Abstract Thrifty methods to represent and store three dimensional objects are important. Two diff... more Abstract Thrifty methods to represent and store three dimensional objects are important. Two different methods for describing voxel-based objects (VBOs) by means of edging (ETs) and intersecting (ITs) trees are demonstrated. Each tree comes from a different kind of border of the underlying VBO, and both trees are one dimensional alternative descriptors to skeletons for VBOs representation. Vertices in the trees correspond to the vertices of the VBO enclosing surface where some surface vertices have been conveniently suppressed. These descriptors are computed using a base-five digit chain code (combined with parentheses) and has been used to illustrate three dimensional curves and enclosing trees. The descriptors are invariant under rotation and translation, and preserve the VBO shape. Using either descriptor, the description of the mirror image of a VBO is easily obtained. The proposed descriptor notation is a good tool for storing VBOs, and intersecting trees providing further storage savings.
A method is described for representing voxel-based objects by means of enclosing trees. An enclos... more A method is described for representing voxel-based objects by means of enclosing trees. An enclosing tree is a tree which totally covers a voxel-based object, the vertices of the enclosing tree correspond to the vertices of the enclosing surface of the analyzed voxel-based object. An enclosing tree is represented by a chain of base-five digit strings suitably combined by means of parentheses. The enclosing-tree notation is invariant under rotation and translation. Furthermore, using this notation it is possible to obtain the mirror image of any voxel-based object with ease. The enclosing-tree notation preserves the shape of voxel-based objects, allowing us to know some of their topological and geometrical properties. Also, the proposed enclosing-tree notation is a good tool for storing of voxel-based objects.
Our experiments with 21 subjects showed that the SWLDA’s performance using our shape-feature vect... more Our experiments with 21 subjects showed that the SWLDA’s performance using our shape-feature vector was , that is, higher than the one obtained with BCI2000-feature’s vector. The shape-feature vector is 34-dimensional for every electrode; however, it is possible to significantly reduce its dimensionality while keeping a high sensitivity.
"The shape number of a curve is derived for two-dimensional non-intersecting closd curves that ar... more "The shape number of a curve is derived for two-dimensional non-intersecting closd 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 shapes of these two curves. Again, a procedure is given to compute it, without need for such list or grammatical parsing or lease 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 qualitative study of shape.
An algorithm for constructing fractal trees is presented. Fractal trees are represented by means ... more An algorithm for constructing fractal trees is presented. Fractal trees are represented by means of the notation called the unique tree descriptor [E. Bribiesca, A method for representing 3D tree objects using chain coding, J. Vis. Commun. Image R. 19 (2008) 184198]. In this manner, we only have a onedimensional representation by each fractal tree via a chain of basefive digit strings suitably combined by means of parentheses. The unique treedescriptor notation is invariant under rotation and translation. Furthermore, using this descriptor it is possible to obtain the mirror image of any fractal tree with ease. In this paper, we focus on fractal planefilling trees and spacefilling trees.
Journal of Visual Communication and Image Representation, 2015
ABSTRACT Many applications in fields as diverse as computer graphics, medical imaging or pattern ... more ABSTRACT Many applications in fields as diverse as computer graphics, medical imaging or pattern recognition require the usage of the boundary of digital objects, or discrete surface. A discrete surface is a set of orthogonal quadrilaterals connected to each other that is typically represented either as a face adjacency graph or as a polygon mesh. In this work we propose a new method, named surface trees, to represent discrete surfaces. Surface trees allow the representation of any discrete surface by coding a tree structure contained in the face adjacency graph. This method uses an alphabet of nine symbols, in addition to the parenthesis notation, to codify trees of maximum degree four. Surface trees are a compact way of representing any discrete surface at the same time they preserve geometrical information and provide invariance under translation and rotation. We demonstrate our method on synthetic surfaces as well as others obtained from real data.
