Abstract-Multiresolution representations are very effective for ana-lyzing the information content of images. We study the properties of the operator which approximates a signal at a given resolution. We show that the difference of information between the approximation of a signal at the resolutions 2 ’ + ’ and 2jcan be extracted by decomposing this signal on a wavelet orthonormal basis of L*(R”). In LL(R), a wavelet orthonormal basis is a family of functions ( @ w (2’ ~-n)),,,“jEZt, which is built by dilating and translating a unique function t+r (xl. This decomposition defines an orthogonal multiresolution rep-resentation called a wavelet representation. It is computed with a py-ramidal algorithm based on convolutions with quadrature mirror lil-ters. For images, the wavelet representation differentiates several spatial orientations. We study the application of this representation to data compression in image coding, texture discrimination and fractal analysis. Index Terms-Coding, fractals, multiresolution pyramids, quadra-ture mirror filters, texture discrimination, wavelet transform.

Most of a signal information is often found in irregular structures and transient phenomena. We review the mathematical characterization of singularities with Lipschitz exponents. The main theorems that estimate local Lipschitz exponents of functions, from the evolution across scales of their wavelet transform are explained. We then prove that the local maxima of a wavelet transform detect the location of irregular structures and provide numerical procedures to compute their Lipschitz exponents. The wavelet transform of singularities with fast oscillations have a different behavior that we study separately. We show that the size of the oscillations can be measured from the wavelet transform local maxima. It has been shown that one and two-dimensional signals can be reconstructed from the local maxima of their wavelet transform [14]. As an application, we develop an algorithm that removes white noises by discriminating the noise and the signal singularities through an analysis of their ...

When extracting features from image data, the type of information that can be extracted may be strongly dependent on the scales at which the feature detectors are applied. This article presents a systematic methodology for addressing this problem. A mechanism is presented for automatic selection of scale levels when detecting one-dimensional features, such as edges and ridges. Anovel concept of a scale-space edge is introduced, defined as a connected set of points in scale-space at which: (i) the gradient magnitude assumes a local maximum in the gradient direction, and (ii) a normalized measure of the strength of the edge response is locally maximal over scales. An important property of this definition is that it allows the scale levels to vary along the edge. Two specific measures of edge strength are analysed in detail. It is shown that by expressing these in terms of γ-normalized derivatives, an immediate consequence of this definition is that fine scales are selected for sharp edges (so as to reduce the shape distortions due to scale-space smoothing), whereas coarse scales are selected for diffuse edges, such that an edge model constitutes a valid abstraction of the intensity profile across the edge. With slight modifications, this idea can be used for formulating a ridge detector with automatic scale selection, having the characteristic property that the selected scales on a scale-space ridge instead reflect the width of the ridge.

: This paper describes a method for the enhancement of curvilinear structures such as vessels and bronchi in 3D medical images. A 3D line enhancement filter is developed with the aim of discriminating line structures from other structures and recovering line structures of various widths. The 3D line filter is based on a combination of the eigenvalues of the 3D Hessian matrix. Multi-scale integration is formulated by taking the maximum among single-scale filter responses, and its characteristics are examined to derive criteria for the selection of parameters in the formulation. The resultant multi-scale line-filtered images provide significantly improved segmentation and visualization of curvilinear structures. The usefulness of the method is demonstrated by the segmentation and visualization of brain vessels from MRI (magnetic resonance imaging) and MRA (magnetic resonance angiography), bronchi from a chest CT, and liver vessels (portal veins) from an abdominal CT. Keywords: 3D image ...

Image segmentation is very essential and critical to image processing and pattern recognition. This survey provides a summary of color image segmentation techniques available now. Basically, color segmentation approaches are based on monochrome segmentation approaches operating in di erent color spaces. Therefore, we rst discuss the major segmentation approaches for segmenting monochrome images: histogram thresholding, characteristic feature clustering, edge detection, region-based methods, fuzzy techniques, neural networks, etc. � then review some major color representation methods and their advantages/disadvantages� nally summarize the color image segmentation techniques using di erent color representations. The usage of color models for image segmentation is also discussed. Some novel approaches such as fuzzy method and physics based method are investigated as well.

This paper presents a novel, parameter-free technique for the segmentation and local description of line structures on multiple scales, both in 2-D and 3-D. The algorithm is based on a nonlinear combination of linear filters and searches for elongated, symmetric line structures, while suppressing the response to edges. The filtering process creates one sharp maximum across the line-feature profile and across scalespace. The multiscale response reflects local contrast and is independent of the local width.

In computer vision and image processing, edge detection concerns the localization of significant variations of the grey level image and the identification of the physical phenomena that originated them. This information is very useful for applications in 3D reconstruction, motion, recognition, image enhancement and restoration, image registration, image compression, and so on. Usually, edge detection requires smoothing and differentiation of the image. Differentiation is an ill-conditioned problem and smoothing results in a loss of information. It is difficult to design a general edge detection algorithm which performs well in many contexts and captures the requirements of subsequent processing stages. Consequently, over the history of digital image processing a variety of edge detectors have been devised which differ in their mathematical and algorithmic properties. This paper is an account of the current state of our understanding of edge detection. We propose an overview of research...

Multiscale image analysis has been used successfully in a number of applications to classify image features according to their relative scales. As a consequence, much has been learned about the scale-space behavior of intensity extrema, edges, intensity ridges, and grey-level blobs. In this paper, we investigate the multiscale behavior of gradient watershed regions. These regions are defined in terms of the gradient properties of the gradient magnitude of the original image. Boundaries of gradient watershed regions correspond to the edges of objects in an image. Multiscale analysis of intensity minima in the gradient magnitude image provides a mechanism for imposing a scale-based hierarchy on the watersheds associated with these minima. This hierarchy can be used to label watershed boundaries according to their scale. This provides valuable insight into the multiscale properties of edges in an image without following these curves through scale-space. In addition, the gradient watershed region hierarchy can be used for automatic or interactive image segmentation. By selecting subtrees of the region hierarchy, visually sensible objects in an image can be easily constructed.

We propose a geometric smoothing method based on local curvature in shapes and images which is governed by the geometric heat equation and is a special case of the reaction-diffusion framework proposed by [28]. For shapes, the approach is analogous to the classical heat equation smoothing, but with a renormalization by arclength at each infinitesimal step. For images, the smoothing is similar to anisotropic diffusion in that, since the component of diffusion in the direction of the brightness gradient is nil, edge location and sharpness are left intact. We present several properties of curvature deformation smoothing of shape: it preserves inclusion order, annihilates extrema and inflection points without creating new ones, decreases total curvature, satisfies the semigroup property allowing for local iterative computations, etc. Curvature deformation smoothing of an image is based on viewing it as a collection of iso-intensity level sets, each of which is smoothed by curvature and the...

Early on, computer vision researchers have realized that multiscale transforms are important to analyze the information content of images. The wavelet theory gives a stable mathematical foundation to understand the properties of such multiscale algorithms. This tutorial describes major applications to multiresolution search, multiscale edge detection and texture discrimination.