This paper is essentially another version of Horikawa's analysis [1]. The purpose here i... more This paper is essentially another version of Horikawa's analysis [1]. The purpose here is to obtain a complete list of ("minimal") projective degen-erations analogous to the list obtained by Morrison [2] by the analytic method of Kulikov, Persson and Pinkham. (Morrison's list is reproduced ...
Computer Vision and Pattern Recognition, 1991. …, 1991
Page 1. SEGMENTATION BY NONLINEAR DIFFUSION Jayant Shah Mathematics Department, Northeastem Unive... more Page 1. SEGMENTATION BY NONLINEAR DIFFUSION Jayant Shah Mathematics Department, Northeastem University, Boston, Mass. 021 15 ABSTRACT' One of the basic problems in Computer Vision is the problem of segmenting an image into meaningful re-gions. ...
Abstract A new method for determining skeletons of 3D shapes is described. It is a combination of... more Abstract A new method for determining skeletons of 3D shapes is described. It is a combination of the approach based on the "grass-fire" technique and Zhu's approach based on first finding portions of the shape where its width is approximately constant. The method specifically does not re-quire presmoothing of the shape and is robust in the presence of noise. In
A new approach is presented for recovering shapes from noisy images. First, an edge-strength func... more A new approach is presented for recovering shapes from noisy images. First, an edge-strength function, v, at every pixel is determined by implementing a segmenta- tion functional. Then, the zero-crossings of the laplacian of the smoothed image are allowed to evolve under the influence of v. Each point on the zero-crossings moves in the direction of the normal with a
This paper applies a unified approach to variational smoothing and segmentation to brain diffusio... more This paper applies a unified approach to variational smoothing and segmentation to brain diffusion tensor image data along user-selected attributes derived from the tensor, with the aim of extracting detailed brain structure information. The application of this framework simultaneously segments and denoises to produce edges and smoothed regions within the white matter of the brain that are relatively homogeneous with respect to the diffusion tensor attributes of choice. The approach enables the visualization of a, smoothed, scale invariant representation of the tensor data field in a variety of diverse forms. In addition to known attributes such as fractional anisotropy, these representations include selected directional tensor components and, additionally associated continuous valued edge fields that may be used for further segmentation. A comparison is presented of the results of three different data model selections with respect to their ability to resolve white matter structure. The resulting images are integrated to provide better perspective of the model properties (edges, smoothed image, etc.) and their relationship to the underlying brain anatomy. The improvement in brain image quality is illustrated both qualitatively and quantitatively, and the robust performance of the algorithm in the presence of added noise is shown. Smoothing occurs without loss of edge features due to the simultaneous segmentation aspect of the variational approach, and the output enables better delineation of tensors representative of local and long range association, projection and commissural fiber systems.
Journal of Visual Communication and Image Representation, 2002
A system of coupled differential equations that learns priors for modeling “preattentive” texture... more A system of coupled differential equations that learns priors for modeling “preattentive” textures is formulated. Learning is driven by the feature residuals computed from the observed values and the values calculated by the system from a synthesized image that is ...
Journal of Visual Communication and Image Representation, 2002
During the past decade, curve evolution has been applied to shape recovery, shape analysis, image... more During the past decade, curve evolution has been applied to shape recovery, shape analysis, image smoothing, and image segmentation. Almost all of these applications are based on curve evolution which minimizes the total length of the curve. The curve moves ...
Journal of Visual Communication and Image Representation, 2000
The method of curve evolution is a popular method for recovering shape boundaries. However isotro... more The method of curve evolution is a popular method for recovering shape boundaries. However isotropic metrics have always been used to induce the flow of the curve and potential steady states tend to be difficult to determine numerically, especially in noisy or ...
This paper presents a new approach to prior shape and appearance modeling for use in curve evolut... more This paper presents a new approach to prior shape and appearance modeling for use in curve evolution-based seg-mentation. The new method is based on the unified use of feature distributions and allows the incorporation of coupled prior information about shape and appearance ...
