Spectral calibration of digital cameras based on the spectral data of commercially available calibration charts is an ill-conditioned problem that has an infinite number of solutions. We introduce a method to estimate the sensor's spectral sensitivity function based on metamers. For a given patch on the calibration chart we construct numerical metamers by computing convex linear combinations of spectra from calibration chips with lower and higher sensor response values. The difference between the measured reflectance spectrum and the numerical metamer lies in the null space of the sensor. For each measured spectrum we use this procedure to compute a collection of color signals that lie in the null space of the sensor. For a collection of such spaces we compute the robust principal components, and we obtain an estimate of the sensor by computing the common null space spanned by these vectors. Our approach has a number of advantages over standard techniques: It is robust to outliers and is not dominated by larger response values, and it offers the ability to evaluate the goodness of the solution where it is possible to show that the solution is optimal, given the data, if the calculated range is one dimensional. © 2006 Optical Society of America.
Spectral calibration of digital cameras based on the spectral data of commercially available calibration charts is an illconditioned problem which has an infinite number of solutions. To improve upon the estimate, different constraints are commonly employed. Traditionally such constraints include: nonnegativity, smoothness, uni-modality and that the estimated sensors results in as good as possible response fit. In this paper, we introduce a novel method to solve a general ill-conditioned linear system with special focus on the solution of spectral calibration. We introduce a new approach based on metamerism. We observe that the difference between two metamers (spectra that integrate to the same sensor response) is in the null-space of the sensor. These metamers are used to robustly estimate the sensor-s null-space. Based on this nullspace, we derive projection operators to solve for the range of the unknown sensor. Our new approach has a number of advantages over standard techniques: It involves no minimization which means that the solution is robust to outliers and is not dominated by larger response values. It also offers the ability to evaluate the goodness of the solution where it is possible to show that the solution is optimal, given the data, if the calculated range is one dimensional. When comparing the new algorithm with the truncated singular value decomposition and Tikhonov regularization we found that the new method performs slightly better for the training set with noticeable improvements for the test data.
Level set methods are a popular way to solve the image segmentation problem in computer image analysis. A contour is implicitly represented by the zero level of a signed distance function, and evolved according to a motion equation in order to minimize a cost function. This function defines the objective of the segmentation problem and also includes regularization constraints. Gradient descent search is the de facto method used to solve this optimization problem. Basic gradient descent methods, however, are sensitive for local optima and often display slow convergence. Traditionally, the cost functions have been modified to avoid these problems. In this work, we instead propose using a modified gradient descent search based on resilient propagation (Rprop), a method commonly used in the machine learning community. Our results show faster convergence and less sensitivity to local optima, compared to traditional gradient descent.
Level set methods are a popular way to solve the image segmentation problem. The solution contour is found by solving an optimization problem where a cost functional is minimized. Gradient descent methods are often used to solve this optimization problem since they are very easy to implement and applicable to general nonconvex functionals. They are, however, sensitive to local minima and often display slow convergence. Traditionally, cost functionals have been modified to avoid these problems. In this paper, we instead propose using two modified gradient descent methods, one using a momentum term and one based on resilient propagation. These methods are commonly used in the machine learning community. In a series of 2-D/3-D-experiments using real and synthetic data with ground truth, the modifications are shown to reduce the sensitivity for local optima and to increase the convergence rate. The parameter sensitivity is also investigated. The proposed methods are very simple modifications of the basic method, and are directly compatible with any type of level set implementation. Downloadable reference code with examples is available online.
