which can be either unsupervised (without additional user interactions) or supervised (with additional scribbles). We also propose a weighted loss function considering network and interaction-based uncertainty for the fine tuning. We applied this framework to two applications: 2-D segmentation of multiple organs from fetal magnetic resonance (MR) slices, where only two types of these organs were annotated for training and 3-D segmentation of brain tumor core (excluding edema) and whole brain tumor (including edema) from different MR sequences, where only the tumor core in one MR sequence was annotated for training. Experimental results show that: 1) our model is more robust to segment previously unseen objects than state-of-the-art CNNs; 2) image-specific fine tuning with the proposed weighted loss function significantly improves segmentation accuracy; and 3) our method leads to accurate results with fewer user interactions and less user time than traditional interactive segmentation methods.
205007680
Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. Final report
New numerical algorithms are devised (PSC algorithms) for following fronts propagating with curvature-dependent speed. The speed may be an arbitrary function of curvature, and the front can also be passively advected by an underlying flow. These algorithms approximate the equations of motion, which resemble Hamilton-Jacobi equations with parabolic right-hand-sides, by using techniques from the hyperbolic conservation laws. Non-oscillatory schemes of various orders of accuracy are used to solve the equations, providing methods that accurately capture the formation of sharp gradients and cusps in the moving fronts. The algorithms handle topological merging and breaking naturally, work in any number of space dimensions, and do not require that the moving surface be written as a function. The methods can be used also for more general Hamilton-Jacobi-type problems. The algorithms are demonstrated by computing the solution to a variety of surface motion problems.
43183166
Weighted variational model for selective image segmentation with application to medical images
Abstract Selective image segmentation is an important topic in medical imaging and real applications. In this paper, we propose a weighted variational selective image segmentation model which contains two steps. The first stage is to obtain a smooth approximation related to Mumford-Shah model to the target region in the input image. Using weighted function, the approximation provides a larger value for the target region and smaller values for other regions. In the second stage, we make use of this approximation and perform a thresholding procedure to obtain the object of interest. The approximation can be obtained by the alternating direction method of multipliers and the convergence analysis of the method can be established. Experimental results for medical image selective segmentation are given to demonstrate the usefulness of the proposed method. We also do some comparisons and show that the performance of the proposed method is more competitive than other testing methods.
10143244
A fast marching level set method for monotonically advancing fronts.
A fast marching level set method is presented for monotonically advancing fronts, which leads to an extremely fast scheme for solving the Eikonal equation. Level set methods are numerical techniques for computing the position of propagating fronts. They rely on an initial value partial differential equation for a propagating level set function and use techniques borrowed from hyperbolic conservation laws. Topological changes, corner and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. This paper describes a particular case of such methods for interfaces whose speed depends only on local position. The technique works by coupling work on entropy conditions for interface motion, the theory of viscosity solutions for Hamilton-Jacobi equations, and fast adaptive narrow band level set methods. The technique is applicable to a variety of problems, including shape-from-shading problems, lithographic development calculations in microchip manufacturing, and arrival time problems in control theory.
1708973
A Linear Time Algorithm for Computing Exact Euclidean Distance Transforms of Binary Images in Arbitrary Dimensions
A sequential algorithm is presented for computing the exact Euclidean distance transform (DT) of a k-dimensional binary image in time linear in the total number of voxels N. The algorithm, which is based on dimensionality reduction and partial Voronoi diagram construction, can be used for computing the DT for a wide class of distance functions, including the L/sub p/ and chamfer metrics. At each dimension level, the DT is computed by constructing the intersection of the Voronoi diagram whose sites are the feature voxels with each row of the image. This construction is performed efficiently by using the DT in the next lower dimension. The correctness and linear time complexity are demonstrated analytically and verified experimentally. The algorithm may be of practical value since it is relatively simple and easy to implement and it is relatively fast (not only does it run in O(N) time but the time constant is small). A simple modification of the algorithm computes the weighted Euclidean DT, which is useful for images with anisotropic voxel dimensions. A parallel version of the algorithm runs in O(N/p) time with p processors.
122019197
Image selective smoothing and edge detection by nonlinear diffusion. II
A new version of the Perona and Malik theory for edge detection and image restoration is proposed. This new version keeps all the improvements of the original model and avoids its drawbacks: it is proved to be stable in presence of noise, with existence and uniqueness results. Numerical experiments on natural images are presented.