Rendiconti Lincei - Matematica e Applicazioni, 2000
This paper studies a specific metric on plane curves that has the property of being isometric to ... more This paper studies a specific metric on plane curves that has the property of being isometric to classical manifold (sphere, complex projective, Stiefel, Grassmann) modulo change of parametrization, each of these classical manifolds being associated to specific qualifications of the space of curves (closed-open, modulo rotation etc. . . ) Using these isometries, we are able to explicitely describe the geodesics, first in the parametric case, then by modding out the paremetrization and considering horizontal vectors. We also compute the sectional curvature for these spaces, and show, in particular, that the space of closed curves modulo rotation and change of parameter has positive curvature. Experimental results that explicitly compute minimizing geodesics between two closed curves are finally provided Date: May 5, 2008. 1991 Mathematics Subject Classification. Primary 58B20, 58D15, 58E40.
Since lunar and solar parallax play a crucial role in predicting solar eclipses, the focus of thi... more Since lunar and solar parallax play a crucial role in predicting solar eclipses, the focus of this paper is on the computation of parallax. A brief history of parallax computation in India and China is traced. Predictions of solar eclipses based Nilakantha's Tantrasangraha are statistically analyzed. They turn out to be remarkably accurate, but there is a pronounced bias towards predicting false positives rather than false negatives. The false positives occur more to the south of the ecliptic at northerly terrestrial latitudes and more to the north of the ecliptic at southerly latitudes. A very similar bias is found in Chinese astronomy providing another hint at possible links between Indian and Chinese astronomy. The Chinese have traditionally used different values for the eclipse limit north and south of the ecliptic, perhaps to compensate for the southward bias.
In recent years, curve evolution has been applied to smoothing of shapes and shape analysis with ... more In recent years, curve evolution has been applied to smoothing of shapes and shape analysis with considerable success, especially in biomedical image analysis. The multiscale analysis provides information regarding parts of shapes, their axes or centers and shape skeletons. In this paper, we show that the level sets of an edge-strength function provide essentially the same shape analysis as provided by curve evolution. The new method has several advantages over the method of curve evolution. Since the governing equation is linear, the implementation is simpler and faster. The same equation applies to problems of higher dimension. An important advantage is that unlike the method of curve evolution, the new method is applicable to shapes which may have junctions such as triple points. The edge-strength may be calculated from raw images without first extracting the shape outline. Thus the method can be applied to raw images. The method provides a way to approach the segmentation problem and shape analysis within a common integrated framework.
This paper studies a specific metric on plane curves that has the property of being isometric to ... more This paper studies a specific metric on plane curves that has the property of being isometric to classical manifold (sphere, complex projective, Stiefel, Grassmann) modulo change of parametrization, each of these classical manifolds being associated to specific qualifications of the space of curves (closed-open, modulo rotation etc. . . ) Using these isometries, we are able to explicitely describe the geodesics, first in the parametric case, then by modding out the paremetrization and considering horizontal vectors. We also compute the sectional curvature for these spaces, and show, in particular, that the space of closed curves modulo rotation and change of parameter has positive curvature. Experimental results that explicitly compute minimizing geodesics between two closed curves are finally provided Date: April 24, 2012. 1991 Mathematics Subject Classification. Primary 58B20, 58D15, 58E40.
Shape skeletons have been used in Computer Vision to represent shapes and discover their salient ... more Shape skeletons have been used in Computer Vision to represent shapes and discover their salient features. Earlier attempts were based on morphological approach in which a shape is eroded successively and uniformly until it is reduced to its skeleton. The main difficulty with this approach is its sensitivity to noise and several approaches have been proposed for dealing with this problem. In this paper, we propose a new method based on diffusion to smooth out the noise and extract shape skeletons in a robust way. In the process, we also obtain segmentation of the shape into parts. The main tool for shape analysis is a function called the "edge-strength" function. Its level curves are smoothed analogs of the successive shape outlines obtained during the morphological erosion. The new method is closely related to the popular method of curve evolution, but has several advantages over it. Since the governing equation is linear, the implementation is simpler and faster. The same equation applies to problems in higher dimension. Unlike most other methods, the new method is applicable to shapes which may have junctions such as triple points. Another advantage is that the method is robust with respect to gaps in the shape outline. Since it is seldom possible to extract complete shape outlines from a noisy grayscale image, this is obviously a very important feature. The key point is that the edge-strength may be calculated from grayscale images without first extracting the shape outline. Thus the method can be directly applied to grayscale images.
