The invariance and covariance of extracted features from an object under certain transformation play quite important roles in the fields of pattern recognition and image understanding. For instance, in order to recognize a three dimentional object, we need specific features extracted from a given object. These features should be independent of the pose and the location of an object. To extract such feature, The authors have presented the three dimensional vector autoregressive model (3D VAR model). This 3D VAR model is constructed on the quaternion, which is the basis of SU(2) (the rotation group in two dimensional complex space). Then the 3D VAR model is defined by the external products of 3D sequential data and the autoregressive(AR) coefficients, unlike the usual AR models. Therefore the 3D VAR model has some prominent features. For example, The AR coefficients of the 3D VAR model behave like vectors under any three dimensional rotation. In this paper, we derive the invariance from 3D VAR coefficients by inner product of each 3D VAR coefficient. These invariants make it possible to recognize the three dimensional curves.

The general problem of single-view recognition is central to man image understanding and computer vision tasks; so central, that it has been characterized as the holy grail of computer vision. In previous work, we have shown how to approach the general problem of recognizing three dimensional geometric configurations (such as arrangements of lines, points, and conics) from a single two dimensional view, in a manner that is view independent. Our methods make use of advanced mathematical techniques from algebraic geometry, notably the theory of correspondences, and a novel equivariant geometric invariant theory. The machinery gives us a way to understand the relationship that exists between the 3D geometry and its residual in a 2D image. This relationship is shown to be a correspondence in the technical sense of algebraic geometry. Exploiting this, one can compute a set of fundamental equations in 3D and 2D invariants which generate the ideal of the correspondence, and which completely describe the mutual 3D/2D constraints. We have chosen to call these equations object/image equations. They can be exploited in a number of ways. For example, from a given 2D configuration, we can determine a set of non-linear constraints on the geometric invariants of a 3D configurations capable of imaging to the given 2D configuration (features on an object), we can derive a set of equations that constrain the images of that object; helping us to determine if that particular object appears in various images. One previous difficulty has been that the usual numerical geometric invariants get expressed as rational functions of the geometric parameters. As such they are not always defined. This leads to degeneracies in algorithms based on these invariants. We show how to replace these invariants by certain toric subvarieties of Grassmannians where the object/image equations become resultant like expressions for the existence of a non- trivial intersection of these subvarieties with certain Schubert varieties in the Grassmannian. We call this approach the global invariant approach. It greatly increases the robustness and numerical stability of the methods. This approach also has advantages when considering issue sin geometric computation, notably geometric hashing. Here we can exploit the natural metric on the Grassmannian to measure distances between objects and images. Our ultimate aim is the development of new algorithms for geometric content-based retrieval. Content-based retrieval of information from large-scale databases, particularly visual/geometric information contained in images, schematics, design drawings, and geometric models of environments, mechanical parts, or molecules, etc., will play an important role in future distributed information and knowledge system.

In this work we proposed an elegant theory of invariants of color 3D images. Our theory is based on the theory of triplet numbers and quaternions. We propose triplet-quaternion-valued invariants, which are related to the descriptions of objects as the zero sets of implicit polynomials. These are global invariants which show great promise for recognition of complicated objects. Triplet-quaternion-valued invariants have good discriminating power for computer recognition of 3D colour objects using statistical pattern recognition methods. For fast computation of triplet--quaternion--valued invariants we use modular arithmetic of Galois fields and rings, which maps calculation of invariants to fast number theoretical Fourier--Galois--Hamilton--transform.

Highly detailed geometric models, which are represented as dense triangular meshes are becoming popular in computer graphics. Since such 3D meshes often have huge information, we require some methods to treat them efficiently in the 3D mesh processing such as, surface simplification, subdivision surface, curved surface approximation and morphing. In these applications, we often extract features of 3D meshes such as feature vertices and feature edges in preprocessing step. An automatic extraction method of feature edges is treated in this study. In order to realize the feature edge extraction method, we first introduce the concavity and convexity evaluation value. Then the histogram of the concavity and convexity evaluation value is used to separate the feature edge region. We apply a thinning algorithm, which is used in 2D binary image processing. It is shown that the proposed method can extract appropriate feature edges from 3D meshes.

We present a new surface smoothing method. The method enhances sharp variation points of surface normals and, therefore, is good for stable detection of salient surface creases and natural shape segmentation. The method is based on local weighted averaging (diffusion) of surface normals where weights depend nonlinearly on surface curvatures.

