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16
Range Estimation by Optical Differentiation
 JOURNAL OF THE OPTICAL SOCIETY OF AMERICA, 15(7):17771786
, 1988
"... We describe a novel formulation of the range recovery problem based on computation of the differential variation in image intensities with respect to changes in camera position. This method uses a single stationary camera and a pair of calibrated optical masks to directly measure this differential q ..."
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Cited by 37 (0 self)
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We describe a novel formulation of the range recovery problem based on computation of the differential variation in image intensities with respect to changes in camera position. This method uses a single stationary camera and a pair of calibrated optical masks to directly measure this differential quantity. We also describe a variant based on changes in aperture size. The subsequent computation of the range image involves simple arithmetic operations, and is suitable for realtime implementation. We present the theory of this technique and show results from a prototype camera which we have constructed.
The Graycode filter kernels
 IEEE Trans. Pattern Anal. Mach. Intell
, 2007
"... Abstract In this paper we introduce a family of filter kernels the GrayCode Kernels (GCK) and demonstrate their use in image analysis. Filtering an image with a sequence of GrayCode Kernels is highly efficient and requires only 2 operations per pixel for each filter kernel, independent of the si ..."
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Cited by 18 (1 self)
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Abstract In this paper we introduce a family of filter kernels the GrayCode Kernels (GCK) and demonstrate their use in image analysis. Filtering an image with a sequence of GrayCode Kernels is highly efficient and requires only 2 operations per pixel for each filter kernel, independent of the size or dimension of the kernel. We show that the family of kernels is large and includes the WalshHadamard kernels amongst others. The GCK can be used to approximate any desired kernel and as such forms a complete representation. The efficiency of computation using a sequence of GCK filters can be exploited for various realtime applications, such as, pattern detection, feature extraction, texture analysis, texture synthesis, and more.
Design of multiparameter steerable functions using cascade basis reduction
, 1996
"... A new cascade basis reduction method of computing the optimal leastsquares set of basis functions to steer a given function locally is presented. The method combines the Lie grouptheoretic and the singular value decomposition approaches such that their respective strengths complement each other. S ..."
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Cited by 11 (2 self)
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A new cascade basis reduction method of computing the optimal leastsquares set of basis functions to steer a given function locally is presented. The method combines the Lie grouptheoretic and the singular value decomposition approaches such that their respective strengths complement each other. Since the Lie grouptheoretic approach isused, the sets of basis and steering functions computed can be expressed in analytic form. Because the singular value decomposition method is used, these sets of basis and steering functions are optimal in the leastsquares sense. Most importantly, the computational complexity in designing the basis functions for transformation groups with large numbers of parameters is significantly reduced. The efficiency of the cascade basis reduction method is demonstrated by designing a set of basis functions to steer a Gabor function under the fourparameter linear transformation group.
Invariant modules and the reduction of nonlinear partial differential equations to dynamical systems
 Adv. Math
, 2000
"... Abstract. We completely characterize all nonlinear partial differential equations leaving a given finitedimensional vector space of analytic functions invariant. Existence of an invariant subspace leads to a reduction of the associated dynamical partial differential equations to a system of ordinar ..."
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Cited by 11 (3 self)
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Abstract. We completely characterize all nonlinear partial differential equations leaving a given finitedimensional vector space of analytic functions invariant. Existence of an invariant subspace leads to a reduction of the associated dynamical partial differential equations to a system of ordinary differential equations, and provide a nonlinear counterpart to quasiexactly solvable quantum Hamiltonians. These results rely on a useful extension of the classical Wronskian determinant condition for linear independence of functions. In addition, new approaches to the characterization of the annihilating differential operators for spaces of analytic functions are presented.
Representing edge models via local principal component analysis
 7th European Conference on Computer Vision
, 2002
"... Abstract. Edge detection depends not only upon the assumed model of what an edge is, but also on how this model is represented. The problem of how to represent the edge model is typically neglected, despite the fact that the representation is a bottleneck for both computational cost and accuracy. We ..."
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Cited by 6 (1 self)
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Abstract. Edge detection depends not only upon the assumed model of what an edge is, but also on how this model is represented. The problem of how to represent the edge model is typically neglected, despite the fact that the representation is a bottleneck for both computational cost and accuracy. We propose to represent edge models by a partition of the edge manifold corresponding to the edge model, where each local element of the partition is described by its principal components. We describe the construction of this representation and demonstrate its benefits for various edge models. 1
Wavelet steerability and the higherorder Riesz transform
 ALDROUBI, “A REVIEW OF WAVELETS IN BIOMEDICAL APPLICATIONS,” PROC. OF THE IEEE
, 2010
"... Our main goal in this paper is to set the foundations of a general continuousdomain framework for designing steerable, reversible signal transformations (a.k.a. frames) in multiple dimensions (). To that end, we introduce a selfreversible, thorder extension of the Riesz transform. We prove that ..."