This work provides a variational framework for fusing range and intensity data for recovering reg... more This work provides a variational framework for fusing range and intensity data for recovering regularized surfaces. It is shown that this framework provides natural boundary conditions for the shape-from-shading problem, results in a new shape-from-shading formulation in the absence of range data, and provides a new fusion paradigm when range data is incorporated. The approach is demonstrated on simulated range and intensity images; error analysis with respect to the ground truth surface is presented. It is shown that the formulation performs well even in very noisy images
In 718 CE Ch'üt'an Hsi-ta (Gautama Siddha − rtha), an Indian astronomer who was appointed an"astr... more In 718 CE Ch'üt'an Hsi-ta (Gautama Siddha − rtha), an Indian astronomer who was appointed an"astronomer royal" in the T'ang court compiled a compendium of omens and divinations, called K'aiyü an Chan-ching, analogous to Varahamihira's Br . hatsam . hita. The 104th volume of this work, Chiu-chih li (Nine Upholders Calendrical System) on astronomy was entirely based on the Indian astronomy of the 7th century which in turn was based on the geometric astronomy of the Greeks. A few years later, the emperor asked I-hsing (Yixing in pinyin), a buddhist monk, an astronomer and a mathematician, to overhaul the traditional Chinese astronomy. He submitted an astronomical system, called Ta-yen li (Grand Expansion Calendrical System, Dayan li in pinyin) in 727 CE Only Ta-yen li was officially adopted.
This paper is essentially another version of Horikawa's analysis [1]. The purpose here i... more This paper is essentially another version of Horikawa's analysis [1]. The purpose here is to obtain a complete list of ("minimal") projective degen-erations analogous to the list obtained by Morrison [2] by the analytic method of Kulikov, Persson and Pinkham. (Morrison's list is reproduced ...
Computer Vision and Pattern Recognition, 1991. …, 1991
Page 1. SEGMENTATION BY NONLINEAR DIFFUSION Jayant Shah Mathematics Department, Northeastem Unive... more Page 1. SEGMENTATION BY NONLINEAR DIFFUSION Jayant Shah Mathematics Department, Northeastem University, Boston, Mass. 021 15 ABSTRACT' One of the basic problems in Computer Vision is the problem of segmenting an image into meaningful re-gions. ...
Abstract A new method for determining skeletons of 3D shapes is described. It is a combination of... more Abstract A new method for determining skeletons of 3D shapes is described. It is a combination of the approach based on the "grass-fire" technique and Zhu's approach based on first finding portions of the shape where its width is approximately constant. The method specifically does not re-quire presmoothing of the shape and is robust in the presence of noise. In
A new approach is presented for recovering shapes from noisy images. First, an edge-strength func... more A new approach is presented for recovering shapes from noisy images. First, an edge-strength function, v, at every pixel is determined by implementing a segmenta- tion functional. Then, the zero-crossings of the laplacian of the smoothed image are allowed to evolve under the influence of v. Each point on the zero-crossings moves in the direction of the normal with a
This paper applies a unified approach to variational smoothing and segmentation to brain diffusio... more This paper applies a unified approach to variational smoothing and segmentation to brain diffusion tensor image data along user-selected attributes derived from the tensor, with the aim of extracting detailed brain structure information. The application of this framework simultaneously segments and denoises to produce edges and smoothed regions within the white matter of the brain that are relatively homogeneous with respect to the diffusion tensor attributes of choice. The approach enables the visualization of a, smoothed, scale invariant representation of the tensor data field in a variety of diverse forms. In addition to known attributes such as fractional anisotropy, these representations include selected directional tensor components and, additionally associated continuous valued edge fields that may be used for further segmentation. A comparison is presented of the results of three different data model selections with respect to their ability to resolve white matter structure. The resulting images are integrated to provide better perspective of the model properties (edges, smoothed image, etc.) and their relationship to the underlying brain anatomy. The improvement in brain image quality is illustrated both qualitatively and quantitatively, and the robust performance of the algorithm in the presence of added noise is shown. Smoothing occurs without loss of edge features due to the simultaneous segmentation aspect of the variational approach, and the output enables better delineation of tensors representative of local and long range association, projection and commissural fiber systems.