We have previously shown that it is possible to construct a coordinate system in the space of illumination spectra such that the coordinate vectors of the illuminants are located in a cone. Changes in the space of illuminants can then be described by an intensity related scaling and a transformation in the Lorentz group SU(1,1). In practice it is often difficult and expensive to measure these coordinate vectors. Therefore it is of interest to estimate the characteristics of an illuminant from an RGB image captured by a camera. In this paper we will investigate the relation between sequences of illuminants and statistics computed from RGB images of scenes illuminated by these illuminants. As a typical example we will study sequences of black body radiators of varying temperature. We have shown earlier that black body radiators in the mired parametrization can be described by one-parameter groups of the Lorentz group SU(1,1). In this paper we will show that this group theoretical structure of the illuminant space induces a similar structure in spaces of statistical descriptors of the resulting RGB images. We show this relation for mean vectors of RGB images, for RGB histograms and for histograms of images obtained by applying certain spatiospectral linear filters to the RGB images. As a result we obtain estimates of the color temperature of the illuminant from sequences of RGB images of scenes under these illuminants.
We consider several collections of multispectral color signals and describe how linear and non-linear methods can be used to investigate their internal structure. We use databases consisting of blackbody radiators, approximated and measured daylight spectra, multispectral images of indoor and outdoor scenes under different illumination conditions and numerically computed color signals. We apply Principal Components Analysis, group-theoretical methods and three manifold learning methods: Laplacian Eigenmaps, ISOMAP and Conformal Component Analysis. Identification of low-dimensional structures in these databases is important for analysis, model building and compression and we compare the results obtained by applying the algorithms to the different databases.
In this paper, we use Riemann geometry to develop a generalframework for the characterization of and mapping betweencolor spaces. Within this framework we show how to constructmaps, so-called isometries, between two color spaces that preservecolor differences. We illustrate applications of this frameworkby constructing a uniform color space and developing algorithmsfor color reproduction on different printers and correctionof color-vision for color-weak observers.
Visual feature descriptors are essential elements in most computer and robot vision systems. They typically lead to an abstraction of the input data, images, or video, for further processing, such as clustering and machine learning. In clustering applications, the cluster center represents the prototypical descriptor of the cluster and estimates the corresponding signal value, such as color value or dominating flow orientation, by decoding the prototypical descriptor. Machine learning applications determine the relevance of respective descriptors and a visualization of the corresponding decoded information is very useful for the analysis of the learning algorithm. Thus decoding of feature descriptors is a relevant problem, frequently addressed in recent work. Also, the human brain represents sensorimotor information at a suitable abstraction level through varying activation of neuron populations. In previous work, computational models have been derived that agree with findings of neurophysiological experiments on the represen-tation of visual information by decoding the underlying signals. However, the represented variables have a bias toward centers or boundaries of the tuning curves. Despite the fact that feature descriptors in computer vision are motivated from neuroscience, the respec-tive decoding methods have been derived largely independent. From first principles, we derive unbiased decoding schemes for biologically motivated feature descriptors with a minimum amount of redundancy and suitable invariance properties. These descriptors establish a non-parametric density estimation of the underlying stochastic process with a particular algebraic structure. Based on the resulting algebraic constraints, we show formally how the decoding problem is formulated as an unbiased maximum likelihood estimator and we derive a recurrent inverse diffusion scheme to infer the dominating mode of the distribution. These methods are evaluated in experiments, where stationary points and bias from noisy image data are compared to existing methods.
Techniques from the theory of partial differential equations are often used to design filter methods that are locally adapted to the image structure. These techniques are usually used in the investigation of gray-value images. The extension to color images is non-trivial, where the choice of an appropriate color space is crucial. The RGB color space is often used although it is known that the space of human color perception is best described in terms of non-euclidean geometry, which is fundamentally different from the structure of the RGB space. Instead of the standard RGB space, we use a simple color transformation based on the theory of finite groups. It is shown that this transformation reduces the color artifacts originating from the diffusion processes on RGB images. The developed algorithm is evaluated on a set of real-world images, and it is shown that our approach exhibits fewer color artifacts compared to state-of-the-art techniques. Also, our approach preserves details in the image for a larger number of iterations.