14502908
Scale-Space and Edge Detection Using Anisotropic Diffusion
A new definition of scale-space is suggested, and a class of algorithms used to realize a diffusion process is introduced. The diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing. It is shown that the 'no new maxima should be generated at coarse scales' property of conventional scale space is preserved. As the region boundaries in the approach remain sharp, a high-quality edge detector which successfully exploits global information is obtained. Experimental results are shown on a number of images. Parallel hardware implementations are made feasible because the algorithm involves elementary, local operations replicated over the image. >
120853571
A parallel splitting up method and its application to Navier-Stokes equations
Abstract A parallel splitting-up method (or the so called alternating-direction method) is proposed in this paper. The method not only reduces the original linear and nonlinear problems into a series of one dimensional linear problems, but also enables us to compute all these one dimensional linear problems by parallel processors. Applications of the method to linear parabolic problem, steady state and nonsteady state Navier-Stokes problems are given.
15230541
Efficient and reliable schemes for nonlinear diffusion filtering
Nonlinear diffusion filtering in image processing is usually performed with explicit schemes. They are only stable for very small time steps, which leads to poor efficiency and limits their practical use. Based on a discrete nonlinear diffusion scale-space framework we present semi-implicit schemes which are stable for all time steps. These novel schemes use an additive operator splitting (AOS), which guarantees equal treatment of all coordinate axes. They can be implemented easily in arbitrary dimensions, have good rotational invariance and reveal a computational complexity and memory requirement which is linear in the number of pixels. Examples demonstrate that, under typical accuracy requirements, AOS schemes are at least ten times more efficient than the widely used explicit schemes.
17434449
On Two Multigrid Algorithms for Modeling Variational Multiphase Image Segmentation
In this paper, we present two related multigrid algorithms for multiphase image segmentation. Algorithm I solves the model by Vese-Chan. We first generalize our recently developed multigrid method to this multiphase segmentation model (MG1); we also give a local Fourier analysis for the local smoother which leads to a new and more effective smoother. Although MG1 is found many magnitudes faster than the fast method of additive operator splitting (AOS), both algorithms are not robust with regard to the initial guess. To overcome this dependence on the initial guess, we consider a hierarchical segmentation model which achieves multiphase segmentation by repeated use of the Chan-Vese two-phase model; our algorithm II solves this model by a multigrid algorithm (MG2). Numerical experiments show that both algorithms are efficient and in particular MG2 is more robust than MG1 with respect to initial guesses. AMS subject classifications: 68U10, 65F10, 65K10.
12054143
Nonlinear oblique derivative problems for singular degenerate parabolic equations on a general domain
Abstract We establish comparison and existence theorems of viscosity solutions of the initial-boundary value problem for some singular degenerate parabolic partial differential equations with nonlinear oblique derivative boundary conditions. The theorems cover the capillary problem for the mean curvature flow equation and apply to more general Neumann-type boundary problems for parabolic equations in the level set approach to motion of hypersurfaces with velocity depending on the normal direction and curvature.
10233402
TO VISCOSITY SOLUTIONS OF SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS
The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions. Introduction The theory of viscosity solutions applies to certain partial differential equations of the form F(x ,u,Du, D2u) = 0 where F:RN xRxRN x S"(N) -* R and S"(N) is the set of symmetric N x N matrices. The primary virtues of this theory are that it allows merely continuous functions to be solutions of fully nonlinear equations of second order, that it provides very general existence and uniqueness theorems and that it yields precise formulations of general boundary conditions. Moreover, these features go hand-in-hand with a great flexibility in passing to limits in various settings and relatively simple proofs. In the expression F(x, u, Du, D2u) u will be a real-valued function defined on some subset cf of R^, Du corresponds to the gradient of u and D2u corresponds to the matrix of second derivatives of u. However, as explained below, Du and D2u will not have classical meanings and, in fact, the examples exhibited in § 1 will convince the reader that the theory encompasses classes of equations that have no solutions that are differentiable in the classical sense. In order that the theory apply to a given equation F = 0, we will require F to satisfy a fundamental monotonicity condition (0.1) F(x, r, p, X) < F(x, s, p, Y) whenever r < s and Y < X ; where r,s eR, x,p eRN, X,Y e S"{N) and S"(N) is equipped with its usual order. Received by the editors November 16, 1990. 1991 Mathematics Subject Classification. Primary 35D05, 35B50, 35J60, 35K55; Secondary 35B05, 35B25, 35F20, 35J25, 35J70, 35K20, 35K15, 35K65.
