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in industrial inspection , there is an ever - growing demand for highly accurate , non - destructive measurements of three - dimensional object geometries . a variety of optical sensors have been developed to meet these demands  @xcite . these sensors satisfy the requirements at least partially . numerous applications , however , still wait for a capable metrology . the limitations of those sensors emerge from physics and technology  the physical limits are determined by the wave equation and by coherent noise , while the technological limits are mainly due to the space - time - bandwidth product of electronic cameras . closer consideration reveals that the technological limits are basically of information - theoretical nature . the majority of the available optical 3d sensors need large amounts of raw data in order to obtain the shape . a lot of redundant information is acquired and the expensive channel capacity of the sensors is used inefficiently  @xcite . a major source of redundancy is the shape of the object itself : if the object surface @xmath0 is almost planar , there is similar height information at each pixel . in terms of information theory the surface points of such objects are `` correlated '' ; their power spectral density @xmath1 decreases rapidly . in order to remove redundancy , one can apply spatial differentiation to whiten the power spectral density ( see fig . [ spectra ] ) . fortunately , there are optical systems that perform such spatial differentiation . indeed , sensors that acquire just the local slope instead of absolute height values are much more efficient in terms of exploiting the available channel capacity . further , reconstructing the object height from slope data reduces the high - frequency noise since integration acts as a low - pass filter . there are several sensor principles that acquire the local slope : for _ rough surfaces _ , it is mainly the principle of shape from shading  @xcite . for _ specular surfaces _ , there are differentiating sensor principles like the differential interference contrast microscopy or deflectometry  @xcite . deflectometric scanning methods allow an extremely precise characterization of optical surfaces by measuring slope variations as small as @xmath2arcsec  @xcite . full - field deflectometric sensors acquire the two - dimensional local gradient of a ( specular ) surface . using `` phase - measuring deflectometry '' ( pmd )  @xcite , for example , one can measure the local gradient of an object at one million sample points within a few seconds . the repeatability of the sensor described in  @xcite is below @xmath3arcsec with an absolute error less than @xmath4arcsec , on a measurement field of @xmath5 mm  @xmath6  @xmath5 mm and a sampling distance of @xmath7 mm .    in several cases it is sufficient to know the local gradient or the local curvature ; however , most applications demand the height information as well . as an example we consider eyeglass lenses . in order to calculate the local surface power of an eyeglass lens by numerical differentiation , we only need the surface slope and the lateral sampling distance . but for quality assurance in an industrial setup , it is necessary to adjust the production machines according to the measured shape deviation . this requires height information of the surface . another application is the measurement of precision optics . for the optimization of these systems sensors are used to measure the local gradient of wavefronts  @xcite . to obtain the shape of these wavefronts , a numerical shape reconstruction method is needed .      in the previous section we stated that measuring the gradient instead of the object height is more efficient from an information - theoretical point of view , since redundant information is largely reduced . using numerical integration techniques , the shape of the object can be reconstructed locally with high accuracy . for example , a full - field deflectometric sensor allows the detection of local height variations as small as _ a few nanometers_.    however , if we want to reconstruct the global shape of the object , low - frequency information is essential . acquiring _ solely the slope _ of the object reduces the low - frequency information substantially ( see fig . [ spectra ] ) . in other words , we have a lot of _ local _ information while lacking _ global _ information , because we reduced the latter by optical differentiation . as a consequence , small measuring errors in the low - frequency range will have a strong effect on the overall reconstructed surface shape . this makes the reconstruction of the global shape a difficult task . furthermore , one - dimensional integration techniques can not be easily extended to the two - dimensional case . in this case , one has to choose a path of integration . unfortunately , noisy data leads to different integration results depending on the path@xcite . therefore , requiring the integration to be path independent becomes an important condition ( `` integrability condition '' ) for developing an optimal reconstruction algorithm ( see sections  [ problem ] and  [ optimal ] ) . we consider an _ object surface _ to be a twice continuously differentiable function @xmath8 on some compact , simply connected region @xmath9 . the integrability condition implies that the gradient field @xmath10 is _ curl free _ , i.e. every path integral between two points yields the same