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N.: A LogEuclidean framework for statistics on diffeomorphisms
 In: Proc. MICCAI’06. (2006) 924–931
"... Abstract. In this article, we focus on the computation of statistics of invertible geometrical deformations (i.e., diffeomorphisms), based on the generalization to this type of data of the notion of principal logarithm. Remarkably, this logarithm is a simple 3D vector field, and is welldefined for ..."
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Cited by 101 (44 self)
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Abstract. In this article, we focus on the computation of statistics of invertible geometrical deformations (i.e., diffeomorphisms), based on the generalization to this type of data of the notion of principal logarithm. Remarkably, this logarithm is a simple 3D vector field, and is welldefined for diffeomorphisms close enough to the identity. This allows to perform vectorial statistics on diffeomorphisms, while preserving the invertibility constraint, contrary to Euclidean statistics on displacement fields. We also present here two efficient algorithms to compute logarithms of diffeomorphisms and exponentials of vector fields, whose accuracy is studied on synthetic data. Finally, we apply these tools to compute the mean of a set of diffeomorphisms, in the context of a registration experiment between an atlas an a database of 9 T1 MR images of the human brain. 1
Supplement for realtime soft shadows in dynamic scenes using spherical harmonic exponentiation
 Microsoft Corporation. available on the SIGGRAPH 2006 Conference DVD
, 2006
"... Previous methods for soft shadows numerically integrate over many light directions at each receiver point, testing blocker visibility in each direction. We introduce a method for realtime soft shadows in dynamic scenes illuminated by large, lowfrequency light sources where such integration is impr ..."
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Cited by 44 (9 self)
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Previous methods for soft shadows numerically integrate over many light directions at each receiver point, testing blocker visibility in each direction. We introduce a method for realtime soft shadows in dynamic scenes illuminated by large, lowfrequency light sources where such integration is impractical. Our method operates on vectors representing lowfrequency visibility of blockers in the spherical harmonic basis. Blocking geometry is modeled as a set of spheres; relatively few spheres capture the lowfrequency blocking effect of complicated geometry. At each receiver point, we compute the product of visibility vectors for these blocker spheres as seen from the point. Instead of computing an expensive SH product per blocker as in previous work, we perform inexpensive vector sums to accumulate the log of blocker visibility. SH exponentiation then yields the product visibility vector over all blockers. We show how the SH exponentiation required can be approximated accurately and efficiently for loworder SH, accelerating previous CPUbased methods by a factor of 10 or more, depending on blocker complexity, and allowing realtime GPU implementation.
Error estimation and evaluation of matrix functions via the Faber transform
 SIAM J. Numer. Anal
"... Abstract. The need to evaluate expressions of the form f(A) orf(A)b, wheref is a nonlinear function, A is a large sparse n × n matrix, and b is an nvector, arises in many applications. This paper describes how the Faber transform applied to the field of values of A can be used to determine improved ..."
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Cited by 39 (12 self)
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Abstract. The need to evaluate expressions of the form f(A) orf(A)b, wheref is a nonlinear function, A is a large sparse n × n matrix, and b is an nvector, arises in many applications. This paper describes how the Faber transform applied to the field of values of A can be used to determine improved error bounds for popular polynomial approximation methods based on the Arnoldi process. Applications of the Faber transform to rational approximation methods and, in particular, to the rational Arnoldi process also are discussed.
A NEW SCALING AND SQUARING ALGORITHM FOR THE MATRIX EXPONENTIAL
, 2009
"... The scaling and squaring method for the matrix exponential is based on the approximation eA ≈ (rm(2−sA)) 2s, where rm(x) is the [m/m] Padé approximant to ex and the integers m and s are to be chosen. Several authors have identified a weakness of existing scaling and squaring algorithms termed overs ..."
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Cited by 36 (23 self)
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The scaling and squaring method for the matrix exponential is based on the approximation eA ≈ (rm(2−sA)) 2s, where rm(x) is the [m/m] Padé approximant to ex and the integers m and s are to be chosen. Several authors have identified a weakness of existing scaling and squaring algorithms termed overscaling, in which a value of s much larger than necessary is chosen, causing a loss of accuracy in floating point arithmetic. Building on the scaling and squaring algorithm of Higham [SIAM J. Matrix Anal. Appl., 26 (2005), pp. 1179–1193], which is used by the MATLAB function expm, we derive a new algorithm that alleviates the overscaling problem. Two key ideas are employed. The first, specific to triangular matrices, is to compute the diagonal elements in the squaring phase as exponentials instead of from powers of rm. The second idea is to base the backward error analysis that underlies the algorithm on members of the sequence {‖Ak‖1/k} instead of ‖A‖, since for nonnormal matrices it is possible that ‖Ak‖1/k is much smaller than ‖A‖, andindeed this is likely when overscaling occurs in existing algorithms. The terms ‖Ak‖1/k are estimated without computing powers of A by using a matrix 1norm estimator in conjunction with a bound of the form ‖Ak‖1/k ≤ max ( ‖Ap‖1/p, ‖Aq‖1/q) that holds for certain fixed p and q less than k. The improvements to the truncation error bounds have to be balanced by the potential for a large ‖A‖
A Fast and LogEuclidean Polyaffine Framework for Locally Linear Registration
 JOURNAL OF MATHEMATICAL IMAGING AND VISION
"... In this article, we focus on the parameterization of nonrigid geometrical deformations with a small number of flexible degrees of freedom. In previous work, we proposed a general framework called polyaffine to parameterize deformations with a finite number of rigid or affine components, while guar ..."
