Showing papers in "Journal of Mathematical Imaging and Vision in 1997"

The Dynamics of Nonlinear Relaxation Labeling Processes

Abstract: We present some new results which definitively explain the behavior of the classical, heuristic nonlinear relaxation labeling algorithm of Rosenfeld, Hummel, and Zucker in terms of the Hummel-Zucker consistency theory and dynamical systems theory. In particular, it is shown that, when a certain symmetry condition is met, the algorithm possesses a Liapunov function which turns out to be (the negative of) a well-known consistency measure. This follows almost immediately from a powerful result of Baum and Eagon developed in the context of Markov chain theory. Moreover, it is seen that most of the essential dynamical properties of the algorithm are retained when the symmetry restriction is relaxed. These properties are also shown to naturally generalize to higher-order relaxation schemes. Some applications and implications of the presented results are finally outlined.

Regularization, Scale-Space, and Edge Detection Filters

Abstract: Computational vision often needs to deal with derivatives of digital images. Such derivatives are not intrinsic properties of digital data; a paradigm is required to make them well-defined. Normally, a linear filtering is applied. This can be formulated in terms of scale-space, functional minimization, or edge detection filters. The main emphasis of this paper is to connect these theories in order to gain insight in their similarities and differences. We do not want, in this paper, to take part in any discussion of how edge detection must be performed, but will only link some of the current theories. We take regularization (or functional minimization) as a starting point, and show that it boils down to Gaussian scale-space if we require scale invariance and a semi-group constraint to be satisfied. This regularization implies the minimization of a functional containing terms up to infinite order of differentiation. If the functional is truncated at second order, the Canny-Deriche filter arises. It is also shown that higher dimensional regularization boils down to a rotated version of the one dimensional case, when Cartesian invariance is imposed and the image is vanishing at the borders. This means that the results from 1D regularization can be easily generalized to higher dimensions. Finally we show how an efficient implementation of regularization of order n can be made by recursive filtering using 2n multiplications and additions per output element without introducing any approximation.

Image Processing with Complex Daubechies Wavelets

Abstract: Analyses based on Symmetric Daubechies Wavelets (SDW) lead to complex-valued multiresolution representations of real signals. After a recall of the construction of the SDW, we present some specific properties of these new types of Daubechies wavelets. We then discuss two applications in image processing: enhancement and restoration. In both cases, the efficiency of this multiscale representation relies on the information encoded in the phase of the complex wavelet coefficients.

The Logarithmic Image Processing Model: Connections with Human Brightness Perception and Contrast Estimators

Abstract: The logarithmic image processing (LIP) model is a mathematical framework based on abstract linear mathematics which provides a set of specific algebraic and functional operations that can be applied to the processing of intensity images valued in a bounded range. The LIP model has been proved to be physically justified in the setting of transmitted light and to be consistent with several laws and characteristics of the human visual system. Successful application examples have also been reported in several image processing areas, e.g., image enhancement, image restoration, three-dimensional image reconstruction, edge detection and image segmentation. The aim of this article is to show that the LIP model is a tractable mathematical framework for image processing which is consistent with several laws and characteristics of human brightness perception. This is a survey article in the sense that it presents (almost) previously published results in a revised, refined and self-contained form. First, an introduction to the LIP model is exposed. Emphasis will be especially placed on the initial motivation and goal, and on the scope of the model. Then, an introductory summary of mathematical fundamentals of the LIP model is detailed. Next, the article aims at surveying the connections of the LIP model with several laws and characteristics of human brightness perception, namely the brightness scale inversion, saturation characteristic, Weber‘s and Fechner‘s laws, and the psychophysical contrast notion. Finally, it is shown that the LIP model is a powerful and tractable framework for handling the contrast notion. This is done through a survey of several LIP-model-based contrast estimators associated with special subparts (point, pair of points, boundary, region) of intensity images, that are justified both from a physical and mathematical point of view.