Computer models of real world objects and scenes are essential in a large and rapidly growing number of applications, hence motivating the automatic generation of models from images. While the completeness and accuracy of extracted models may be essential in some cases, in applications such as image-based view synthesis in which the goal is to produce new views of a scene, partial models with limited accuracy may produce satisfactory results. In this paper a method is described for partial image based-modeling which relies on a sparse set of matching points between several views. While a sparse set of matching points may be obtained more reliably, it provides only partial information on the reconstructed scene and uses only a small subset of the information contained in the images. Consequently, in the proposed approach, correlation constraints are used in order to test hypotheses in projective space so as to improve the correctness of the reconstructed model. The correlation constraints are based on all the image pixels belonging to the convex hull of the matched point set, thus utilizing a large amount of the information contained in the images. The same constraints are then used to modify the reconstructed model by detecting zones in which the model should be broken into several parts in order to accommodate occlusions in the scene and in order to smooth planar surfaces composed of several polygons. The paper provides demonstration of the application of the proposed approach to image-based view synthesis and geometric distortion correction in document images.

Perceptual geometry is an emerging field of interdisciplinary research whose objectives focus on study of geometry from the perspective of visual perception, and in turn, apply such geometric findings to the ecological study of vision. Perceptual geometry attempts to answer fundamental questions in perception of form and representation of space through synthesis of cognitive and biological theories of visual perception with geometric theories of the physical world. Perception of form and space are among fundamental problems in vision science. In recent cognitive and computational models of human perception, natural scenes are used systematically as preferred visual stimuli. Among key problems in perception of form and space, we have examined perception of geometry of natural surfaces and curves, e.g. as in the observer's environment. Besides a systematic mathematical foundation for a remarkably general framework, the advantages of the Gestalt theory of natural surfaces include a concrete computational approach to simulate or recreate images whose geometric invariants and quantities might be perceived and estimated by an observer. The latter is at the very foundation of understanding the nature of perception of space and form, and the (computer graphics) problem of rendering scenes to visually invoke virtual presence.

We demonstrate how to derive morphological information from micrographs, i.e., grey-level images, of polymeric foams. The segmentation of the images is performed by applying a pulse-coupled neural network. This processing generates blobs of the foams walls/struts and voids, respectively. The contours of the blobs and their corresponding points form the input to a constrained Delaunay tessellation, which provides an unstructured grid of the material under consideration. The subsequently applied Chordal Axis Transform captures the intrinsic shape characteristics, and facilitates the identification and localization of key morphological features. While stochastic features of the polymeric foams struts/walls such as areas, aspect ratios, etc., already can be computed at this stage, the foams voids require further geometric processing. The voids are separated into single foam cells. This shape manipulation leads to a refinement of the initial blob contours, which then requires the repeated application of the constrained Delaunay tessellation and Chordal Axis Transform, respectively. Using minimum enclosing rectangles for each foam cell, finally the stochastic features of the foam voids are computed.

This work studies the generation of a ruled surface from scattered data, which can be obtained by a shape- from - X procedure or can be an experimental output. The ruled surface is generated by the so- called Ferguson Curve Model. This model is an interpolation technique based on parametric cubic polynomials, and thus guaranteeing continuity of the curvature in each point of the initial set. We show how morphological operations - esp. openings and closings can be used to obtain good smoothness of the surface in practice. An application to robot motion planning is presented.

A new method for text extraction from binary images with a textured background is proposed. Text extraction in such a case is very important for successful character recognition, because many character recognition methods expect text printed on a uniform (and typically white) background and their performance significantly degrades if this condition is not satisfied. The methods that have been already proposed to solve this problem, attempt to extract primitives or elements composing the textured background in order to separate text from them. From experiments with commercial character recognition software we observed that such an approach easily leads to the significant growth of errors in character recognition because of degradations in extracted characters, introduced during text extraction. On the other hand, it is hardly possible to reconstruct (more or less precisely) the degraded characters without knowing their class labels and this information is not yet available at this stage. In contrast, we explore another approach similar to symbolic compression of text, which is implemented as a morphological filter using the top-hat transform. This approach detects characters having similar shapes from an original image and it thus avoids character degradations. As a result, the accuracy of character recognition can be improved.

