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 use methods from Riemann geometry to investigate transformations between the color spaces of color-normal and color-weak observers. The two main applications are the simulation of the perception of a color weak observer for a color-normal observer, and the compensation of color images in a way that a color-weak observer has approximately the same perception as a color-normal observer. The metrics in the color spaces of interest are characterized with the help of ellipsoids defined by the just-noticeable-differences between the colors which are measured with the help of color-matching experiments. The constructed mappings are the isometries of Riemann spaces that preserve the perceived color differences for both observers. Among the two approaches to build such an isometry, we introduce normal coordinates in Riemann spaces as a tool to construct a global color-weak compensation map. Compared with the previously used methods, this method is free from approximation errors due to local linearizations, and it avoids the problem of shifting locations of the origin of the local coordinate system. We analyze the variations of the Riemann metrics for different observers obtained from new color-matching experiments and describe three variations of the basic method. The performance of the methods is evaluated with the help of semantic differential tests.

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.

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.

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.

A search engine for font recognition is presented and evaluated. The intended usage is the search in very large font databases. The input to the search engine is an image of a text line, and the output is the name of the font used when rendering the text. After pre-processing and segmentation of the input image, a local approach is used, where features are calculated for individual characters. The method is based on eigenimages calculated from edge filtered character images, which enables compact feature vectors that can be computed rapidly. In this study the database contains 2763 different fonts for the English alphabet. To resemble a real life situation, the proposed method is evaluated with printed and scanned text lines and character images. Our evaluation shows that for 99 % of the queries, the correct font name can be found within the five best matches.

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.

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.

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 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.

Color is widely used for content-based image retrieval. In these applications the color properties of an image are characterized by the probability distribution of the colors in the image. These probability distributions are very often estimated by histograms although the histograms have many drawbacks compared to other estimators such as kernel density methods. In this article we investigate whether using kernel density estimators instead of histograms could give better retrieval results based on hue descriptors of color images. In this article we introduce the Fourier series coefficients as descriptors of hue distributions. We argue that under certain conditions these coefficients are optimal in a least squared error sense. We will also apply Parseval formula to compute the similarity of two distributions directly from these Fourier coefficients. Our experiments show that this modification of the kernel based similarity estimation has better retrieval performance than the histogram methods and we will also show that the method is insensitive to parameter changes as long as they are selected in a reasonable range. © 2005, IS&T - The Society for Imaging Science and Technology.

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.

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.

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 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 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.

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.

Visualization of contrast enhanced blood vessels in CT angiography data presents a challenge due to varying concentration of the contrast agent. The purpose of this work is to evaluate the correctness (effectiveness) in visualizing the vessel lumen using two different 3D visualization strategies, thereby assessing the feasibility of using such visualizations for diagnostic decisions. We compare a standard visualization approach with a recent method which locally adapts to the contrast agent concentration. Both methods are evaluated in a parallel setting where the participant is instructed to produce a complete visualization of the vessel lumen, including both large and small vessels, in cases of calcified vessels in the legs. The resulting visualizations are thereafter compared in a slice viewer to assess the correctness of the visualized lumen. The results indicate that the participants generally overestimated the size of the vessel lumen using the standard visualization, whereas the locally adaptive method better conveyed the true anatomy. The participants did find the interpretation of the locally adaptive method to be less intuitive, but also noted that this did not introduce any prohibitive complexity in the work flow. The observed trends indicate that the visualized lumen strongly depends on the width and placement of the applied transfer function and that this dependency is inherently local rather than global. We conclude that methods that permit local adjustments, such as the method investigated in this study, can be beneficial to certain types of visualizations of large vascular trees