Adaptive Pattern Recognition

Abstract: A long term goal of research in artificial intelligence is to determine and to implement principles which permit a movable machine to direct its actions depending upon sensory feed-back from its environment. This paper concentrates on spatial sensors which input images (2-dimensional arrays). A proposal is put forward in which the machine adapts to the actual data, and examples are given of input prediction, of detection of unexpected events, and of recognition of spatial patterns. The image sequence is locally partitioned into temporally contiguous subsequences of a fixed spatial extent. The spatial extent is constant over time and the temporal extent of a subsequence is maximized subject to the condition that the subsequence has occurred previously. The principle is illustrated on image sequences. It is further demonstrated on images which are structured as pseudo-temporal sequences of their rows. The demonstrations use diverse complex and simple examples to illustrate the versatility of the method. The demonstrations show that to a large degree it is not necessary for the user to supply explicit models for different pattern recognition tasks.

Uniqueness in Shape from Shading Revisited

Abstract: We analyse the problem of representing solutions of first-order partial differential equations in terms of complete integrals and envelopes. In this context, we revisit the uniqueness results already existing in the shape-from-shading literature that concern eikonal equations corresponding to the images of a Lambertian hemi-sphere and a Lambertian plane. We show that the approach adopted by Brooks in [2, 3] is incomplete and subsequently re-establish its uniqueness claims.

Locality and Adjacency Stability Constraints for MorphologicalConnected Operators

Abstract: This paper investigates two constraints for the connected operator class. For binary images, connected operators are those that treat grains and pores of the input in an all or nothing way, and therefore they do not introduce discontinuities. The first constraint, called connected-component (c.c.) locality, constrains the part of the input that can be used for computing the output of each grain and pore. The second, called adjacency stability, establishes an adjacency constraint between connected components of the input set and those of the output set. Among increasing operators, usual morphological filters can satisfy both requirements. On the other hand, some (non-idempotent) morphological operators such as the median cannot have the adjacency stability property. When these two requirements are applied to connected and idempotent morphological operators, we are lead to a new approach to the class of filters by reconstruction. The important case of translation invariant operators and the relationships between translation invariance and connectivity are studied in detail. Concepts are developed within the binary (or set) framework; however, conclusions apply as well to flat non-binary (gray-level) operators.

Set-Valued Means of Random Particles

Abstract: Planar images of powder particles or sand grains can be interpreted as “figures”, i.e., equivalence classes of directly congruent compact sets. The paper introduces a concept of set-valued means and real-valued variances for samples of such figures. In obtaining these results, the images are registered to have similar locations and orientations. The method is applied to find a mean figure of a sample of polygonal particles.

On Projective Invariant Smoothing and Evolutions of PlanarCurves and Polygons

Abstract: Several recently introduced and studied planar curve evolution equations turn out to be iterative smoothing procedures that are invariant under the actions of the Euclidean and affine groups of continuous transformations. This paper discusses possible ways to extend these results to the projective group of transformations. Invariant polygon evolutions are also investigated.

Optimal Binary Morphological Bandpass Filters Induced byGranulometric Spectral Representation

Abstract: Euclidean granulometries are used to decompose a binary image into a disjoint union based on interaction between image shape and the structuring elements generating the granulometry. Each subset of the resulting granulometric spectral bands composing the union defines a filter by passing precisely the bands in the subset. Given an observed image and an ideal image to be estimated, an optimal filter must minimize the expected symmetric-difference error between the ideal image and filtered observed image. For the signal-union-noise model, and for both discrete and Euclidean images, given a granulometry, a procedure is developed for finding a filter that optimally passes bands of the observed noisy image. The key is characterization of an optimal filter in the Euclidean case. Optimization is achieved by decomposing the mean functions of the signal and noise size distributions into singular and differentiable parts, deriving an error representation based on the decomposition, and describing optimality in terms of generalized derivatives for the singular parts and ordinary derivatives for the differentiable parts. Owing to the way in which spectral bands are optimally passed, there are strong analogies with the Wiener filter.