The problem of estimating spectral reflectances from the responses of a digital camera has received considerable attention recently This problem can be cast as a regularized regression problem or as a statistical inversion problem. We discuss some previously suggested estimation methods based on critically undersampled RGB measurements and describe some relations between them. We concentrate mainly on those models that are using a priori information in the form of high-resolution measurements. We use the "kernel machine" framework in our evaluations and concentrate on the use of multiple illuminations and on the investigation of the performance of global and locally adapted estimation methods. We also introduce a nonlinear transformation of reflectance values to ensure that the estimated reflection spectra fulfill physically motivated boundary conditions. The reported experimental results are derived from measured and simulated camera responses from the Mansell Matte, NCS, and Pantone data sets.
Using ordinary digital cameras as relatively cheap measurementdevices for estimating spectral color properties has becomean interesting alternative to making pointwise high precisionspectral measurements with special equipments like photospectrometers.The results obtained with these methods cannotcompete with the quality of the traditional high resolutiondevices but they are very attractive since the equipment is relativelycheap and instant measurements are obtained for millionsof measurement points.In this paper we investigate the problem of estimating reflectancespectra from measurements taken with ordinary digitalRGB cameras. We study the effects of using multiple illuminationsand treat the estimation of the reflectance spectra asa regression or a statistical inversion problem. We use both,linear- and non-linear estimation methods where we focus onusing reproducing kernels to avoid explicit formulation of nonlinearities.We also include non-linear conditions based on theproperties of the reflection spectra. Munsell Matte color andPantone are used as data sets to support the proposed methods.The experiments show that the proposed methods improve the estimationresults when compared to standard linear methods.
In this paper we discuss the role of curvature in the context of color spaces. Curvature is a differential geometric property of color spaces that has attracted less attention than other properties like the metric or geodesics. In this paper we argue that the curvature of a color space is important since curvature properties are essential in the construction of color coordinate systems. Only color spaces with negative or zero curvature everywhere allow the construction of Munsell-like coordinates with geodesics, shortest paths between two colors, that never intersect. In differential geometry such coordinate systems are known as Riemann coordinates and they are generalizations of the well-known polar coordinates. We investigate the properties of two measurement sets of just-noticeable-difference (jnd) ellipses and color coordinate systems constructed from them. We illustrate the role of curvature by investigating Riemann normal coordinates in CIELUV and CIELAB spaces. An algorithsm is also shown to build multipatch Riemann coordinates for spaces with the positive curvature.
We extend a method for color weak compensation based on the criterion of preservation of subjective color differences between color normal and color weak observers presented in [2]. We introduce a new algorithm for color weak compensation using local affine maps between color spaces of color normal and color weak observers. We show howto estimate the local affine map and how to determine correspondences between the origins of local coordinates in color spaces of color normal and color weak observers. We also describe a new database of measured color discrimination threshold data. The new measurements are obtained at different lightness levels in CIELUV space. They are measured for color normal and color weak observers. The algorithms are implemented and evaluated using the Semantic Differential method.
We apply a general form of affine transformation model to compensate illumination variations in a series of multispectral images of a static scene and compare it to a particular affine and a diagonal transformation models. These models operate in the original multispectral space or in a lower-dimensional space obtained by Singular Value Decomposition (SVD) of the set of images. We use a system consisting of a multispectral camera and a light dome that allows the measurement of multispectral data under carefully controlled illumination conditions to generate a series of multispectral images of a static scene under varying illumination conditions. We evaluate the compensation performance using the CIELAB colour difference between images. The experiments show that the first 2 models perform satisfactorily in the original and lower dimensional spaces.
Methods from the representation theory of finite groups are used to construct efficient processing methods for the special geometries related to the finite subgroups of the rotation group. We motivate the use of these subgroups in computer vision, summarize the necessary facts from the representation theory and develop the basics of Fourier theory for these geometries. We illustrate its usage for data compression in applications where the processes are (on average) symmetrical with respect to these groups. We use the icosahedral group as an example since it is the largest finite subgroup of the 3D rotation group. Other subgroups with fewer group elements can be studied in exactly the same way.