3759644
Segmentation of complex geophysical structures with well data
In many problems of geophysical interest, when trying to segment images (i.e., to locate interfaces between different regions on the images), one has to deal with data that exhibit very complex structures. This occurs, for instance, when describing complex geophysical images (with layers, faults,...); in that case, segmentation is very difficult. Moreover, the segmentation process requires to take into account well data to interpolate, which implies integrating interpolation condition in the mathematical model.More precisely, let $I:\Omega\rightarrow\Re$ be a given bounded image function, where Ω is an open and bounded domain that belongs to $\Re^{n}$. Let $S=\left\{ x_{i}\right\} _{i}\in\Omega$ be a finite set of given points (well data). The aim is to find a contour Γ⊂Ω such that Γ is an object boundary interpolating the points from S. To do that, we combine the ideas of the geodesic active contour (Caselles et al., Int. J. Comput. Vision 22-1:61-87, 1997) and of interpolation of points (Zhao et al., Comput. Vis. Image Understand. 80:295-314, 1986) in a Level Set approach developed by Osher and Sethian (J. Comput. Phys. 79:12-49, 1988). We present modelling of the proposed method. Both theoretical results (viscosity solution) and numerical results (on a velocity model for a real seismic line) are given.
123749007
New second order Mumford–Shah model based on Γ-convergence approximation for image processing
Abstract In this paper, a second order variational model named the Mumford–Shah total generalized variation (MSTGV) is proposed for simultaneously image denoising and segmentation, which combines the original Γ-convergence approximated Mumford–Shah model with the second order total generalized variation (TGV). For image denoising, the proposed MSTGV can eliminate both the staircase artefact associated with the first order total variation and the edge blurring effect associated with the quadratic H 1 regularization or the second order bounded Hessian regularization. For image segmentation, the MSTGV can obtain clear and continuous boundaries of objects in the image. To improve computational efficiency, the implementation of the MSTGV does not directly solve its high order nonlinear partial differential equations and instead exploits the efficient split Bregman algorithm. The algorithm benefits from the fast Fourier transform, analytical generalized soft thresholding equation, and Gauss–Seidel iteration. Extensive experiments are conducted to demonstrate the effectiveness and efficiency of the proposed model.
7740699
Image Segmentation Using Euler’s Elastica as the Regularization
The active contour segmentation model of Chan and Vese has been widely used and generalized in different contexts in the literature. One possible modification is to employ Euler’s elastica as the regularization of active contour. In this paper, we study the new effects of this modification and validate them numerically using the augmented Lagrangian method.
17603209
Existence and uniqueness results for the gradient vector flow and geodesic active contours mixed model
This article deals with the so called GVF (Gradient Vector Flow) 
introduced by C. Xu, J.L. Prince [26, 27]. We give existence and uniqueness 
results for the front propagation flow for boundary extraction that was initiated 
by Paragios, Mellina-Gottardo and Ralmesh [22, 23]. This model combines the 
geodesic active contour flow and the GVF to determine the geometric flow. 
The motion equation is considered within a level set formulation to result an 
Hamilton-Jacobi equation.
60147867
Variational models and numerical algorithms for selective image segmentation
This thesis deals with the numerical solution of nonlinear partial differential equations and their application in image processing. The differential equations we deal with 
here arise from the minimization of variational models for image restoration techniques (such as denoising) and recognition of objects techniques (such as segmentation). Image denoising is a technique aimed at restoring a digital image that has been contaminated by noise while segmentation is a fundamental task in image analysis responsible for partitioning an image as sub-regions or representing the image into something that is more meaningful and easier to analyze such as extracting one or more specific objects of interest in images based on relevant information or a desired feature. 
Although there has been a lot of research in the restoration of images, the performance of such methods is still poor, especially when the images have a high level of 
noise or when the algorithms are slow. Task of the segmentation is even more challenging problem due to the difficulty of delineating, even manually, the contours of the objects of interest. The problems are often due to low contrast, fuzzy contours, similar intensities with adjacent objects, or the objects to be extracted having no real contours. 
The first objective of this work is to develop fast image restoration and segmentation methods which provide better denoising and fast and robust performance for image segmentation. The contribution presented here is the development of a restarted homotopy analysis method which has been designed to be easily adaptable to various types of image processing problems. As a second research objective we propose a framework for image selective segmentation which partitions an image based on the 
information known in advance of the object/objects to be extracted (for example the left kidney is the target to be extracted in a CT image and the prior knowledge is a 
few markers in this object of interest). This kind of segmentation appears especially in medical applications. Medical experts usually estimate and manually draw the boundaries of the organ/organs based on their experience. Our aim is to introduce automatic segmentation of the object of interest as a contribution not only to the way doctors and surgeons diagnose and operate but to other fields as well. The proposed methods showed success in segmenting different objects and perform well in different types of images not only in two-dimensional but in three-dimensional images as well.