On the Precision in Estimating the Location of Edges and Corners

Abstract: Recently, in Rohr [13], we analyzed the systematic localization errors introduced by local operators for detecting grey-value corners. These errors are inherently due to the differential structure of the operators and, in general, are enlarged by discretization and noise effects. Here, we take the statistical point of view to analyze the localization errors caused by noisy data. We consider a continuous image model that represents the blur as well as noise introduced by an imaging system. In general, the systematic intensity variations are nonlinear functions of the location parameters. For this model we derive analytic results stating lower bounds for the location uncertainty of image features. The lower bounds are evaluated for explicit edge and corner models. We show that the precision of localization in general depends on the noise level, on the size of the observation window, on the width of the intensity transitions, as well as on other parameters describing the systematic intensity variations. We also point out that the uncertainty lower bounds in localizing these image features can in principle be attained by fitting parametric models directly to the image intensities. To give an impression of the achievable accuracy numerical examples are presented.

Minimization of MRF Energy with Relaxation Labeling

Abstract: Recently, there has been increasing interest in Markov random field (MRF) modeling for solving a variety of computer vision problems formulated in terms of the maximum a posteriori (MAP) probability. When the label set is discrete, such as in image segmentation and matching, the minimization is combinatorial. The objective of this paper is twofold: Firstly, we propose to use the continuous relaxation labeling (RL) as an alternative approach for the minimization. The motivation is that it provides a good compromise between the solution quality and the computational cost. We show how the original combinatorial optimization can be converted into a form suitable for continuous RL. Secondly, we compare various minimization algorithms, namely, the RL algorithms proposed by Rosenfeld et al., and by Hummel and Zucker, the mean field annealing of Peterson and Soderberg, simulated annealing of Kirkpatrick, the iterative conditional modes (ICM) of Besag and an annealing version of ICM proposed in this paper. The comparisons are in terms of the minimized energy value (i.e., the solution quality), the required number of iterations (i.e., the computational cost), and also the dependence of each algorithm on heuristics.

Novel Neural Network Models for Computing HomotheticInvariances: An Image Algebra Notation

Abstract: In this paper we propose a theoretical approach to invariant perception. Invariant perception is an important aspect in both natural and artificial perception systems, and it remains an important unsolved problem in heuristically based pattern recognition. Our approach is based on a general theory of neural networks and studies of invariant perception by the cortex. The neural structures that we propose uphold both the architecture and functionality of the cortex as currently understood. The formulation of the proposed neural structures is in the language of image algebra, a mathematical environment for expressing image processing algorithms. Thus, an additional benefit of our study is the implication that image algebra provides an excellent environment for expressing and developing artificial perception systems. The focus of our study is on invariances that are expressible in terms of affine transformations, specifically, homothetic transformations. Our discussion will include both one-dimensional and two-dimensional signal patterns. The main contribution of this paper is the formulation of several novel morphological neural networks that compute homothetic auditory and visual invariances. With respect to the latter, we employ the theory and trends of currently popular artificial vision systems.

Convex Set Symmetry Measurement via Minkowski Addition

Abstract: We introduce and investigate measures of rotation and reflection symmetries for compact convex sets The appropriate symmetrization transformations are used to transform original sets into symmetrical ones Symmetry measures are defined as the ratio of volumes (Lebesgue measure) of the original set and the corresponding symmetrical set For the case of rotation symmetry we use as a symmetrization transformation a generalization of the Minkowski symmetric set (a difference body) for a cyclic group of rotations For the case of reflection symmetry we investigate Blaschke symmetrization Given a convex set and a hyperplane E in R^n we get a set symmetrical with respect to this hyperplane Analyzing all hyperplanes containing the coordinate center one gets the measure of reflection symmetry We discuss the lower bounds of this symmetry measure and also a derived symmetry measure In the two dimensional case a perimetric measure representation of convex sets is applied for convex sets symmetrization as well as for the symmetry measure calculation The perimetric measure allows also to perform a decomposition of a compact convex set into Minkowski sum of two sets The first one is rotationally symmetrical and the second one is completely asymmetrical in the sense that it does not allow such a decomposition We discuss a problem of the fast computation of symmetrization transformations Minkowski addition of two sets is reduced to the convolution of their characteristic functions Therefore, in the case of the discrete plane Z^2, Fast Fourier Transformation can be applied for the fast computation of symmetrization transformations