Journal of Visual Communication and Image Representation, 2002
A system of coupled differential equations that learns priors for modeling “preattentive” texture... more A system of coupled differential equations that learns priors for modeling “preattentive” textures is formulated. Learning is driven by the feature residuals computed from the observed values and the values calculated by the system from a synthesized image that is ...
Journal of Visual Communication and Image Representation, 2002
During the past decade, curve evolution has been applied to shape recovery, shape analysis, image... more During the past decade, curve evolution has been applied to shape recovery, shape analysis, image smoothing, and image segmentation. Almost all of these applications are based on curve evolution which minimizes the total length of the curve. The curve moves ...
Journal of Visual Communication and Image Representation, 2000
The method of curve evolution is a popular method for recovering shape boundaries. However isotro... more The method of curve evolution is a popular method for recovering shape boundaries. However isotropic metrics have always been used to induce the flow of the curve and potential steady states tend to be difficult to determine numerically, especially in noisy or ...
This paper presents a new approach to prior shape and appearance modeling for use in curve evolut... more This paper presents a new approach to prior shape and appearance modeling for use in curve evolution-based seg-mentation. The new method is based on the unified use of feature distributions and allows the incorporation of coupled prior information about shape and appearance ...
Rendiconti Lincei - Matematica e Applicazioni, 2000
This paper studies a specific metric on plane curves that has the property of being isometric to ... more This paper studies a specific metric on plane curves that has the property of being isometric to classical manifold (sphere, complex projective, Stiefel, Grassmann) modulo change of parametrization, each of these classical manifolds being associated to specific qualifications of the space of curves (closed-open, modulo rotation etc. . . ) Using these isometries, we are able to explicitely describe the geodesics, first in the parametric case, then by modding out the paremetrization and considering horizontal vectors. We also compute the sectional curvature for these spaces, and show, in particular, that the space of closed curves modulo rotation and change of parameter has positive curvature. Experimental results that explicitly compute minimizing geodesics between two closed curves are finally provided Date: May 5, 2008. 1991 Mathematics Subject Classification. Primary 58B20, 58D15, 58E40.
Since lunar and solar parallax play a crucial role in predicting solar eclipses, the focus of thi... more Since lunar and solar parallax play a crucial role in predicting solar eclipses, the focus of this paper is on the computation of parallax. A brief history of parallax computation in India and China is traced. Predictions of solar eclipses based Nilakantha's Tantrasangraha are statistically analyzed. They turn out to be remarkably accurate, but there is a pronounced bias towards predicting false positives rather than false negatives. The false positives occur more to the south of the ecliptic at northerly terrestrial latitudes and more to the north of the ecliptic at southerly latitudes. A very similar bias is found in Chinese astronomy providing another hint at possible links between Indian and Chinese astronomy. The Chinese have traditionally used different values for the eclipse limit north and south of the ecliptic, perhaps to compensate for the southward bias.
In recent years, curve evolution has been applied to smoothing of shapes and shape analysis with ... more In recent years, curve evolution has been applied to smoothing of shapes and shape analysis with considerable success, especially in biomedical image analysis. The multiscale analysis provides information regarding parts of shapes, their axes or centers and shape skeletons. In this paper, we show that the level sets of an edge-strength function provide essentially the same shape analysis as provided by curve evolution. The new method has several advantages over the method of curve evolution. Since the governing equation is linear, the implementation is simpler and faster. The same equation applies to problems of higher dimension. An important advantage is that unlike the method of curve evolution, the new method is applicable to shapes which may have junctions such as triple points. The edge-strength may be calculated from raw images without first extracting the shape outline. Thus the method can be applied to raw images. The method provides a way to approach the segmentation problem and shape analysis within a common integrated framework.