We present a discrete compactness (DC) index, together with a classification scheme, based both o... more We present a discrete compactness (DC) index, together with a classification scheme, based both on the size and shape features extracted from brain volumes, to determine different aging stages: healthy controls (HC), mild cognitive impairment (MCI), and Alzheimer's disease (AD). A set of 30 brain magnetic resonance imaging (MRI) volumes for each group was segmented and two indices were measured for several structures: three-dimensional DC and normalized volumes (NVs). The discrimination power of these indices was determined by means of the area under the curve (AUC) of the receiver operating characteristic, where the proposed compactness index showed an average AUC of 0.7 for HC versus MCI comparison, 0.9 for HC versus AD separation, and 0.75 for MCI versus AD groups. In all cases, this index outperformed the discrimination capability of the NV. Using selected features from the set of DC and NV measures, three support vector machines were optimized and validated for the pairwise separation of the three classes. Our analysis shows classification rates of up to 98.3% between HC and AD, 85% between HC and MCI, and 93.3% for MCI and AD separation. These results outperform those reported in the literature and demonstrate the viability of the proposed morphological indices to classify different aging stages.
2009 IEEE 12th International Conference on Computer Vision Workshops, ICCV Workshops, 2009
This paper describes using 3D chain expressions for encoding unit-width curve skeletons and measu... more This paper describes using 3D chain expressions for encoding unit-width curve skeletons and measuring shape dissimilarity. By integrating a robust skeletonization technique with 3D chain coding, the proposed algorithm can compare not only the skeleton topology but also the ...
ABSTRACT A method for representing 2D (two-dimensional) tree objects is described. This represent... more ABSTRACT A method for representing 2D (two-dimensional) tree objects is described. This representation is based on a chain code, which is called the Slope Chain Code (SCC). Thus, 2D tree objects are described by means of a chain of element strings suitably combined by means of parentheses. These 2D tree objects correspond to naturally existing 2D tree structures. This tree notation preserves the shape of trees (and the shape of their branches), allows us to know their topological and geometrical properties. The proposed notation of 2D tree objects is invariant under translation, rotation and, optionally, under scaling. Also, it is possible to define a unique start vertex for each tree via the unique path in the tree. Using this notation it is possible to obtain the mirror image of any tree with ease. Furthermore, two interesting properties of trees are presented: the accumulated slope and the tortuosity. Tortuosity is a very important property of trees and has many applications in different fields. In order to prove our method for representing 2D tree objects, we obtain some tree descriptors of tree objects and compute their measures of accumulated slope and tortuosity. Finally, we present some examples of 2D trees from the real world about echinoderm species identification.
ABSTRACT A measure of tortuosity for 2D curves is presented. Tortuosity is a very important prope... more ABSTRACT A measure of tortuosity for 2D curves is presented. Tortuosity is a very important property of curves and has many applications, such as: how to measure the tortuosity of retinal blood vessels, intracerebral vasculature, aluminum foams, etc. The measure of tortuosity proposed here is based on a chain code called Slope Chain Code (SCC). The SCC uses some ideas which were described in [A geometric structure for 2D shapes and 3D surfaces, Pattern Recognition 25 (1992) 483-496]. The SCC of a curve is obtained by placing straight-line segments of constant length around the curve (the endpoints of the straight-line segments always touching the curve), and calculating the slope changes between contiguous straight-line segments scaled to a continuous range from −1 to 1. The SCC of a curve is independent of translation, rotation, and optionally, of scaling, which is an important advantage for computing tortuosity. Also, the minimum and maximum values of tortuosity for curves and a measure of normalized tortuosity are described. Finally, an application of the proposed measure of tortuosity is presented which corresponds to the computation of retinal blood vessel tortuosity.
Abstract A method is described for representing voxel-based objects by means of enclosing trees. ... more Abstract A method is described for representing voxel-based objects by means of enclosing trees. An enclosing tree is a tree which totally covers a voxel-based object, the vertices of the enclosing tree correspond to the vertices of the enclosing surface of the analyzed voxel-...
Journal of Knot Theory and Its Ramifications, 2002
A method for representing knots by means of a chain code is presented. Knots which are digitalize... more A method for representing knots by means of a chain code is presented. Knots which are digitalized and represented by the orthogonal direction change chain code are called discrete knots. Discrete knots are composed of constant straight-line segments using only orthogonal directions. ...
Journal of Knot Theory and Its Ramifications, 2006
An easy and fast algorithm for obtaining minimal discrete knots is presented. A minimal discrete ... more An easy and fast algorithm for obtaining minimal discrete knots is presented. A minimal discrete knot is the digitalized representation of a knot, which is composed of the minimum number of constant orthogonal straight-line segments and is represented by the knot-number ...