The Gabor wavelet is well-known tool in the various fields, such as computational neuroscience, multi-resolutional analysis and so on. The Gabor wavelet is a kind of the Gaussian modulated sinusoidal wave or a kind of windowed Fourier transformation with the Gaussian kernel window. The Gabor wavelet attains the minimum of the uncertainty relation. However the width and the height of the time-frequency window do not change in their lengths depending on the analyzing frequency. This makes the application area of the Gabor wavelet narrow. On the other hand, instead of using the conventional Gaussian distribution as a kernel of the Gabor wavelet, if the q-normal distribution is used, we can get the q-Gabor wavelet as a possible generalization of the Gabor wavelet. The q-normal distribution, which is given by the author, is one of the generalized Gaussian distribution. In this paper, we give the definitions of the q-Gabor wavelet for continuous version and discrete version. The discrete version consists of two different types of the q-Gabor wavelet. One is similar to the conventional Gabor wavelet with respect to the width and the height of the time-frequency window, the other is similar to the conventional discrete wavelet system such that the width and the height of the time-frequency window change in their lengths depending on the frequency. The mother wavelet is also given for the orthonormal q-Gabor wavelet with some approximation.

This paper proposes a fast, effective and also very adaptable incremental learning system for identifying textures based on features extracted from Gabor space. The Gabor transform is a useful technique for feature extraction since it exhibits properties that are similar to biologically visual sensory systems such as those found in the mammalian visual cortex. Although two-dimensional Gabor filters have been applied successfully to a variety of tasks such as text segmentation, object detection and fingerprint analysis, the work of this paper extends previous work by incorporating incremental learning to facilitate easier training. The proposed system transforms textural images into Gabor space and a non-linear threshold function is then applied to extract feature vectors that bear signatures of the textural images. The mean and variance of each training group is computed followed by a technique that uses the Kohonen network to cluster these features. The centers of these clusters form the basis of an incremental learning paradigm that allows new information to be integrated into the existing knowledge. A number of experiments are conducted for real-time identification or discrimination of textural images.

As we published in the last few years, for a binary neural network pattern recognition system to learn a given mapping {U_m mapped to V_m, m=1 to M} where u_m is an N- dimension analog (pattern) vector, V_m is a P-bit binary (classification) vector, the if-and-only-if (IFF) condition that this network can learn this mapping is that each i-set in {Y_mi, m=1 to M} (where Y_mithere existsV_miU_m and V_mi=+1 or -1, is the i-th bit of VR-m).)(i=1 to P and there are P sets included here.) Is POSITIVELY, LINEARLY, INDEPENDENT or PLI. We have shown that this PLI condition is MORE GENERAL than the convexity condition applied to a set of N-vectors. In the design of old learning machines, we know that if a set of N-dimension analog vectors form a convex set, and if the machine can learn the boundary vectors (or extreme edges) of this set, then it can definitely learn the inside vectors contained in this POLYHEDRON CONE. This paper reports a new method and new algorithm to find the boundary vectors of a convex set of ND analog vectors.

We present a method based on active contours to track and segment biological cells in large image sequences obtained by video-microscopy. This task is facilitated by good time resolution and global contrast, but important obstacles are the low contrast boundary deformations known as pseudopods, as well as cell aggregations and divisions. In order to allow better detection of local boundary deformations, we adopted the gradient vector flow (GVF) model of Xu and Prince, which is defined as the steady-state solution of a reaction-diffusion problem. We discuss an undesirable effect of boundary competition in the GVF that can lead to incorrect segmentations for grey-level images. We propose to replace the steady-state solution by a transient solution of the diffusion equation to alleviate this effect, which also allows significant gains in computation time. To enhance pseudopods over texture features, we use a binary edge map obtained from a Canny-Deriche filter followed by hysteresis thresholding. We use topological operators to efficiently detect intersections and maintain contour separation between aggregating cells. Cell divisions are automatically handled by this method. We discuss limits and possible improvements of this work.

A problem with wavefront reconstruction is that smaller details are often smeared out in the background of larger noise. A least squares fit tries to distribute measurement errors over the entire region. Features of the size of the RMS error tend to get obscured. We present a reconstruction algorithm and framework that redresses some of these problems. Small patches are reconstructed and then joined together. We discuss how this is implemented.

The aims of this series of papers are: (a) to formulate a geometric framework for non-linear analysis of global features of massive data sets; and (b) to quantify non-linear dependencies among (possibly) uncorrelated parameters that describe the data. In this paper, we consider an application of the methods to extract and characterize nonlinearities in the functional magnetic resonance imaging data and EEG of human brain (fMRI). A more general treatment of this theory applies to a wider variety of massive data sets; however, the usual technicalities for computation and accurate interpretation of abstract concepts remain a challenge for each individual area of application.