value . this is equivalent to the requirement that there exists a potential @xmath11 to the gradient field @xmath12 which is unique up to a constant . most object surfaces measurable by deflectometric sensors fulfill these requirements , or at least they can be decomposed into simple surface patches showing these properties . measuring the gradient @xmath12 at each sample point @xmath13 yields a discrete vector field @xmath14 . these measured gradient values usually are contaminated by noise the vector field is not necessarily curl free . hence , there might not exist a potential @xmath11 such that @xmath15 for all @xmath16 . in that case , we seek a least - squares approximation , i.e.  a surface representation @xmath11 such that the following error functional is minimized  @xcite : @xmath17 ^ 2 + \left[z_y({\bf x}_i)-q({\bf x}_i)\right]^2.\ ] ] in the case of one - dimensional data , integration is a rather straightforward procedure which has been investigated quite extensively  @xcite . in case of two - dimensional data , there exist mainly two different approaches to solve the stated problem  @xcite . on the one hand , there are _ local methods _ : they integrate along predetermined paths . the advantage of these methods is that they are simple and fast , and that they reconstruct small local height variations quite well . however , they propagate both the measurement error and the discretization error along the path . therefore , they introduce a global shape deviation . this effect is even worse if the given gradient field is not guaranteed to be curl free : in this case , the error also depends on the chosen path . on the other hand , there are _ global methods _ : they try to minimize @xmath18 by solving its corresponding euler - lagrange equation@xcite @xmath19 where @xmath20 and @xmath21 denote the numerical @xmath22- and @xmath23-derivative of the measured data @xmath24 and @xmath25 , respectively . the advantage of global methods is that there is no propagation of the error ; instead , the error gets evenly distributed over all sample points . unfortunately , the implementation has certain difficulties . methods based on finite differences are usually inefficient in their convergence when applied to strongly curved surfaces  @xcite . therefore , they are mainly used for nearly planar objects . another approach to solve eq . ( [ poisson ] ) is based on fourier transformation  @xcite . integration in the fourier domain has the advantage of being optimal with respect to information - theoretical considerations  @xcite . however , fourier methods assume a periodic extension on the boundary which can not be easily enforced with irregularly shaped boundaries in a two - dimensional domain .    in general , it is crucial to note that the reconstruction method depends on the slope - measuring sensor and the properties of the acquired data . for example , slope data acquired by shape from shading is rather noisy , exhibits curl , and is usually located on a full grid of millions of points . here , a fast subspace approximation method like the one proposed by frankot and chellappa  @xcite is appropriate . on the other hand , wavefront reconstruction deals with much smaller data sets , and the surface is known to be rather smooth and flat . in this case , a direct finite - difference solver can be applied  @xcite . deflectometric sensors deliver a third type of data : it consists of very large data sets with rather small noise and curl , but the data may not be complete , depending on the local reflectance of the measured surface . furthermore , the measuring field may have an unknown , irregularly shaped boundary . these properties render most of the aforementioned methods unusable for deflectometric data . in the following sections , we will describe a surface reconstruction method which is especially able to deal with slope data acquired by sensors such as phase - measuring deflectometry . the desired surface reconstruction method should have the properties of both _ local and global _ integration methods : it needs to _ preserve local details _ without propagating the error along a certain path . it also needs to minimize the error functional of eq . ( [ approx_functional ] ) , hence yielding a _ globally optimal solution _ in a least - squares sense . further , the method should be able to deal with _ irregularly shaped boundaries _ , _ missing data points _ , and it has to be able to reconstruct surfaces of a large variety of objects with steep slopes and _ high curvature values_. it should also be able to _ handle large data sets _ which may consist of some million sample points . we now show how to meet these challenges using an analytic interpolation approach . a low noise level allows _ interpolation _ of the slope values instead of approximation . interpolation is a special case which has the great advantage that we can ensure that small height variations are preserved . in this paper we will only focus on the interpolation approach as analytic reconstruction method . for other measurement principles like shape from shading , an approximation approach might be more appropriate . the basic idea of the integration method is as follows : we seek an analytic interpolation function such that its gradient interpolates the measured gradient data . once this interpolation is determined , it uniquely defines the surface reconstruction up to an integration constant . to obtain the analytic interpolant , we choose a generalized hermite interpolation approach employing _ radial basis functions _ ( rbfs )  @xcite . this method has the advantage that it can be applied to _ scattered data_. it allows us to integrate data sets with holes , irregular sampling grids , or irregularly shaped boundaries . furthermore , this method allows for an optimal surface recovery in the sense of eq . ( [ approx_functional ] ) ( see section  [ optimal ] below ) .    