Invariants of Families of Coplanar Conics and Their Applicationsto Object Recognition

Abstract: This paper presents mutual invariants of families of coplanar conics. These invariants are compared with the use of invariants of two conics and a case is presented where the proposed invariants have a greater discriminating power than the previously used invariants. The use of invariants for two conics is extended to any number of coplanar conics. A lambda-matrix is associated with each family of coplanar conics. The use of lambda-matrices is extended from the single variable polynomial to multi-variable polynomials. The Segre characteristic and other invariants of the lambda-matrix are used as invariants of the family of conics.

A Wavelet-Based Multiresolution Method to AutomaticallyRegister Images

Abstract: We present a method to automatically register images presenting both global and local deformations. The image registration process is performed by exploiting a multi-level/multi-image approach whereby after having wavelet-transformed the images, the subband images at different levels are used in a non-feature-based way to determine the motion vectors between the reference and the target images. The crude motion field determined by block matching at the coarsest level of the pyramid is successively refined by taking advantage of both the orientation sensitivity of the different subbands and the contribution of the adjacent levels. The final registered image is obtained by applying the motion field to the lowest level of the pyramid and by inversely transforming it.

A Subdivision Scheme for Continuous-Scale B-Splinesand Affine-Invariant Progressive Smoothing

Abstract: Multiscale representations and progressive smoothing constitute an important topic in different fields as computer vision, CAGD, and image processing. In this work, a multiscale representation of planar shapes is first described. The approach is based on computing classical B-splines of increasing orders, and therefore is automatically affine invariant. The resulting representation satisfies basic scale-space properties at least in a qualitative form, and is simple to implement. The representation obtained in this way is discrete in scale, since classical B-splines are functions in {\bf C}^{k-2}, where k is an integer bigger or equal than two. We present a subdivision scheme for the computation of B-splines of finite support at continuous scales. With this scheme, B-splines representations in {\bf C}^r are obtained for any real r in [0, \infty) , and the multiscale representation is extended to continuous scale. The proposed progressive smoothing receives a discrete set of points as initial shape, while the smoothed curves are represented by continuous (analytical) functions, allowing a straightforward computation of geometric characteristics of the shape.

Can the Sun‘s Direction be Estimated from an Image Prior to the Computation of Object Shape?

Abstract: Various computational techniques have been developed that perform reasonably well in inferring shape from shading. However, these techniques typically require substantial prerequisite information if they are to evolve an estimate of surface shape. It is therefore interesting to consider how depth might be inferred from shading information without prior knowledge of various scene conditions. One approach has been to undertake a pre-processing step of estimating the light-source direction, thereby providing input to the computation of shape from shading. In this paper, we present evidence that a versatile light-source-direction estimator is unattainable, and propose that, in the absence of domain-specific knowledge, shape and light-source direction should be determined in a coupled manner

Two Image-Template Operations for Binary Image Processing

Abstract: This paper presents two new image algebra image-template operationsmatch and mismatch derived from the general image-template product. These image algebra operations extend the binary morphological erosion and dilation operations and can be used to express elegantly most of binary image processing algorithms in a more natural way than binary morphological operations from the image processing viewpoint. In addition, the match and mismatch operations are easy to implement efficiently on SIMD bit-serial parallel computers.

Anisotropic Textures with Arbitrary Orientation

Abstract: We present a Markov random field (MRF) model for digital images capable of representing anisotropic textures with arbitrary orientations. The discrete Hamiltonian is obtained through finite difference discretization of a continuous elliptic operator on R ², together with a polynomial perturbation. We present the solution of a non-linear system of algebraic equations that estimates the orientation angle and the elliptic operator parameters in terms of the estimated discrete Hamiltonian parameters. We perform experiments of simulation and retrieval of parameters using, respectively, the Gibbs sampler algorithm and the variational estimators for MRF. We use also the estimation algorithm to identify relative rotation of digital images of the same realistic picture scanned at various different orientations.