Journal of Knot Theory and Its Ramifications, 2005
A method for generating families of particular 3D curves which differs from random self-avoiding ... more A method for generating families of particular 3D curves which differs from random self-avoiding walks is presented. This method is based on a chain code called knot numbers. The knot-number notation describes discrete knots. A discrete knot is the digitalized representation of a ...
Resumen SE PRESENTA UN MÉTODO RÁPIDO Y EFICIENTE PARA GRAFICAR OBJETOS SÓLIDOS Y RÍGIDOS COMPUEST... more Resumen SE PRESENTA UN MÉTODO RÁPIDO Y EFICIENTE PARA GRAFICAR OBJETOS SÓLIDOS Y RÍGIDOS COMPUESTOS POR UN GRAN NUMERO DE VAXELES. ESTE MÉTODO SE BASA EN EL CONCEPTO DEL ÁREA DE SUPERFICIE DE CONTACTO ...
Compactness is an intrinsic property of objects. This feature is often associated with the old an... more Compactness is an intrinsic property of objects. This feature is often associated with the old and classical ratio perimeter2/area. This ra-tio is used in many scientific fields as shape descriptor in shape analysis tasks. However, when we use this ratio for measuring shape ...
Abstract Thrifty methods to represent and store three dimensional objects are important. Two diff... more Abstract Thrifty methods to represent and store three dimensional objects are important. Two different methods for describing voxel-based objects (VBOs) by means of edging (ETs) and intersecting (ITs) trees are demonstrated. Each tree comes from a different kind of border of the underlying VBO, and both trees are one dimensional alternative descriptors to skeletons for VBOs representation. Vertices in the trees correspond to the vertices of the VBO enclosing surface where some surface vertices have been conveniently suppressed. These descriptors are computed using a base-five digit chain code (combined with parentheses) and has been used to illustrate three dimensional curves and enclosing trees. The descriptors are invariant under rotation and translation, and preserve the VBO shape. Using either descriptor, the description of the mirror image of a VBO is easily obtained. The proposed descriptor notation is a good tool for storing VBOs, and intersecting trees providing further storage savings.
A method is described for representing voxel-based objects by means of enclosing trees. An enclos... more A method is described for representing voxel-based objects by means of enclosing trees. An enclosing tree is a tree which totally covers a voxel-based object, the vertices of the enclosing tree correspond to the vertices of the enclosing surface of the analyzed voxel-based object. An enclosing tree is represented by a chain of base-five digit strings suitably combined by means of parentheses. The enclosing-tree notation is invariant under rotation and translation. Furthermore, using this notation it is possible to obtain the mirror image of any voxel-based object with ease. The enclosing-tree notation preserves the shape of voxel-based objects, allowing us to know some of their topological and geometrical properties. Also, the proposed enclosing-tree notation is a good tool for storing of voxel-based objects.
Our experiments with 21 subjects showed that the SWLDA’s performance using our shape-feature vect... more Our experiments with 21 subjects showed that the SWLDA’s performance using our shape-feature vector was , that is, higher than the one obtained with BCI2000-feature’s vector. The shape-feature vector is 34-dimensional for every electrode; however, it is possible to significantly reduce its dimensionality while keeping a high sensitivity.
"The shape number of a curve is derived for two-dimensional non-intersecting closd curves that ar... more "The shape number of a curve is derived for two-dimensional non-intersecting closd 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 shapes of these two curves. Again, a procedure is given to compute it, without need for such list or grammatical parsing or lease 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 qualitative study of shape.
An algorithm for constructing fractal trees is presented. Fractal trees are represented by means ... more An algorithm for constructing fractal trees is presented. Fractal trees are represented by means of the notation called the unique tree descriptor [E. Bribiesca, A method for representing 3D tree objects using chain coding, J. Vis. Commun. Image R. 19 (2008) 184198]. In this manner, we only have a onedimensional representation by each fractal tree via a chain of basefive digit strings suitably combined by means of parentheses. The unique treedescriptor notation is invariant under rotation and translation. Furthermore, using this descriptor it is possible to obtain the mirror image of any fractal tree with ease. In this paper, we focus on fractal planefilling trees and spacefilling trees.