Computerized Tomography medical images contain significant low intensity black regions along image boundaries. This paper discusses the visualization method by using wavelet based data dependent criterion: The wavelet based criterion is minimizing the difference of wavelet coefficients across the common face. In the first procedure, the important data are selected based on the coefficients of wavelet from the cuberille data. Marching cubes method is not appropriate when the data are scattered data. Data dependent terahedrization is one of pre-processing steps for trivariate scattered data interpolation. The quality of an interpolation depends not only on the distribution of the data point in 3D space, but also on the data values. We apply wavelet-based criterion for each tetrahedron. To achieve the smooth transition, we minimize the difference between each adjacent surface to achieve a nearly C¹. Minimizing the difference of wavelet coefficients is related to achieve the smooth transition. Simulated annealing algorithm is employed to achieve the global optimum for a wide class of optimization criteria. The results of trivariated scattered data interpolation is visualized through an iso-surface rendering. The visualization algorithm of this study was implemented on an O₂ workstation of Silicon Graphics Systems.

Principal and Independent Component Analysis (PCA and ICA) are popular and powerful methods for approximation, regression, blind source separation and numerous other statistical tasks. These methods have inherent linearity assumptions that limit their applicability to globally estimate massive and realistic data sets in terms of a few parameters. Global description of such data sets requires more versatile nonlinear methods. Nonetheless, modification of PCA and ICA can be used in a variety of circumstances to discover the underlying non-linear features of the data set. Differential topology and Riemannian geometry have developed systematic methods for local-to-global integration of linearizable features. Numerical methods from approximation theory are applicable to provide a discrete and algorithmic adaptation of continuous topological methods. Such nonlinear descriptions have a far smaller number of parameters than the dimension of the feature space. In addition, it is possible to describe nonlinear relationship between such parameters. We present the mathematical framework for the extension of these methods to a robust estimate for non-linear PCA. We discuss the application of this technique to the study of the topology of the space of parameters in human image databases.

This paper gives a new method for the rectification. The method is based on an examination of the fundamental matrix, which describes the epipolar geometry of the image pair. The approach avoids camera calibration and makes the resampling images extremely simple by using Bresenham Algorithm to extract pixels along the corresponding epipolar line. For a large set of camera motions, remapping to a plane has the drawback of creating rectified images that are potentially infinitely large and presents a loss of pixel information along epipolar lines. In contrast, our method guarantees that the rectified images are bounded for all possible camera motions and minimizes the loss of pixel information along epipolar lines. Furthermore, it never splits the image so that connected regions are no longer connected even if the epipole locates in the image. A large number of intensive experiments have been carried out, and the results show that more accurate matches can be obtained for initial pair of images after the rectification.

The primary combining of remote sensing and GIS is mainly realized by the transforms of data structure. Because of its own limitations, it is in urgent need to investigate the integration of RS and GIS in higher levels. In this paper, we have discussed the different types of combinings between RS and GIS, and proposed that GIS data should be directly brought into image processing from the first. A tentative idea of how to use the method of granularity to study the common processing unit of RS and GIS is described. Some man-made objects and green lands are chosen for their relative importance. The method called (lambda) - f(alpha) _**p representation is presented here for image compositing based on the concepts of connection cost. The example for the determination of granularity of spatial data processing relating to run-length-code line is also given.

This papre addresses the issue of 3D mesh indexation by using shape descriptors (SDs) under constraints of geometric and topological invariance. A new shape descriptor, the Optimized 3D Hough Transform Descriptor (O3HTD) is here proposed. Intrinsically topologically stable, the O3DHTD is not invariant to geometric transformations. Nevertheless, we show mathematically how the O3DHTD can be optimally associated (in terms of compactness of representation and computational complexity) with a spatial alignment procedure which leads to a geometric invariant behavior. Experimental results have been carried out upon the MPEG-7 3D model database consisting of about 1300 meshes in VRML 2.0 format. Objective retrieval results, based upon the definition of a categorized ground truth subset, are reported in terms of Bull Eye Percentage (BEP) score and compared to those obtained by applying the MPEg-7 3D SD. It is shown that the O3DHTD outperforms the MPEg-7 3D SD of up to 28%.