A Measure of Shape Dissimilarity for 3D Curves (original) (raw)

Pattern Recognition Letters, 1982

Two reference points of a region are defined which do not depend on the position, size and orientation of the region. Reference points are used to get borders on the basis of which the shape distance and shape similarity are defined.

Classification/comparison of curves by an infinite family of shape invariants

The First Asian Conference on Pattern Recognition, 2011

In this paper we start with a family of boundary based shape measures IN (γ) = R γ (x(s) 2 + y(s) 2) N ds, N = 1, 2,. .. , defined for every curve γ given in an arc-length parametrisation x = x(s), y = y(s), s ∈ [0, 1] and placed such that the centroid of γ and the origin coincide. We prove IN (γ) ≤ 4 −N , for all N = 1, 2,. .. which implies that the sequence IN (γ) converges quickly to 0 and, therefore the first few measures IN (γ) are most useful to compare shapes and to be applied in tasks like object classification, recognition or identification. In order to overcome such a problem, we modify the family IN (γ) and also introduce a parameter p to define a new family IN,p(γ), N = 1, 2,. .. of shape measures. The new family IN,p(γ) includes an infinite number of measures which range over intervals wide enough to provide a discrimination capacity enough to distinguish among the shapes. The role of the parameter p is to provide tuning possibilities for the modified family and to expand the number of applications where the measures can be used efficiently. A set of experimental results are provided in order to justify the theoretical considerations.

A method of optimum transformation of 3D objects used as a measure of shape dissimilarity

Image and Vision Computing, 2003

In this work, we present a method which transforms an object into another. The computation of this transformation is used as a measure of shape-of-object dissimilarity. The considered objects are composed of voxels. Thus, the shape difference of two objects can be ascertained by counting how many voxels we have to move and how far to change one object into another. This work is based on the method presented in [Pattern Recognition 29 (1996) 1117], and our contributions to such a work are a method of optimum transformation of objects and a proposed method of principal axes, which is used to orientate objects. The proposed method is applied to global data. Finally, we present some results using objects of the real world. q

Non-Rigid Shape Comparison of Plane Curves in Images

2002

A mathematical theory for establishing correspondences between curves and for non-rigid shape comparison is developed in this paper. The proposed correspondences, called bimorphisms, are more general than those obtained from one-to-one functions. Their topology is investigated in detail. A new criterion for non-rigid shape comparison using bimorphisms is also proposed. The criterion avoids many of the mathematical problems of previous approaches by comparing shapes non-rigidly from the bimorphism. Geometric invariants are calculated for curves whose shapes can be exactly matched with a bimorphism. The invariants are related to the concave and convex segments of a curve and provide justification for parsing the curve into such segments.

A measure of 2D shape-of-object dissimilarity

Applied Mathematics Letters, 1997

We describe an approach for measuring 2D shape-of-object dissimilarity. The shape of the different objects to be compared is mapped to a representation invariant under translation, rotation, and area. Thus, the shapes will have the same amount of information to describe them (equal number of pixels). The measure of dissimilarity is based on the transformation of one shape into another. This transformation is performed by moving pixels. Thus, the shape difference could be ascertained by counting how many pixels we have to move and how far to change one shape into another. When the shape transformation is performed, the distribution of the shape difference is computed, which permits an improvement in shape comparison.

Contributions to Statistics, 2008

The choice of a dissimilarity measure between curves is a key point for clustering functional data. Functions are usually pointwise compared and, in many situations, this approach is not appropriate. Mathematical Morphology provides us with a toolbox to overcome this problem. We propose some dissimilarity measures based on morphological granulometries and their performance is evaluated on some functional datasets.

IEEE Transactions on Pattern Analysis and Machine Intelligence, 2000

ÐA cognitively motivated similarity measure is presented and its properties are analyzed with respect to retrieval of similar objects in image databases of silhouettes of 2D objects. To reduce influence of digitization noise, as well as segmentation errors, the shapes are simplified by a novel process of digital curve evolution. To compute our similarity measure, we first establish the best possible correspondence of visual parts (without explicitly computing the visual parts). Then, the similarity between corresponding parts is computed and aggregated. We applied our similarity measure to shape matching of object contours in various image databases and compared it to well-known approaches in the literature. The experimental results justify that our shape matching procedure gives an intuitive shape correspondence and is stable with respect to noise distortions. Index TermsÐShape representation, shape similarity measure, visual parts, discrete curve evolution. ae 1 INTRODUCTION A shape similarity measure useful for shape-based retrieval in image databases should be in accord with our visual perception. This basic property leads to the following requirements:

Journal of Symbolic Computation

We provide an algorithm to check whether two rational space curves are related by a similarity. The algorithm exploits the relationship between the curvatures and torsions of two similar curves, which is formulated in a computer algebra setting. Helical curves, where curvature and torsion are proportional, need to be distinguished as a special case. The algorithm is easy to implement, as it involves only standard computer algebra techniques, such as greatest common divisors and resultants, and Gröbner basis for the special case of helical curves. Details on the implementation and experimentation carried out using the computer algebra system Maple 18 are provided.

How to describe pure form and how to measure differences in shapes using shape numbers

Pattern Recognition, 1980

The shape number of a curve is derived for two-dimensional non-intersecting closed curves that are the boundary of simply connected regions. This description is independent of their size, orientation and position, but it depends on their shape. Each curve carries “within it” its own shape number. The order of the shape number indicates the precision with which that number describes the shape of the curve. For a curve, the order of its shape number is the length of the perimeter of a ‘discrete shape’ (a closed curve formed by vertical and horizontal segments, all of equal length) closely corresponding to the curve. A procedure is given that deduces, without table look-up, string matching or correlations, the shape number of any order for an arbitrary curve. To find out how close in shape two curves are, the degree of similarity between them is introduced; dissimilar regions will have a low degree of similarity, while analogous shapes will have a high degree of similarity. Informally speaking, the degree of similarity between the shapes of two curves tells how deep it is necessary to descend into a list of shapes, before being able to differentiate between the shape of those two curves. Again, a procedure is given to compute it, without need for such list or grammatical parsing or least square curve or area fitting. The degree of similarity maps the universe of curves into a tree or hierarchy of shapes. The distance between the shapes of any two curves, defined as the inverse of their degree of similarity, is found to be an ultradistance over this tree. The shape number is a description that changes with skewing, anisotropic dilation and mirror images, as the intuitive psychological concept of “shape” demands. Nevertheless, at the end of the paper a related Theory “B” of shapes is introduced that allows anisotropic changes of scale, thus permitting for instance a rectangle and a square to have the same B shape. These definitions and procedures may facilitate a quantitative study of shape.

A skeletal measure of 2D shape similarity

Computer Vision and Image Understanding, 2004

This paper presents a geometric measure that can be used to gauge the similarity of 2D shapes by comparing their skeletons. The measure is defined to be the rate of change of boundary length with distance along the skeleton. We demonstrate that this measure varies continuously when the shape undergoes deformations. Moreover, we show that ligatures are associated with low values of the shape-measure. The measure provides a natural way of overcoming a number of problems associated with the structural representation of skeletons. The first of these is that it allows us to distinguish between perceptually distinct shapes whose skeletons are ambiguous. Second, it allows us to distinguish between the main skeletal structure and its ligatures, which may be the result of local shape irregularities or noise. We illustrate how the new shapemeasure can be used for the purposes of clustering shock-trees of the same shape class.

A Measure of Shape Dissimilarity for 3D Curves (original) (raw)

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