in more detail : assuming that the object surface fulfills the requirements described in section  [ problem ] , the data is given as pairs @xmath26 , where @xmath27 and @xmath28 are the measured slopes of the object at @xmath29 in @xmath22- and @xmath23-direction , respectively , for @xmath30 . we define the interpolant to be @xmath31 where @xmath32 and @xmath33 , for @xmath34 , are coefficients and @xmath35 is a radial basis function . hereby , @xmath36 and @xmath37 denote the analytic derivative of @xmath38 with respect to @xmath22 and @xmath23 , respectively . this interpolant is specifically tailored for gradient data  @xcite . to obtain the coefficients in eq . ( [ interpolant ] ) we match the analytic derivatives of the interpolant with the measured derivatives : @xmath39 this leads to solving the following system of linear equations  @xcite : @xmath40      \\[-0.7em ]      $ \phi_{x y}(\mathbf{x}_i-\mathbf{x}_j)$ & $ \phi_{y y}(\mathbf{x}_i-\mathbf{x}_j)$\\      \end{tabular } }      \right)}_{a \ , \in \ , m^{2n\times2n } }      \underbrace{\left (      \mbox{\begin{tabular}{c }       $ \alpha_i$\\      \\[-0.7em ]        \\[-0.7em ]        $ \beta_i$      \end{tabular } }      \right)}_{\alpha \ , \in \ , m^{2n\times1 } }      =      \underbrace{\left (      \mbox{\begin{tabular}{c }       $ p(\mathbf{x}_j)$\\          \\[-0.7em ]        \\[-0.7em ]        $ q(\mathbf{x}_j)$      \end{tabular } }      \right)}_{d \ , \in \ , m^{2n\times1}}.\ ] ] + using the resulting coefficients @xmath41 we then can apply the interpolant in eq . ( [ interpolant ] ) to reconstruct the object surface . for higher noise levels an _ approximation _ approach is recommended . in this case , we simply reduce the number of basis functions so that they do not match the number of data points any more . the system @xmath42 in eq . ( [ linear_system ] ) then becomes overdetermined and can be solved in a least - squares sense . the interpolation approach employing radial basis functions has the advantage that it yields a _ unique _ solution to the surface recovery problem : within this setup , the interpolation matrix in eq . ( [ linear_system ] ) is always symmetric and positive definite . further , the solution satisfies a _ minimization principle _ in the sense that the resulting analytic surface function has minimal energy  @xcite . we choose @xmath38 to be a wendland s function  @xcite , @xmath43 , with @xmath44 this has two reasons : first , wendland s functions allow to choose their continuity according to the smoothness of the given data . the above wendland s function leads to an interpolant which is three - times continuously differentiable , hence guaranteeing the integrability condition . second , the compact support of the function allows to adjust the support size in such a way that the solution of eq . ( [ linear_system ] ) is stable in the presence of noise . it turns out that the support size has to be chosen rather large in order to guarantee a good surface reconstruction  @xcite . the amount of data commonly acquired with a pmd sensor in a single measurement is rather large : it consists of about one million sample points . this amount of data , which results from a measurement with high lateral resolution , would require the inversion of a matrix with @xmath45 entries ( eq .  ( [ linear_system ] ) ) . since we choose a large support size for our basis functions to obtain good numerical stability the corresponding matrix is not sparse . it is obvious that this large amount of data can not be handled directly by inexpensive computing equipment in reasonable time .    to cope with such large data sets we developed a method which first splits the data field into a set of overlapping rectangular patches . we interpolate the data on each patch separately . if the given data were height information only , this approach would yield the complete surface reconstruction . for slope data , we interpolate the data and obtain a surface reconstruction _ up to a constant of integration _ ( see fig . [ mirror_fit](a ) ) on each patch . in order to determine the missing information we apply the following fitting scheme : let us denote two adjacent patches as @xmath46 and @xmath47 and the resulting interpolants as @xmath48 and @xmath49 , respectively . since the constant of integration is still unknown the two interpolants might be on different height levels . generally , we seek a correcting function @xmath50 by minimizing @xmath51 this fitting scheme is then propagated from the center toward the borders of the data field to obtain the reconstructed surface on the entire field ( see fig .  [ mirror_fit](b ) ) .    in the simplest case , the functions @xmath52 are chosen to be constant on each patch , representing the missing constant of integration . if the systematic error of the measured data is small , the constant fit method is appropriate since it basically yields no error propagation . for very noisy data sets it might be better to use a planar fit , i.e.  @xmath53 , to avoid discontinuities at the patch boundaries . this modification , however , introduces a propagation of the error along the patches . the correction angle required on each patch to minimize eq . ( [ lsqfit ] ) depends on the noise of the data . numerical experiments have shown that in most cases the correction angle is at least ten times smaller than the noise level of the measured data . using this information we can estimate the global height error which , by error propagation , may sum up toward the borders of the measuring field  @xcite : @xmath54 where @xmath55 is the standard deviation of the correction angles , @xmath56 is the patch size , and @xmath57 is the number of patches . suppose we want to integrate over a field of @xmath5 mm ( which corresponds to a typical eyeglass diameter ) , assuming a realistic noise level of @xmath58arcsec and a patch size ( not including its overlaps ) of @xmath59 mm . with our setup , this results in @xmath60 patches , with a maximal tipping of @xmath61arcsec per patch . according to eq . ( [ globaltilt ] ) , the resulting global error caused by propagation of the correction angles is _ only @xmath62nm_.    we choose the size of the patches as big as possible , provided that a single patch can still be handled efficiently . for the patch size in the example , @xmath63 points ( including @xmath64 overlap ) correspond to a @xmath65 interpolation matrix that can be inverted quickly using standard numerical methods like cholesky decomposition . a final remark concerning the runtime complexity of the method described above : the complexity can be further reduced in case the sampling grid is regular . since the patches all have the same size and the matrix entries in eq . ( [ linear_system ] ) only depend on the distances between sample points , the matrix can be inverted once for all patches and then applied to varying data on different patches , as long as the particular data subset is complete . note that eq . ( [ interpolant ] ) can be written as @xmath66 , where @xmath67 is the evaluation matrix . then , by applying eq . ( [ linear_system ] ) we obtain @xmath68 , where the matrix @xmath69 needs to be calculated only once for all complete patches . if samples are missing , however , the interpolation yields different coefficients @xmath70 and hence forces to recompute @xmath69 for this particular patch . using these techniques , the reconstruction of @xmath71 surface values from their gradients takes about @xmath72 minutes on a current personal computer . first , we investigated the stability of our method with respect to noise . we simulated realistic gradient data of a sphere ( with @xmath5 mm radius , @xmath5 mm  @xmath6  @xmath5 mm field with sampling distance @xmath73 mm , see figure  [ noise_sphere](a ) ) and added uniformly distributed noise of different levels , ranging from @xmath74 to @xmath75arcsec . we reconstructed the surface of the sphere using the interpolation method described in section  [ method ] . hereby , we aligned the patches by only adding a constant to each patch . the reconstruction was performed for @xmath76 statistically independent slope data sets for each noise level .    depicted in figure  [ noise_sphere](b ) is a cross - section of the absolute error of the surface reconstruction from the ideal sphere , for a realistic noise level of @xmath58arcsec . the absolute error is less than @xmath77 nm _ on the entire measurement field_. the local height error corresponding to this noise level is about @xmath78 nm . this demonstrates that the dynamic range of the global absolute error with respect to the height variance ( @xmath79 mm ) of the considered sphere is about @xmath80 . the graph in figure  [ noise_sphere](c ) depicts the mean value and the standard deviation ( black error bars ) of the absolute error of the reconstruction , for @xmath76 different data sets and for different noise levels . it demonstrates that for increasing noise level the absolute error grows only linearly ( linear fit depicted in gray ) , and even for a noise level being the _ fifty - fold of the typical sensor noise _ the global absolute error remains in the _ sub - micrometer regime_. this result implies that the reconstruction error is smaller than most technical applications require . another common task in quality assurance is the detection and quantification of surface defects like scratches or grooves . we therefore tested our method for its ability to reconstruct such local defects that may be in a range of only a few nanometers . for this purpose , we considered data from a pmd sensor for small , specular objects . the sensor has a resolvable distance of @xmath81 m  laterally and a local angular uncertainty of about @xmath76arcsec  @xcite . in order to quantify the deviation of the perfect shape , we again simulated a sphere ( this time with @xmath76 mm radius and @xmath82 mm  @xmath6  @xmath82 mm data field size ) . we added parallel , straight grooves of varying depths from @xmath83 to @xmath4 nm and of @xmath84 m  width and reconstructed the surface from the modified gradients . the perfect sphere was then subtracted from the reconstructed surface . the resulting reconstructed grooves are depicted in figure  [ groove_sphere](a ) . the grooves ranging from @xmath4 down to @xmath72 nm depth are clearly distinguishable from the plane . figure  [ groove_sphere](b ) shows that all reconstructed depths agree fairly well with the actual depths . note that each groove is determined by