This paper studies a specific metric on plane curves that has the property of being isometric to ... more This paper studies a specific metric on plane curves that has the property of being isometric to classical manifold (sphere, complex projective, Stiefel, Grassmann) modulo change of parametrization, each of these classical manifolds being associated to specific qualifications of the space of curves (closed-open, modulo rotation etc. . . ) Using these isometries, we are able to explicitely describe the geodesics, first in the parametric case, then by modding out the paremetrization and considering horizontal vectors. We also compute the sectional curvature for these spaces, and show, in particular, that the space of closed curves modulo rotation and change of parameter has positive curvature. Experimental results that explicitly compute minimizing geodesics between two closed curves are finally provided Date: April 24, 2012. 1991 Mathematics Subject Classification. Primary 58B20, 58D15, 58E40.
Shape skeletons have been used in Computer Vision to represent shapes and discover their salient ... more Shape skeletons have been used in Computer Vision to represent shapes and discover their salient features. Earlier attempts were based on morphological approach in which a shape is eroded successively and uniformly until it is reduced to its skeleton. The main difficulty with this approach is its sensitivity to noise and several approaches have been proposed for dealing with this problem. In this paper, we propose a new method based on diffusion to smooth out the noise and extract shape skeletons in a robust way. In the process, we also obtain segmentation of the shape into parts. The main tool for shape analysis is a function called the "edge-strength" function. Its level curves are smoothed analogs of the successive shape outlines obtained during the morphological erosion. The new method is closely related to the popular method of curve evolution, but has several advantages over it. Since the governing equation is linear, the implementation is simpler and faster. The same equation applies to problems in higher dimension. Unlike most other methods, the new method is applicable to shapes which may have junctions such as triple points. Another advantage is that the method is robust with respect to gaps in the shape outline. Since it is seldom possible to extract complete shape outlines from a noisy grayscale image, this is obviously a very important feature. The key point is that the edge-strength may be calculated from grayscale images without first extracting the shape outline. Thus the method can be directly applied to grayscale images.
This work provides a variational framework for fusing range and intensity data for recovering reg... more This work provides a variational framework for fusing range and intensity data for recovering regularized surfaces. It is shown that this framework provides natural boundary conditions for the shape-from-shading problem, results in a new shape-from-shading formulation in the absence of range data, and provides a new fusion paradigm when range data is incorporated. The approach is demonstrated on simulated range and intensity images; error analysis with respect to the ground truth surface is presented. It is shown that the formulation performs well even in very noisy images
In 718 CE Ch'üt'an Hsi-ta (Gautama Siddha − rtha), an Indian astronomer who was appointed an"astr... more In 718 CE Ch'üt'an Hsi-ta (Gautama Siddha − rtha), an Indian astronomer who was appointed an"astronomer royal" in the T'ang court compiled a compendium of omens and divinations, called K'aiyü an Chan-ching, analogous to Varahamihira's Br . hatsam . hita. The 104th volume of this work, Chiu-chih li (Nine Upholders Calendrical System) on astronomy was entirely based on the Indian astronomy of the 7th century which in turn was based on the geometric astronomy of the Greeks. A few years later, the emperor asked I-hsing (Yixing in pinyin), a buddhist monk, an astronomer and a mathematician, to overhaul the traditional Chinese astronomy. He submitted an astronomical system, called Ta-yen li (Grand Expansion Calendrical System, Dayan li in pinyin) in 727 CE Only Ta-yen li was officially adopted.
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Papers by Jayant Shah
out to be remarkably accurate, but there is a pronounced bias towards predicting false positives rather than false negatives. The false positives occur more to the south of the ecliptic at northerly terrestrial latitudes and more to the north of the ecliptic at southerly latitudes. A very similar bias is found in Chinese astronomy providing another hint at possible links between Indian and Chinese astronomy. The Chinese have traditionally used different values for the eclipse limit north and south of the ecliptic, perhaps to compensate for the southward bias.
out to be remarkably accurate, but there is a pronounced bias towards predicting false positives rather than false negatives. The false positives occur more to the south of the ecliptic at northerly terrestrial latitudes and more to the north of the ecliptic at southerly latitudes. A very similar bias is found in Chinese astronomy providing another hint at possible links between Indian and Chinese astronomy. The Chinese have traditionally used different values for the eclipse limit north and south of the ecliptic, perhaps to compensate for the southward bias.