The description of the relation between the one-parameter subgroups of a group and the differential operators in the Lie-algebra of the group is one of the major topics in Lie-theory. In this paper, we use this framework to derive a partial differential equation which describes the relation between the time-change of the spectral characteristics of the illumination source and the change of the color pixels in an image. In the first part of the paper, we introduce and justify the usage of conical coordinate systems in color space. In the second part we derive the differential equation describing the illumination change and in the last part we illustrate the algorithm with some simulation examples.
The human visual system uses eye movements to gather visual information. They act as visual scanning processes and can roughly be divided into two different types: small movements around fixation points and larger movements between fixation points. The processes are often modeled as random walks, and recent models based on heavy tail distributions, also known as Lev\&\#x00FD; flights, have been used in these investigations. In contrast to these approaches we do not model the stochastic processes, but we will show that the step lengths of the movements between fixation points follow generalized Pareto distributions (GPDs). We will use general arguments from the theory of extreme value statistics to motivate the usage of the GPD and show empirically that the GPDs provide good fits for measured eye tracking data. In the framework of information geometry the GPDs with a common threshold form a two-dimensional Riemann manifold with the Fisher information matrix as a metric. We compute the Fisher information matrix for the GPDs and introduce a feature vector describing a GPD by its parameters and different geometrical properties of its Fisher information matrix. In our statistical analysis we use eye tracker measurements in a database with 15 observers viewing 1003 images under free-viewing conditions. We use Matlab functions with their standard parameter settings and show that a naive Bayes classifier using the eigenvalues of the Fisher information matrix provides a high classification rate identifying the 15 observers in the database.
We introduce the generalized extreme value distributions asdescriptors of edge-related visual appearance properties. Theoreticallythese distributions are characterized by their limitingand stability properties which gives them a role similarto that of the normal distributions. Empirically we will showthat these distributions provide a good fit for images from alarge database of microscopy images with two visually verydifferent types of images. The generalized extreme value distributionsare transformed exponential distributions for whichanalytical expressions for the Fisher matrix are available. Wewill show how the determinant of the Fisher matrix and thegradient of the determinant of the Fisher matrix can be usedas sharpness functions and a combination of the determinantand the gradient information can be used to improve the qualityof the focus estimation.
We introduce the generalized Pareto distributions as a statistical model to describe thresholded edge-magnitude image filter results. Compared to the more common Weibull or generalized extreme value distributions these distributions have at least two important advantages, the usage of the high threshold value assures that only the most important edge points enter the statistical analysis and the estimation is computationally more efficient since a much smaller number of data points have to be processed. The generalized Pareto distributions with a common threshold zero form a two-dimensional Riemann manifold with the metric given by the Fisher information matrix. We compute the Fisher matrix for shape parameters greater than -0.5 and show that the determinant of its inverse is a product of a polynomial in the shape parameter and the squared scale parameter. We apply this result by using the determinant as a sharpness function in an autofocus algorithm. We test the method on a large database of microscopy images with given ground truth focus results. We found that for a vast majority of the focus sequences the results are in the correct focal range. Cases where the algorithm fails are specimen with too few objects and sequences where contributions from different layers result in a multi-modal sharpness curve. Using the geometry of the manifold of generalized Pareto distributions more efficient autofocus algorithms can be constructed but these optimizations are not included here.