Image Compression and Transmission Through a Low-RateUltrasonic Link in Subsea Telerobotic Applications

Abstract: This paper describes a progressive data compression algorithm for subsea gray-level image (data) transmission through a low rate ultrasonic link in a telerobotics system. The proposed image compression algorithm is based on JPEG, and has been modified for a specific subsea application, where the communication bit rate is confined to 100–200 bits/s and a frequent updating of reconstructed images is required. The experimental result based on 23 real images shows that the proposed image data compression algorithm performs better than JPEG when images are reconstructed from a small amount of data. The transmission error effect and computational complexity have also been analysed with respect to the proposed algorithm.

Feature Mapping and View Planning with Localized Surface Parameters

Abstract: Object recognition is imperative in industry automation since it empowers robots with the perceptual capability of understanding the three-dimensional(3-D) environment by means of sensory devices. Considering object recognition as a mapping between object models and a partial description of an object, this paper introduces a three-phase filtering method that eliminates candidate models when their differences with the object show up. Throughout the process, a view-insensitive modeling method, namely localized surface parameters, is employed. Surface matching is carried out in the first phase to match models with the object by comparing their localized surface descriptions. A model is a candidate of the object only if every object surface matches locally with at least one of the model surfaces. Since the topological relationship between surfaces specifies the global shape of the object and models, it is then checked in the next phase with local coordinate systems to make sure that a candidate model has the identical structure as the object. Because the information of an object cannot be complete in a single viewing direction, the first two conditions can only determine if a candidate has the same portion as the object. The selected model may still be bigger than the object. To avoid the part-to-whole confusion, in the third phase, a back projection from candidate models is performed to ensure that no unmatched model features become visible when a model is virtually brought to the object‘s orientation. In case multiple models are selected as a result of the insufficient information, disambiguating features and their visible directions are derived to verify the expected feature. In addition to the view independent object recognition under even ambiguous situations, the filtering method has a low computational complexity upper bounded byO(m^2n^2) and lower bounded by O(mn), where m and n are the numbers of model and object features. The three-phase object recognition has been exercised with real and synthesized range images. Experiment results are given in the paper.

Typology of Spatial Structures of Images having the Same Color Set

Abstract: A previous study proposed a new model for generating random spatial patterns for modelling the dispersion of different colors in an image (Chiarello et al., 1996). These simulations represented spatial structures in the sense of landscape ecology and they had to be compared to a real image. Thus, the aim of the present study was to measure the discrepancy between a set of simulations of multicolored mosaics and an observed pattern in order to build a Monte Carlo test. The multicolored mosaics were considered as random closed sets and described with the hitting function for all pairs of colors. This description provided large three-dimensional data tables (distances × color pairs × images) that were analyzed with the help of multiway data analyses. The partial triadic analysis was used. It provided a synthesis of the hitting function since the intrastructure enabled a typology of the distances for each image: the factorial coordinates of supplementary columns were plotted as ordinates against distances as abscissa. This synthetic descriptor provided a graphic tool for measuring the differences between the spatial dispersions of a same set of colors in several images.

Efficient Parallel Manipulations of IBB Coded Imageson Meshes with Multiple Broadcasting

Abstract: The Interpolation-Based Bintree (IBB) is a storage-saving encoding scheme for representing binary images In this paper, we present efficient parallel algorithms for important manipulations on IBB coded images (also called bincodes) Given a set of bincodes, eg,B with size n, the 4-neighbor finding and the diagonal-neighbor finding algorithms onB can be accomplished in O(1) time on ann x n mesh computer with multiple broadcasting (MMB) Given two sets of bincodes, B_1 andB_2 , with size n and m ≤ n, respectively, the intersection and union operations for B_1 and B_2 can be performed in O(1) time on an MMB using (n+m)^2 processors With n^2 processors, the complement operation for B can be performed in O(Log n) time