Journal of Visual Communication and Image Representation, 2015
ABSTRACT Many applications in fields as diverse as computer graphics, medical imaging or pattern ... more ABSTRACT Many applications in fields as diverse as computer graphics, medical imaging or pattern recognition require the usage of the boundary of digital objects, or discrete surface. A discrete surface is a set of orthogonal quadrilaterals connected to each other that is typically represented either as a face adjacency graph or as a polygon mesh. In this work we propose a new method, named surface trees, to represent discrete surfaces. Surface trees allow the representation of any discrete surface by coding a tree structure contained in the face adjacency graph. This method uses an alphabet of nine symbols, in addition to the parenthesis notation, to codify trees of maximum degree four. Surface trees are a compact way of representing any discrete surface at the same time they preserve geometrical information and provide invariance under translation and rotation. We demonstrate our method on synthetic surfaces as well as others obtained from real data.
We present a discrete compactness (DC) index, together with a classification scheme, based both o... more We present a discrete compactness (DC) index, together with a classification scheme, based both on the size and shape features extracted from brain volumes, to determine different aging stages: healthy controls (HC), mild cognitive impairment (MCI), and Alzheimer's disease (AD). A set of 30 brain magnetic resonance imaging (MRI) volumes for each group was segmented and two indices were measured for several structures: three-dimensional DC and normalized volumes (NVs). The discrimination power of these indices was determined by means of the area under the curve (AUC) of the receiver operating characteristic, where the proposed compactness index showed an average AUC of 0.7 for HC versus MCI comparison, 0.9 for HC versus AD separation, and 0.75 for MCI versus AD groups. In all cases, this index outperformed the discrimination capability of the NV. Using selected features from the set of DC and NV measures, three support vector machines were optimized and validated for the pairwise separation of the three classes. Our analysis shows classification rates of up to 98.3% between HC and AD, 85% between HC and MCI, and 93.3% for MCI and AD separation. These results outperform those reported in the literature and demonstrate the viability of the proposed morphological indices to classify different aging stages.
2009 IEEE 12th International Conference on Computer Vision Workshops, ICCV Workshops, 2009
This paper describes using 3D chain expressions for encoding unit-width curve skeletons and measu... more This paper describes using 3D chain expressions for encoding unit-width curve skeletons and measuring shape dissimilarity. By integrating a robust skeletonization technique with 3D chain coding, the proposed algorithm can compare not only the skeleton topology but also the ...
ABSTRACT A method for representing 2D (two-dimensional) tree objects is described. This represent... more ABSTRACT A method for representing 2D (two-dimensional) tree objects is described. This representation is based on a chain code, which is called the Slope Chain Code (SCC). Thus, 2D tree objects are described by means of a chain of element strings suitably combined by means of parentheses. These 2D tree objects correspond to naturally existing 2D tree structures. This tree notation preserves the shape of trees (and the shape of their branches), allows us to know their topological and geometrical properties. The proposed notation of 2D tree objects is invariant under translation, rotation and, optionally, under scaling. Also, it is possible to define a unique start vertex for each tree via the unique path in the tree. Using this notation it is possible to obtain the mirror image of any tree with ease. Furthermore, two interesting properties of trees are presented: the accumulated slope and the tortuosity. Tortuosity is a very important property of trees and has many applications in different fields. In order to prove our method for representing 2D tree objects, we obtain some tree descriptors of tree objects and compute their measures of accumulated slope and tortuosity. Finally, we present some examples of 2D trees from the real world about echinoderm species identification.
ABSTRACT A measure of tortuosity for 2D curves is presented. Tortuosity is a very important prope... more ABSTRACT A measure of tortuosity for 2D curves is presented. Tortuosity is a very important property of curves and has many applications, such as: how to measure the tortuosity of retinal blood vessels, intracerebral vasculature, aluminum foams, etc. The measure of tortuosity proposed here is based on a chain code called Slope Chain Code (SCC). The SCC uses some ideas which were described in [A geometric structure for 2D shapes and 3D surfaces, Pattern Recognition 25 (1992) 483-496]. The SCC of a curve is obtained by placing straight-line segments of constant length around the curve (the endpoints of the straight-line segments always touching the curve), and calculating the slope changes between contiguous straight-line segments scaled to a continuous range from −1 to 1. The SCC of a curve is independent of translation, rotation, and optionally, of scaling, which is an important advantage for computing tortuosity. Also, the minimum and maximum values of tortuosity for curves and a measure of normalized tortuosity are described. Finally, an application of the proposed measure of tortuosity is presented which corresponds to the computation of retinal blood vessel tortuosity.