Dihedral filters correspond to the Fourier transform of functions defined on square grids. For gray value images there are six pairs of dihedral edge-detector pairs on 5 5 windows. In low-level image statistics the Weibull-or the generalized extreme value distributions are often used as statistical distributions modeling such filter results. Since only points with high filter magnitudes are of interest we argue that the generalized Pareto distribution is a better choice. Practically this also leads to more efficient algorithms since only a fraction of the raw filter results have to be analyzed. The generalized Pareto distributions with a fixed threshold form a Riemann manifold with the Fisher information matrix as a metric tensor. For the generalized Pareto distributions we compute the determinant of the inverse Fisher information matrix as a function of the shape and scale parameters and show that it is the product of a polynomial in the shape parameter and the squared scale parameter. We then show that this determinant defines a sharpness function that can be used in autofocus algorithms. We evaluate the properties of this sharpness function with the help of a benchmark database of microscopy images with known ground truth focus positions. We show that the method based on this sharpness function results in a focus estimation that is within the given ground truth interval for a vast majority of focal sequences. Cases where it fails are mainly sequences with very poor image quality and sequences with sharp structures in different layers. The analytical structure given by the Riemann geometry of the space of probability density functions can be used to construct more efficient autofocus methods than other methods based on empirical moments.
We describe how Lie-theoretical methods can be used to analyze color related problems in machine vision. The basic observation is that the non-negative nature of spectral color signals restricts these unctions to be members of a limited, conical section of the larger Hilbert space of square-integrable functions. From this observation we conclude that the space of color signals can be equipped with a coordinate system consisting of a half-axis and a unit ball with the Lorentz groups as natural trans-formation group. We introduce the theory of the Lorentz group SU(1; 1) as a natural tool for analyzing color image processing problems and derive some descriptions and algorithms that are useful in the investigation of dynamical color changes. We illustrate the usage of these results by describing how to use compress, interpolate, extrapolate and compensate image sequences generated by dynamical color changes.
The description of the relation between the one-parameter groups of a group and the differential operators in the Lie-algebra of the group is one of the major topics in Lie-theory. In this paper we use this framework to derive a partial differential equation which describes the relation between the time-change of the spectral characteristics of the illumination source and the change of the color pixels in an image. In the first part of the paper we introduce and justify the usage of conical coordinate systems in color space. In the second part we derive the differential equation describing the illumination change and in the last part we illustrate the algorithm with some simulation examples.
Eigenvector expansions and perspective projections are used to decompose a space of positive functions into a product of a half-axis and a solid unit ball. This is then used to construct a conical coordinate system where one component measures the distance to the origin, a radial measure of the distance to the axis and a unit vector describing the position on the surface of the ball. A Lorentz group is selected as symmetry group of the unit ball which leads to the Mehler-Fock transform as the Fourier transform of functions depending an the radial coordinate only. The theoretical results are used to study statistical properties of edge magnitudes computed from databases of image patches. The constructed radial values are independent of the orientation of the incoming light distribution (since edge-magnitudes are used), they are independent of global intensity changes (because of the perspective projection) and they characterize the second order statistical moment properties of the image patches. Using a large database of images of natural scenes it is shown that the generalized extreme value distribution provides a good statistical model of the radial components. Finally, the visual properties of textures are characterized using the Mehler-Fock transform of the probability density function of the generalized extreme value distribution.
We introduce the Siegel upper half-space with its symplectic geometry as a framework for low-level image processing. We characterize properties of images with the help of six parameters: two spatial coordinates, the pixel value, and the three parameters of a symmetric positive-definite (SPD) matrix such as the metric tensor. We construct a mapping of these parameters into the Siegel upper half-space. From the general theory, it is known that there is a distance on this space that is preserved by the symplectic transformations. The construction provides a mapping that has relatively simply transformation properties under spatial rotations, and the distance values can be computed with the help of closed-form expressions which allow an efficient implementation. We illustrate the properties of this geometry by considering a special case where we compute for every pixel its symplectic distance to its four spatial neighbors and we show how spatial distances, pixel value changes, and texture properties are described in this unifying symplectic framework.
"Advances in Imaging and Electron Physics" merges two long-running serials-"Advances in Electronics and Electron Physics" and "Advances in Optical and Electron Microscopy". This series features extended articles on the physics of electron devices (especially semiconductor devices), particle optics at high and low energies, microlithography, image science and digital image processing, electromagnetic wave propagation, electron microscopy, and the computing methods used in all these domains.
