Image-processing transforms must satisfy a list of formal requirements. We discuss these requirements and classify them into three categories: “architectural requirements” like locality, recursivity and causality in the scale space, “stability requirements” like the comparison principle and “morphological requirements”, which correspond to shape-preserving properties (rotation invariance, scale invariance, etc.). A complete classification is given of all image multiscale transforms satisfying these requirements. This classification yields a characterization of all classical models and includes new ones, which all are partial differential equations. The new models we introduce have more invariance properties than all the previously known models and in particular have a projection invariance essential for shape recognition. Numerical experiments are presented and compared. The same method is applied to the multiscale analysis of movies. By introducing a property of Galilean invariance, we find a single multiscale morphological model for movie analysis.

Axioms and fundamental equations of image processing

A survey of shape analysis techniques

This paper presents an algorithm based on mathematical morphology and curvature evaluation for the detection of vessel-like patterns in a noisy environment. Such patterns are very common in medical images. Vessel detection is interesting for the computation of parameters related to blood flow. Its tree-like geometry makes it a usable feature for registration between images that can be of a different nature. In order to define vessel-like patterns, segmentation is performed with respect to a precise model. We define a vessel as a bright pattern, piece-wise connected, and locally linear, mathematical morphology is very well adapted to this description, however other patterns fit such a morphological description. In order to differentiate vessels from analogous background patterns, a cross-curvature evaluation is performed. They are separated out as they have a specific Gaussian-like profile whose curvature varies smoothly along the vessel. The detection algorithm that derives directly from this modeling is based on four steps: (1) noise reduction; (2) linear pattern with Gaussian-like profile improvement; (3) cross-curvature evaluation; (4) linear filtering. We present its theoretical background and illustrate it on real images of various natures, then evaluate its robustness and its accuracy with respect to noise.

/pdf/segmentation-of-vessel-like-patterns-using-mathematical-u31a33sacz.pdf

Segmentation of vessel-like patterns using mathematical morphology and curvature evaluation

Presents a new method to match a 2D image to a translated, rotated and scaled reference image. The approach consists of two steps: the calculation of a Fourier-Mellin invariant (FMI) descriptor for each image to be matched, and the matching of the FMI descriptors. The FMI descriptor is translation invariant, and represents rotation and scaling as translations in parameter space. The matching of the FMI descriptors is achieved using symmetric phase-only matched filtering (SPOMF). The performance of the FMI-SPOMF algorithm is the same or similar to that of phase-only matched filtering when dealing with image translations. The significant advantage of the new technique is its capability to match rotated and scaled images accurately and efficiently. The innovation is the application of SPOMF to the FMI descriptors, which guarantees high discriminating power and excellent robustness in the presence of noise. This paper describes the principle of the new method and its discrete implementation for either image detection problems or image registration problems. Practical results are presented for various applications in medical imaging, remote sensing, fingerprint recognition and multiobject identification. >

Symmetric phase-only matched filtering of Fourier-Mellin transforms for image registration and recognition

The distance transform (DT) is a general operator forming the basis of many methods in computer vision and geometry, with great potential for practical applications. However, all the optimal algorithms for the computation of the exact Euclidean DT (EDT) were proposed only since the 1990s. In this work, state-of-the-art sequential 2D EDT algorithms are reviewed and compared, in an effort to reach more solid conclusions regarding their differences in speed and their exactness. Six of the best algorithms were fully implemented and compared in practice.

2D Euclidean distance transform algorithms: A comparative survey

An algorithm is presented to recognize and locate partially distorted 2D shapes without regard to their orientation, location, and size. The algorithm first calculates the curvature function from the digitized image of an object. The points of local maxima and minima extracted from the smooth curvature are used as control points to segment the boundary and to guide the boundary-matching procedure. The boundary-matching procedure considers two shapes at a time, one shape from the template databank, and the other from the object being classified. The procedure tries to match the control points in the unknown shape to those of a shape from the template databank, and estimates the translation, rotation, and scaling factors to be used to normalize the boundary of the unknown shape. The chamfer 3/4 distance transformation and a partial distance measurement scheme constitute the final step in measuring the similarity between the two shapes. The unknown shape is assigned to the class corresponding to the minimum distance. The algorithm has been successfully tested on partial shapes using two sets of data, one with sharp corners and the other with curve segments. This algorithm not only is computationally simple, but also works reasonably well in the presence of a moderate amount of noise. >

Partial shape classification using contour matching in distance transformation

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Partial shape classification using contour matching in distance transformation