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Cited by 36 (8 self)
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In this article, we focus on the parameterization of nonrigid geometrical deformations with a small number of flexible degrees of freedom. In previous work, we proposed a general framework called polyaffine to parameterize deformations with a finite number of rigid or affine components, while guaranteeing the invertibility of global deformations. However, this framework lacks some important properties: the inverse of a polyaffine transformation is not polyaffine in general, and the polyaffine fusion of affine components is not invariant with respect to a change of coordinate system. We present here a novel general framework, called LogEuclidean polyaffine, which overcomes these defects. We also detail a simple algorithm, the Fast Polyaffine Transform, which allows to compute very efficiently LogEuclidean polyaffine transformations and their inverses on regular grids. The results presented here on real 3D locally affine registration suggest that our novel framework provides a general and efficient way of fusing local rigid or affine deformations into a global invertible transformation without introducing artifacts, independently of the way local deformations are first estimated.
Deformable medical image registration: A survey
 IEEE TRANSACTIONS ON MEDICAL IMAGING
, 2013
"... Deformable image registration is a fundamental task in medical image processing. Among its most important applications, one may cite: i) multimodality fusion, where information acquired by different imaging devices or protocols is fused to facilitate diagnosis and treatment planning; ii) longitudin ..."
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Cited by 34 (1 self)
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Deformable image registration is a fundamental task in medical image processing. Among its most important applications, one may cite: i) multimodality fusion, where information acquired by different imaging devices or protocols is fused to facilitate diagnosis and treatment planning; ii) longitudinal studies, where temporal structural or anatomical changes are investigated; and iii) population modeling and statistical atlases used to study normal anatomical variability. In this paper, we attempt to give an overview of deformable registration methods, putting emphasis on the most recent advances in the domain. Additional emphasis has been given to techniques applied to medical images. In order to study image registration methods in depth, their main components are identified and studied independently. The most recent techniques are presented in a systematic fashion. The contribution of this paper is to provide an extensive account of registration techniques in a systematic manner.
COMPUTING THE ACTION OF THE MATRIX EXPONENTIAL, WITH AN APPLICATION TO EXPONENTIAL INTEGRATORS
, 2010
"... A new algorithm is developed for computing etAB, where A is an n × n matrix and B is n×n0 with n0 ≪ n. The algorithm works for any A, its computational cost is dominated by the formation of products of A with n × n0 matrices, and the only input parameter is a backward error tolerance. The algorithm ..."
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Cited by 28 (9 self)
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A new algorithm is developed for computing etAB, where A is an n × n matrix and B is n×n0 with n0 ≪ n. The algorithm works for any A, its computational cost is dominated by the formation of products of A with n × n0 matrices, and the only input parameter is a backward error tolerance. The algorithm can return a single matrix etAB or a sequence etkAB on an equally spaced grid of points tk. It uses the scaling part of the scaling and squaring method together with a truncated Taylor series approximation to the exponential. It determines the amount of scaling and the Taylor degree using the recent analysis of AlMohy and Higham [SIAM J. Matrix Anal. Appl. 31 (2009), pp. 970989], which provides sharp truncation error bounds expressed in terms of the quantities ‖Ak‖1/k for a few values of k, where the norms are estimated using a matrix norm estimator. Shifting and balancing are used as preprocessing steps to reduce the cost of the algorithm. Numerical experiments show that the algorithm performs in a numerically stable fashion across a wide range of problems, and analysis of rounding errors and of the conditioning of the problem provides theoretical support. Experimental comparisons with two Krylovbased MATLAB codes show the new algorithm to be sometimes much superior in terms of computational cost and accuracy. An important application of the algorithm is to exponential integrators for ordinary differential equations. It is shown that the sums of the form ∑p k=0 ϕk(A)uk that arise in exponential integrators, where the ϕk are related to the exponential function, can be expressed in terms of a single exponential of a matrix of dimension n + p built by augmenting A with additional rows and columns, and the algorithm of this paper can therefore be employed.
Computing the Fréchet derivative of the matrix exponential, with an application to condition number estimation
, 2008
"... The matrix exponential is a muchstudied matrix function having many applications. The Fréchet derivative of the matrix exponential describes the first order sensitivity of e A to perturbations in A and its norm determines a condition number for e A. Among the numerous methods for computing e A the ..."
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Cited by 22 (15 self)
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The matrix exponential is a muchstudied matrix function having many applications. The Fréchet derivative of the matrix exponential describes the first order sensitivity of e A to perturbations in A and its norm determines a condition number for e A. Among the numerous methods for computing e A the scaling and squaring method is the most widely used. We show that the implementation of the method in [N. J. Higham. The scaling and squaring method for the matrix exponential revisited. SIAM J. Matrix Anal. Appl., 26(4):1179–1193, 2005] can be extended to compute both e A and the Fréchet derivative at A in the direction E, denoted by L(A, E), at a cost about three times that for computing e A alone. The algorithm is derived from the scaling and squaring method by differentiating the Padé approximants and the squaring recurrence, reusing quantities computed during the evaluation of the Padé approximant, and intertwining the recurrences in the squaring phase. To guide the choice of algorithmic parameters an extension of the existing backward error analysis for the scaling and squaring method is developed which shows that, modulo rounding errors, the approximations obtained are e A+∆A and L(A+∆A, E + ∆E), with the same ∆A in both cases, and with computable bounds on �∆A � and �∆E�. The algorithm for L(A, E) is used to develop an algorithm that computes e A together with an estimate of its condition number.