The structure and transformations of spectral color spaces are described. The approach applied for the spectral color spaces is data driven in the sense that the properties of the illumination sources and the reflectance properties of objects are of primary interest. Mainly three types of objects are considered in this approach including illumination spectra, reflectance spectra, and sensor sensitivity functions. It is found that under low-intensity illumination, a different definition has to be used. In the new coordinate system, one finds a good similarity between the Euclidean distance between two CIELAB vectors and the perceptual color difference between the corresponding colors for an average human observer.
We introduce the theory of group representations as a tool to investigate the structure of spaces related to RGB vectors. Our basic assumption is that for many sources of RGB vectors we can assume that the three channels R, G and B are interchangeable. This means that vectors (RGB) and their permutations such as (GRB) appear approximately with the same probability. We introduce the wide-sense-stationary processes as processes whose matrix of second-order moments commute with the permutations of the channels. For such processes the theory of group representations provides the tools to construct a coordinate transformation that block-diagonalizes the corresponding matrices of second-order moments. This coordinate transformation defines therefore a partial principal component analysis. We implemented the transform and investigated its properties with the help of two large databases together containing over one million images. We also introduce a new parametrization of the coefficient space and show that this parametrization can be used to provide information about the internal structure of the RGB histogram space. We also sketch a generalization taking into account the effect of the reducing the number of bins used.
Many signals can be described as functions on the unit disk (ball). In the framework of group representations it is well-known how to construct Hilbert-spaces containing these functions that have the groups SU(1,N) as their symmetry groups. One illustration of this construction is three-dimensional color spaces in which chroma properties are described by points on the unit disk. A combination of principal component analysis and the Perron-Frobenius theorem can be used to show that perspective projections map positive signals (i.e., functions with positive values) to a product of the positive half-axis and the unit ball. The representation theory (harmonic analysis) of the group SU(1,1) leads to an integral transform, the Mehler-Fock-transform (MFT), that decomposes functions, depending on the radial coordinate only, into combinations of associated Legendre functions. This transformation is applied to kernel density estimators of probability distributions on the unit disk. It is shown that the transform separates the influence of the data and the measured data. The application of the transform is illustrated by studying the statistical distribution of RGB vectors obtained from a common set of object points under different illuminants.
We introduce a two-stage analysis of color spectra. In the first processing stage, correlation with the first eigenvector of a spectral database is used to measure the intensity of a color spectrum. In the second step, a perspective projection is used to map the color spectrum to the hyperspace of spectra with first eigenvector coefficient equal to unity. The location in this hyperspace describes the chromaticity of the color spectrum. In this new projection space, a second basis of eigenvectors is computed and the projected spectrum is described by the expansion in this chromaticity basis. This description is possible since the space of color spectra as conical. We compare this two-stage process with traditional principal component analysis and find that the results of the new structure are closer to the structure of traditional chromaticity descriptors than traditional principal component analysis.
In applications of principal component analysis (PCA) it has often been observed that the eigenvector with the largest eigenvalue has only nonnegative entries when the vectors of the underlying stochastic process have only nonnegative values. This has been used to show that the coordinate vectors in PCA are all located in a cone. We prove that the nonnegativity of the first eigenvector follows from the Perron-Frobenius (and Krein-Rutman theory). Experiments show also that for stochastic processes with nonnegative signals the mean vector is often very similar to the first eigenvector. This is not true in general, but we first give a heuristical explanation why we can expect such a similarity. We then derive a connection between the dominance of the first eigenvalue and the similarity between the mean and the first eigenvector and show how to check the relative size of the first eigenvalue without actually computing it. In the last part of the paper we discuss the implication of theoretical results for multispectral color processing. © 2005 Optical Society of America.