Combining the channels of a multi-band image with the help of a pixelwise weighted sum is one of the basic operations in color and multispectral image processing. A typical example is the conversion of RGB- to intensity images. Usually the weights are given by some standard values or chosen heuristically. This does not take into account neither the statistical nature of the image source nor the intended further processing of the scalar image. In this paper we will present a framework in which we specify the statistical properties of the input data with the help of a representative collection of image patches. On the output side we specify the intended processing of the scalar image with the help of a filter kernel with zero-mean filter coefficients. Given the image patches and the filter kernel we use the Fisher information of the manifold of two-parameter Weibull distributions to introduce the trace of the Fisher information matrix as a cost function on the space of weight vectors of unit length. We will illustrate the properties of the method with the help of a database of scanned leaves and some color images from the internet. For the green leaves we find that the result of the mapping is similar to standard mappings like Matlab’s RGB2Gray weights. We then change the colour of the leaf using a global shift in the HSV representation of the original image and show how the proposed mapping adapts to this color change. This is also confirmed with other natural images where the new mapping reveals much more subtle details in the processed image. In the last experiment we show that the mapping emphasizes visually salient points in the image whereas the standard mapping only emphasizes global intensity changes. The proposed approach to RGB filter design provides thus a new methodology based only on the properties of the image statistics and the intended post-processing. It adapts to color changes of the input images and, due to its foundation in the statistics of extreme-value distributions, it is suitable for detecting salient regions in an image.

We use the theory of group representations to construct very fast image descriptors that split the vector space of local RGB distributions into small group-invariant subspaces. These descriptors are group theoretical generalizations of the Fourier Transform and can be computed with algorithms similar to the FFT. Because of their computational efficiency they are especially suitable for retrieval, recognition and classification in very large image datasets. We also show that the statistical properties of these descriptors are governed by the principles of the Extreme Value Theory (EVT). This enables us to work directly with parametric probability distribution models, which offer a much lower dimensionality and higher resolution and flexibility than explore the connection to EVT and analyse the characteristics of these descriptors from a probabilistic viewpoint with the help of large image databases.

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.

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 two large databases consisting of 750 000 and 1.2 million thumbnail-sized images, labeled with emotion-related keywords. The smaller database consists of images from Matton Images, an image provider. The larger database consists of web images that were indexed by the crawler of the image search engine Picsearch. The images in the Picsearch database belong to one of 98 emotion related categories and contain meta-data in the form of secondary keywords, the originating website and some view statistics. We use two psycho-physics related feature vectors based on the emotional impact of color combinations, the standard RGB-histogram and two SIFT-related descriptors to characterize the visual properties of the images. These features are then used in two-class classification experiments to explore the discrimination properties of emotion-related categories. The clustering software and the classifiers are available in the public domain, and the same standard configurations are used in all experiments. Our findings show that for emotional categories, descriptors based on global image statistics (global histograms) perform better than local image descriptors (bag-of-words models). This indicates that content-based indexing and retrieval using emotion-based approaches are fundamentally different from the dominant object- recognition based approaches for which SIFT-related features are the standard descriptors.

We propose a color-based image descriptor that can be used for image indexing based on high-level semantic concepts. The descriptor is based on Kobayashi's Color Image Scale, which is a system that includes 130 basic colors combined in 1170 three-color combinations. Each combination is labeled with one of 180 high-level semantic concepts, like "elegant", "romantic", "provocative", etc. Moreover, words are located in a two-dimensional semantic space, and arranged into groups based on perceived similarity. From a modified approach for statistical analysis of images, involving transformations of ordinary RGB-histograms, a semantic image descriptor is derived, containing semantic information about both color combinations and single colors in the image. We show how the descriptor can be translated into different levels of semantic information, and used in indexing of multi-colored images. Intended applications are, for instance, image labeling and retrieval.

We describe a novel system for interaction with large font databases. The system is an efficient tool for browsing in large font databases and as such it can be used by people in the Graphic Arts industry. The proposed approach is based on shape descriptors developed for visual characterization of character images rendered from different fonts. Here the descriptors are used in a visualization of a large font database containing more than 2700 fonts. By applying geometry preserving linear- and non-linear manifold learning methods, in combination with a refinement process, character images of different fonts are organized on a two-dimensional grid, where fonts are positioned based on visual similarity.