only @xmath72 inner sample points . the simulation results demonstrate that our method is almost free of error propagation while preserving small , local details of only some nanometers height . so far , we used only simulated data to test the reconstruction . now , we want to demonstrate the application of our method to a real measurement . the measurement was performed with a phase - measuring deflectometry sensor for very small , specular objects . it can laterally resolve object points with a distance of @xmath85 nm , while having a local angular uncertainty of about @xmath86arcsec . the object under test is a part of a wafer with about @xmath87 nm height range . the size of the measurement field was @xmath4 m  @xmath6  @xmath5 m . depicted in figure  [ wafer ] is the reconstructed object surface from roughly three million data values . both the global shape and local details could be reconstructed with high precision . we motivated why the employment of optical slope - measuring sensors can be advantageous . we gave a brief overview of existing sensor principles . the question that arose next was how to reconstruct the surface from its slope data . we presented a method based on radial basis functions which enables us to reconstruct the object surface from noisy gradient data . the method can handle large data sets consisting of some million entries . furthermore , the data does not need to be acquired on a regular grid  it can be arbitrarily scattered and it can contain data holes . we demonstrated that , while accurately reconstructing the object s global shape , which may have a _ height range of some millimeters _ , the method preserves _ local height variations on a nanometer scale_.    a remaining challenge is to improve the runtime complexity of the algorithm in order to be able to employ it for inline quality assurance in a production process . the authors thank the deutsche forschungsgemeinschaft for supporting this work in the framework of sfb 603 . i.  weingrtner and m.  schulz , `` novel scanning technique for ultra - precise measurement of slope and topography of flats , aspheres , and complex surfaces , '' in _ optical fabrication and testing _ , ( 1999 ) , no . 3739 in proc . spie , pp . 27482 . m.  c. knauer , j.  kaminski , and g.  husler , `` phase measuring deflectometry : a new approach to measure specular free - form surfaces , '' in _ optical metrology in production engineering _ , , w.  osten and m.  takeda , eds . ( 2004 ) , no . 5457 in proc . spie , pp . 366376 . j.  kaminski , s.  lowitzsch , m.  c. knauer , and g.  husler , `` full - field shape measurement of specular surfaces , '' in _ fringe 2005 , the 5th international workshop on automatic processing of fringe patterns _ , , w.  osten , ed . ( springer , 2005 ) , pp . 372379 . s.  lowitzsch , j.  kaminski , m.  c. knauer , and g.  husler , `` vision and modeling of specular surfaces , '' in _ vision , modeling , and visualization 2005 _ , , g.  greiner , j.  hornegger , h.  niemann , and s.  m. , eds . ( akademische verlagsgesellschaft aka gmbh , 2005 ) , pp . 479486 . j.  pfund , n.  lindlein , j.  schwider , r.  burow , t.  blumel , and k.  e. elssner , `` absolute sphericity measurement : a comparative study of the use of interferometry and a shack - hartmann sensor , '' opt . * 23 * , 742744 ( 1998 ) . a.  agrawal , r.  chellappa , and r.  raskar , `` an algebraic approach to surface reconstruction from gradient fields , '' in `` ieee international conference on computer vision ( iccv ) , '' , vol .  1 ( 2005 ) , vol .  1 , pp . 174181 .          c.  elster and i.  weingrtner , `` high - accuracy reconstruction of a function @xmath88 when only @xmath89 or @xmath90 is known at discrete measurement points , '' in `` x - ray mirrors , crystals , and multilayers ii , '' ( 2002 ) , no . 4782 in proc . spie , pp . 1220 . s.  ettl , j.  kaminski , and g.  husler , `` generalized hermite interpolation with radial basis functions considering only gradient data , '' in _ curve and surface fitting : avignon 2006 _ , , a.  cohen , j .- l . merrien , and l.  l. schumaker , eds . ( nashboro press , brentwood , 2007 ) , pp . 141149 .           of a typical smooth object and ( b ) its derivative @xmath91 , together with ( c ) the power spectral density of the surface @xmath1 and ( d ) of its slope @xmath92 ( all in arbitrary units ) . the power spectral density shows that differentiation reduces redundancy , contained in the low frequencies . , width=627 ]            nm  down to @xmath83 nm . after the reconstruction , the sphere was subtracted to make the grooves visible . the actual , reconstructed grooves are depicted in ( a ) full - field and in ( b ) cross - section.,width=529 ]
we present a novel method for reconstructing the shape of an object from measured gradient data . 
 a certain class of optical sensors does not measure the shape of an object , but its local slope . 
 these sensors display several advantages , including high information efficiency , sensitivity , and robustness . for many applications , however , it is necessary to acquire the shape , which must be calculated from the slopes by numerical integration . 
 existing integration techniques show drawbacks that render them unusable in many cases . 
 our method is based on approximation employing radial basis functions . 
 it can be applied to irregularly sampled , noisy , and incomplete data , and it reconstructs surfaces both locally and globally with high accuracy .