It is known that for every selection of illumination spectra there is a coordinate system such that all coordinate vectors of these illumination spectra are located in a cone. A natural set of transformations of this cone are the Lorentz transformations. In this paper we investigate if sequences of illumination spectra can be described by one-parameter subgroups of Lorentz-transformations. We present two methods to estimate the parameters of such a curve from a set of coordinate points. We also use an optimization technique to approximate a given set of points by a one-parameter curve with a minimum approximation error. In the experimental part of the paper we investigate series of blackbody radiators and sequences of measured daylight spectra and show that one-parameter curves provide good approximations for large sequences of illumination spectra. © 2005 Springer Science + Business Media, Inc.
Understanding the properties of time-varying illumination spectra is of importance in all applications where dynamical color changes due to changes in illumination characteristics have to be analyzed or synthesized. Examples are (dynamical) color constancy and the creation of realistic animations. In this article we show how group theoretical methods can be used to describe sequences of time changing illumination spectra with only few parameters. From the description we can also derive a differential equation that describes the illumination changes. We illustrate the method with investigations of black-body radiation and measured sequences of daylight spectra.
In this paper we introduce the representation theory of the symmetric group S (3) as a tool to investigate the structure of the space of RGB-histograms and to construct fast transforms suitable for search in huge image databases. We show that the theory reveals that typical histogram spaces are highly structured. The algorithms exploit this structure and construct a PCA like decomposition without the need to construct correlation or covariance matrices and their eigenvectors. A hierarchical transform is applied to analyze the internal structure of these histogram spaces. We apply the algorithms to two real-world databases (one from an image provider and one from a image search engine company) containing over one million images.
The octahedral group is one of the finite subgroups of the rotation group in 3-D Euclidean space and a symmetry group of the cubic grid. Compression and filtering of 3-D volumes are given as application examples of its representation theory. We give an overview over the finite subgroups of the 3-D rotation group and their classification. We summarize properties of the octahedral group and basic results from its representation theory. Wide-sense stationary processes are processes with group theoretical symmetries whose principal components are closely related to the representation theory of their symmetry group. Linear filter systems are defined as projection operators and symmetry-based filter systems are generalizations of the Fourier transforms. The algorithms are implemented in Maple/Matlab functions and worksheets. In the experimental part, we use two publicly available MRI volumes. It is shown that the assumption of wide-sense stationarity is realistic and the true principal components of the correlation matrix are very well approximated by the group theoretically predicted structure. We illustrate the nature of the different types of filter systems, their invariance and transformation properties. Finally, we show how thresholding in the transform domain can be used in 3-D signal processing.
In this paper we introduce the representation theory of the symmetric group S(3) as a tool to investigate thestructure of the space of RGB-histograms. We show that the theory reveals that typical histogram spaces arehighly structured and that these structures originate partly in group theoretically defined symmetries. Thealgorithms exploit this structure and constructs a PCA like decomposition without the need to construct correlationor covariance matrices and their eigenvectors. We implemented these algorithms and investigate theirproperties with the help of two real-world databases (one from an image provider and one from a image searchengine company) containing over one million images.
The non-negativity of color signals implies that they span a conical space with a hyperbolic geometry. We use perspective projections to separate intensity from chromaticity, and for 3-D color descriptors the chromatic properties are represented by points on the unit disk. Descriptors derived from the same object point but under different imaging conditions can be joined by a hyperbolic geodesic. The properties of this model are investigated using multichannel images of natural scenes and black body illuminants of different temperatures. We show, over a series of static scenes with different illuminants, how illumination changes influence the hyperbolic distances and the geodesics. Descriptors derived from conventional RGB images are also addressed.