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.

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.

Segmentation of images is often posed as a variational problem. As such, it is solved by formulating an energy functional depending on a contour and other image derived terms. The solution of the segmentation problem is the contour which extremizes this functional. The standard way of solving this optimization problem is by gradient descent search in the solution space, which typically suffers from many unwanted local optima and poor convergence. Classically, these problems have been circumvented by modifying the energy functional. In contrast, the focus of this paper is on alternative methods for optimization. Inspired by ideas from the machine learning community, we propose segmentation based on gradient descent with momentum. Our results show that typical models hampered by local optima solutions can be further improved by this approach. We illustrate the performance improvements using the level set framework.

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.

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.

The correction of the effects of color-weakness is difficultif not impossible. One of the reasons is that are no objectivecriterions available to evaluate corrections since it is difficult toquantify the color perception of an individual. Also the degreeand the character of color-weakness vary widely among individualsand different color stimuli. In this paper, we use Riemanngeometry to construct a new method for color-weakness correction.The basic mechanism is the matching between the discriminationthresholds of color-weak and of color-normal observers.The goal is to provide the same color perception of color-normalobservers to the color-weak observer: it takes into account theindividual differences between observers and since it defines aglobal deformation of the whole color space it is applicable forcorrection of complex visual inputs such as natural 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 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.

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 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.

Satellite images are important sources of information for meteorologists to predict rapid weather changes, for example storms, now and in the near-future (Nowcasting). It is not possible to use traditional numerical weather forecasts for this purpose since these are computed with a time-lag of several hours. This means that the most recent weather changes are not taken into account. This paper presents a method to compute synthetic satellite images from simulated forecast files. The cloud information in numerical forecast data sets is of much more interest if it can be visualized with a well-known representation like the satellite image. The proposed method uses artificial neural network technology to construct a model which is trained with data from numerical forecasts and classified satellite data captured at the same points in time. The cloud cover parameters in the forecast data set are tied to the cloud classification in the satellite image using a point-to-point representation. The results show that this is a useful method to compute synthetic satellite images. The level of detail in the resulting images is lower than in a real satellite image, but detailed enough to provide information about the principal features of the cloud cover.

We present a projective geometry framework for color invariants using the Extended Dichromatic Reflection Model, in which more realistic and complicated illuminations are considered. Many assumptions which have been used by other methods are relaxed in our framework. Specifically some of the proposed invariants do not require any additional assumption except the ones assumed by the Extended Dichromatic Reflection Model. By putting the color invariance into the projective geometry framework, we can generate different types of invariants and clarify the assumptions under which they are valid. Experiments are presented that illustrate the results derived within our framework.

Visualizing a weather prediction data set by actually synthesizing an image of the sky is a difficult problem. In this paper we present a method for synthesizing realistic sky images from weather prediction and climate prediction data. Images of the sky are combined with a number of weather parameters (like pressure and temperature) to train an artificial neural network (ANN) to predict the appearance of the sky from certain weather parameters. Hourly measurements from a period of eight months are used. The principal component analysis (PCA) method is used to decompose images of the sky into their eigen components -- the eigenskies. In this way the image information is compressed into a small number of coefficients while still preserving the main information in the image. This means that the fine details of the cloud cover cannot be synthesized using this method. The PCA coefficients together with measured weather parameters at the same time form a data point that is used to train the ANN. The results show that the method gives adequate results and although some discrepancies exist, the main appearance is correct. It is possible to distinguish between different types of weather. A rainy day looks rainy and a sunny day looks sunny.

We present a framework to compute the distance between color distributions based on differential geometry. We investigate more detailed the case when color distributions are described as linear combinations of a set of pre-computed basic functions. Experiments in our color based image retrieval system, which were done on 1000 images from Corel image database, show the advantage of our method based on the new distance measure and color descriptor.