Ernesto Bribiesca

A new chain code

Abstract A new chain code for shapes composed of regular cells is defined. This boundary chain code is based on the numbers of cell vertices which are in touch with the bounding contour of the shape. This boundary chain code is termed vertex chain code (VCC). The VCC is invariant under translation and rotation. Also, it may be starting point normalized and invariant under mirroring transformation. Using this concept of chain code it is possible to relate the chain length to the contact perimeter , which corresponds to the sum of the boundaries of neighboring cells of the shape (Bribiesca, E., Comp. Math. Appl. 33(11) (1997) 1–9); also, to relate the chain nodes to the contact vertices, which correspond to the vertices of neighboring cells. So, in this way, these relations among the chain and the characteristics of interior of the shape allow us to obtain interesting properties. This work is motivated by the idea of obtaining various shape features computed directly from the VCC without going to Cartesian-coordinate representation. Finally, in order to illustrate the capabilities of the VCC: we present some results using real shapes.

A chain code for representing 3D curves

Abstract A chain code for representing three-dimensional (3D) curves is defined. Any 3D continuous curve can be digitalized and represented as a 3D discrete curve. This 3D discrete curve is composed of constant straight-line segments. Thus, the chain elements represent the orthogonal direction changes of the constant straight-line segments of the discrete curve. The proposed chain code only considers relative direction changes, which allows us to have a curve descriptor invariant under translation and rotation. Also, this curve descriptor may be starting point normalized for open and closed curves and invariant under mirroring transformation. In the content of this work the main characteristics of this chain code are presented. This chain code is inspired by the work of Guzman (MCC Technical Report Number: ACA-254-87, July 13, 1987) for representing 3D Stick Bodies. Finally, we present some results of this chain code to represent and process 3D discrete curves as linear features over the terrain by means of digital elevation model (DEM) data. Also, we use this chain code for representing solids composed of voxels. Thus, each solid represents a DEM which is described by only one chain.

An easy measure of compactness for 2D and 3D shapes

An easy measure of compactness for 2D (two dimensional) and 3D (three dimensional) shapes composed of pixels and voxels, respectively, is presented. The work proposed here is based on the two previous works of the measure of discrete compactness [E. Bribiesca, Measuring 2-D shape compactness using the contact perimeter, Comput. Math. Appl. 33 (1997) 1-9; E. Bribiesca, A measure of compactness for 3D shapes, Comput. Math. Appl. 40 (2000) 1275-1284]. The measure of compactness proposed here improves and simplifies the previous measure of discrete compactness. Now, using this proposed measure of compactness, it is possible to compute measures for any kind of object including porous and fragmented objects. Also, the computation of the measures is very simple by means of the use of only one equation. The measure of compactness proposed here depends in large part on the sum of the contact perimeters of the side-connected pixels for 2D shapes or on the sum of the contact surface areas of the face-connected voxels for 3D shapes. Relations between the perimeter and the contact perimeter for 2D shapes and between the area of the surface enclosing the volume and the contact surface area, are presented. The measure presented here of compactness is invariant under translation, rotation, and scaling. In this work, the term of compactness does not refer to point-set topology, but is related to intrinsic properties of objects. Finally, in order to prove our measure of compactness, we calculate the measures of discrete compactness of different objects. Also, we present an important application for brain structures quantification by means of the use of the new proposed measure of discrete compactness.

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

Abstract 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 measure of compactness for 3D shapes

Abstract A measure of compactness for 3D (three dimensional) shapes composed of voxels, is presented. The work proposed here improves and extends to the measure of discrete compactness [1] from 2D (two dimensional) domain to 3D. The measure of discrete compactness proposed here corresponds to the sum of the contact surface areas of the face-connected voxels of 3D shapes. A relation between the area of the surface enclosing the volume and the contact surface area, is presented. The concept of contact surfaces is extended to 3D shapes composed of different polyhedrons, which divide space generating different 3D lattices. The measure proposed here of discrete compactness is invariant under translation, rotation, and scaling. In this work, the term of compactness does not refer to point-set topology, but is related to intrinsic properties of objects. Finally, in order to prove our measure of compactness, we calculate the measures of discrete compactness of different volcanos (which are compared with their classical measures) from the valley of Mexico using Digital Elevation Model (DEM) data.

Measuring 2-D shape compactness using the contact perimeter

A new perimeter for shapes composed of cells is defined. This perimeter is called the contact perimeter, which corresponds to the sum of the boundaries of neighboring cells of the shape. Also, a relation between the perimeter of the shape and the contact perimeter is presented. The contact perimeter corresponds to the measure of compactness proposed here called discrete compactness. In this case, the term compactness does not refer to point-set topology, but is related to intrinsic properties of objects.

Efficiency of chain codes to represent binary objects

We present a study of compression efficiency for binary objects or bi-level images for different chain-code schemes. Chain-code techniques are used for compression of bi-level images because they preserve information and allow a considerable data reduction. Furthermore, chain codes are the standard input format for numerous shape-analysis algorithms. In this work we apply chain codes to represent object with holes and we compare their compression efficiency for seven chain codes. We have also compared all these chain codes with the JBIG standard for bi-level images.

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

Abstract 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.

A Geometric structure for two-dimensional shapes and three-dimensional surfaces

Abstract A geometric structure called slope change notation (SCN), which describes two-dimensional (2D) shapes and three-dimensional (3D) surfaces in a discrete representation, is presented. The SCN of a curve is obtained by placing constant-length straight-line segments around the curve (the endpoints of the straight-line segments always touching the curve), and calculating the slope changes between contiguous segments scaled to a continuous range from −1 to 1. The SCN is independent of translation and rotation (due to the fact that slope changes around the curve are used), and optionally, of size. The SCN for 2D shapes is 1D. This is an important characteristic, because shapes with particular characteristics are easily generated by numerical sequences; also, it is possible to perform arithmetic operations among shapes and surfaces. The SCN differs from other chain codes, for instance, Freeman chains (Proc. Natn. Electron. Conf.18, 312–324 (1961)), since the proposed notation does not use a grid (and so depends only on itself); its range of slope changes varies continuously from −1 to 1; its vertices always touch the curve, which produces a better description of the shape; and its discrete elements always have the same length. Using this geometric structure only slope changes are variable; the segments size of any shape is always constant. At the end of the paper a related theory “B”, that allows variable segment size as a function of slope changes, is introduced. These ideas are based on previous work (Pattern Recognition13, 123–137 (1981)) and the solutions to many problems which arose are presented in this paper.

A method for representing 3D tree objects using chain coding

We describe a method for representing 3D (three-dimensional) tree objects by means of a chain code. These 3D tree objects correspond to natural existing 3D tree structures, such as: blood vessels, plants, live trees, and so on. Thus, trees are digitalized and represented by a notation called the unique tree descriptor. The unique tree descriptor is invariant under translation and rotation. Furthermore, this descriptor is starting vertex normalized via the unique path in the tree. Also, it is possible to obtain the mirror image of any tree with ease. This unique tree descriptor preserves the shape of trees (and the shape of their branches), allows us to know their geometrical and topological properties. To determine if two 3D tree objects have the same shape, it is only necessary to see if their descriptors are equal. In this manner, graph comparisons and tree searches are eliminated. Also, the proposed tree descriptor is a good tool for storing of 3D tree objects. Finally, in order to prove our method for representing 3D tree objects, we obtain some tree descriptors of objects on real images.