We investigate several topics related to manifoldtechniquesfor signal processing. On the most general levelwe consider manifolds with a Riemannian Geometry. Thesemanifolds are characterized by their inner products on thetangent spaces. We describe the connection between the symmetricpositive-definite matrices defining these inner productsand the Cartan and the Iwasawa decomposition of the generallinear matrix groups. This decomposition gives rise to thedecomposition of the inner product matrices into diagonal matricesand orthonormal and into diagonal and upper triangularmatrices. Next we describe the estimation of the inner productmatrices from measured data as an optimization process onthe homogeneous space of upper triangular matrices. Weshow that the decomposition leads to simple forms of partialderivatives that are commonly used in optimization algorithms.Using the group theoretical parametrization ensures also thatall intermediate estimates of the inner product matrix aresymmetric and positive definite. Finally we apply the methodto a problem from psychophysics where the color perceptionproperties of an observer are characterized with the help ofcolor matching experiments. We will show that measurementsfrom color weak observers require the enforcement of thepositive-definiteness of the matrix with the help of the manifoldoptimization technique.
We investigate several topics related to manifold-techniques for signal processing. On the most general level we consider manifolds with a Riemannian Geometry. These manifolds are characterized by their inner products on the tangent spaces. We describe the connection between the symmetric positive-definite matrices defining these inner products and the Cartan and the Iwasawa decomposition of the general linear matrix groups. This decomposition gives rise to the decomposition of the inner product matrices into diagonal matrices and orthonormal and into diagonal and upper triangular matrices. Next we describe the estimation of the inner product matrices from measured data as an optimization process on the homogeneous space of upper triangular matrices. We show that the decomposition leads to simple forms of gradients and Hessian matrices that are commonly used in optimization algorithms. Using the parametrization from the symmetric space ensures also that all intermediate estimates of the inner product matrix are symmetric and positive definite. Finally we apply the method to a problem from psychophysics where the color perception properties of an observer are characterized with the help of color matching experiments. We will show that measurements from color weak observers require the enforcement of the positive-definiteness of the matrix with the help of the manifold optimization technique.
We introduce an invariant metric in the space of symmetric,positive definite matrices and illustrate the usage of thisspace together with this metric in color processing. For thismetric closed-form expressions for the distances and thegeodesics, (ie. the straight lines in this metric) are availableand we show how to implement them in the case of matricesof size 2x2. In the first illustration we use the framework toinvestigate an interpolation problem related to the ellipsesobtained in the measurements of just-noticeable-distances.For two such ellipses we use the metric to construct an interpolatingsequence of ellipses between them. In the secondapplication construct a texture descriptor for chromaticitydistributions. We describe the probability distributions ofchromaticity vectors by their matrices of second order moments.The distance between these matrices is independentunder linear changes of the coordinate system in the chromaticityspace and can therefore be used to define a distancebetween probability distributions that is independentof the coordinate system used. We illustrate this invariance,by way of an example, in the case of different white pointcorrections.
In this paper we describe illumination changes with the help of elements in the Lorentz group SU(1,1). We show how Lie-theoretical methods can be applied to solve problems related to illumination changes. We derive partial differential equations that describe the changes in the space of color signals. We show how these changes effect the induced variations in the space of RGB vectors. We illustrate the application of these methods with two examples: In the first example we derive a simple linear equation system that links the pointwise pixel changes to the parameters of the illumination change. In the second example we construct operators in the RGB space that either compensate illumination changes or predict the effects of illumination changes.
We introduce a method to combine the color channels of an image to a scalar valued image. Linear combinations of the RGB channels are constructed using the Fisher-Trace-Information (FTI), defined as the trace of the Fisher information matrix of the Weibull distribution, as a cost function. The FTI characterizes the local geometry of the Weibull manifold independent of the parametrization of the distribution. We show that minimizing the FTI leads to contrast enhanced images, suitable for segmentation processes. The Riemann structure of the manifold of Weibull distributions is used to design optimization methods for finding optimal weight RGB vectors. Using a threshold procedure we find good solutions even for images with limited content variation. Experiments show how the method adapts to images with widely varying visual content. Using these image dependent de-colorizations one can obtain substantially improved segmentation results compared to a mapping with pre-defined coefficients.