Abstract A method is described for representing voxel-based objects by means of enclosing trees. ... more Abstract A method is described for representing voxel-based objects by means of enclosing trees. An enclosing tree is a tree which totally covers a voxel-based object, the vertices of the enclosing tree correspond to the vertices of the enclosing surface of the analyzed voxel-...
Journal of Knot Theory and Its Ramifications, 2002
A method for representing knots by means of a chain code is presented. Knots which are digitalize... more A method for representing knots by means of a chain code is presented. Knots which are digitalized and represented by the orthogonal direction change chain code are called discrete knots. Discrete knots are composed of constant straight-line segments using only orthogonal directions. ...
Journal of Knot Theory and Its Ramifications, 2006
An easy and fast algorithm for obtaining minimal discrete knots is presented. A minimal discrete ... more An easy and fast algorithm for obtaining minimal discrete knots is presented. A minimal discrete knot is the digitalized representation of a knot, which is composed of the minimum number of constant orthogonal straight-line segments and is represented by the knot-number ...
Journal of Knot Theory and Its Ramifications, 2005
A method for generating families of particular 3D curves which differs from random self-avoiding ... more A method for generating families of particular 3D curves which differs from random self-avoiding walks is presented. This method is based on a chain code called knot numbers. The knot-number notation describes discrete knots. A discrete knot is the digitalized representation of a ...
Resumen SE PRESENTA UN MÉTODO RÁPIDO Y EFICIENTE PARA GRAFICAR OBJETOS SÓLIDOS Y RÍGIDOS COMPUEST... more Resumen SE PRESENTA UN MÉTODO RÁPIDO Y EFICIENTE PARA GRAFICAR OBJETOS SÓLIDOS Y RÍGIDOS COMPUESTOS POR UN GRAN NUMERO DE VAXELES. ESTE MÉTODO SE BASA EN EL CONCEPTO DEL ÁREA DE SUPERFICIE DE CONTACTO ...
Compactness is an intrinsic property of objects. This feature is often associated with the old an... more Compactness is an intrinsic property of objects. This feature is often associated with the old and classical ratio perimeter2/area. This ra-tio is used in many scientific fields as shape descriptor in shape analysis tasks. However, when we use this ratio for measuring shape ...
Uploads
underlying VBO, and both trees are one dimensional alternative descriptors to skeletons for VBOs representation. Vertices in the trees correspond to the vertices of the VBO enclosing surface where some surface vertices have been conveniently suppressed. These descriptors are computed using a base-five digit chain code (combined with
parentheses) and has been used to illustrate three dimensional
curves and enclosing trees. The descriptors are invariant under rotation and translation, and preserve the VBO shape. Using either descriptor, the description of the mirror image of a VBO is easily obtained. The proposed descriptor notation is a good tool for storing VBOs, and
intersecting trees providing further storage savings.
Key words: curve description, chain encoding, shape code, silhouettes, shape numbers, form similarity, shape comparison, measure of shape difference, binary picture, image processing"
underlying VBO, and both trees are one dimensional alternative descriptors to skeletons for VBOs representation. Vertices in the trees correspond to the vertices of the VBO enclosing surface where some surface vertices have been conveniently suppressed. These descriptors are computed using a base-five digit chain code (combined with
parentheses) and has been used to illustrate three dimensional
curves and enclosing trees. The descriptors are invariant under rotation and translation, and preserve the VBO shape. Using either descriptor, the description of the mirror image of a VBO is easily obtained. The proposed descriptor notation is a good tool for storing VBOs, and
intersecting trees providing further storage savings.
Key words: curve description, chain encoding, shape code, silhouettes, shape numbers, form similarity, shape comparison, measure of shape difference, binary picture, image processing"