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Cited by 6 (3 self)
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Our main goal in this paper is to set the foundations of a general continuousdomain framework for designing steerable, reversible signal transformations (a.k.a. frames) in multiple dimensions (). To that end, we introduce a selfreversible, thorder extension of the Riesz transform. We prove that this generalized transform has the following remarkable properties: shiftinvariance, scaleinvariance, innerproduct preservation, and steerability. The pleasing consequence is that the transform maps any primary wavelet frame (or basis) of into another “steerable” wavelet frame, while preserving the frame bounds. The concept provides a functional counterpart to Simoncelli’s steerable pyramid whose construction was primarily based on filterbank design. The proposed mechanism allows for the specification of wavelets with any order of steerability in any number of dimensions; it also yields a perfect reconstruction filterbank algorithm. We illustrate the method with the design of a novel family of multidimensional RieszLaplace wavelets that essentially behave like the thorder partial derivatives of an isotropic Gaussian kernel.
Rotational invariant operators based on steerable filter banks
"... We introduce a technique for designing rotation invariant operators based on steerable filter banks. Steerable filters are widely used in Computer Vision as local descriptors for texture analysis. Rotation invariance has been shown to improve texturebased classification in certain contexts. Our app ..."
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Cited by 5 (0 self)
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We introduce a technique for designing rotation invariant operators based on steerable filter banks. Steerable filters are widely used in Computer Vision as local descriptors for texture analysis. Rotation invariance has been shown to improve texturebased classification in certain contexts. Our approach to invariance is based on solving the PDE associated with the formulation of invariance in a Lie group framework.
The SVD Approach for Steerable Filter Design
 In IEEE International Symposium on Circuits and Systems, volume V
, 1998
"... The first processing step in computational early vision usually consists of convolutions with a number of kernels. These kernels often are derived from a mother kernel that is rotated, scaled, or deformed with respect to other degrees of freedom. This paper presents an efficient computational approa ..."
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Cited by 5 (2 self)
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The first processing step in computational early vision usually consists of convolutions with a number of kernels. These kernels often are derived from a mother kernel that is rotated, scaled, or deformed with respect to other degrees of freedom. This paper presents an efficient computational approach to calculate the responses of arbitrary mother kernels with arbitrary deformations. Analytical solutions to this problem in most cases are difficult or not possible. Therefore, we present in this paper a numerical approach that emphasizes an algebraical point of view. 1.
Lie generators for computing steerable functions
 Pattern Recognition Letters
, 1998
"... We present a computational, grouptheoretic approach to steerable functions. The approach is grouptheoretic in that the treatment involves continuous transformation groups for which elementary Lie group theory may be applied. The approach is computational in that the theory is constructive and lead ..."
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Cited by 4 (0 self)
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We present a computational, grouptheoretic approach to steerable functions. The approach is grouptheoretic in that the treatment involves continuous transformation groups for which elementary Lie group theory may be applied. The approach is computational in that the theory is constructive and leads directly to a procedural implementation. For functions that are steerable with n basis functions under a kparameter group, the procedure is e cient in that at most nk +1 iterations of the procedure areneeded tocompute all the basis functions. Furthermore, the procedure is guaranteed to return the minimum number of basis functions. If the function is not steerable, a numerical implementation of the procedure could be used to compute basis functions that approximately steer the function over a range of transformation parameters. Examples of both applications are described.
On Rotational Invariance for Texture Recognition
, 2005
"... In this paper we analyze the effect of rotational invariant operators for texture recognition using crossvalidation experiments with different sample sizes. This work presents three main contributions. First, invariant operators for steerable filter banks are derived analytically using the Lie grou ..."
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Cited by 1 (1 self)
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In this paper we analyze the effect of rotational invariant operators for texture recognition using crossvalidation experiments with different sample sizes. This work presents three main contributions. First, invariant operators for steerable filter banks are derived analytically using the Lie group approach. Second, the use of “randomized invariants” for steerable texture analysis is introduced. Randomized invariants produce classification rates that are intermediate between those of noninvariant and of invariant features. Third, a thorough quantitative analysis is presented, highlighting the relationship between classification performances and training sample size, textural characteristics of the data set, and classification algorithm.