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2009

Volume 2, Issue 2, pp. 285-776


Brain Connectivity Mapping Using Riemannian Geometry, Control Theory, and PDEs

Christophe Lenglet, Emmanuel Prados, Jean-Philippe Pons, Rachid Deriche, and Olivier Faugeras

SIAM J. Imaging Sci. 2, pp. 285-322 (38 pages)

Online Publication Date: April 01, 2009

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We introduce an original approach for the cerebral white matter connectivity mapping from diffusion tensor imaging (DTI). Our method relies on a global modeling of the acquired magnetic resonance imaging volume as a Riemannian manifold whose metric directly derives from the diffusion tensor. These tensors will be used to measure physical three-dimensional distances between different locations of a brain diffusion tensor image. The key concept is the notion of geodesic distance that will allow us to find optimal paths in the white matter. We claim that such optimal paths are reasonable approximations of neural fiber bundles. The geodesic distance function can be seen as the solution of two theoretically equivalent but, in practice, significantly different problems in the partial differential equation framework: an initial value problem which is intrinsically dynamic, and a boundary value problem which is, on the contrary, intrinsically stationary. The two approaches have very different properties which make them more or less adequate for our problem and more or less computationally efficient. The dynamic formulation is quite easy to implement but has several practical drawbacks. On the contrary, the stationary formulation is much more tedious to implement; we will show, however, that it has many virtues which make it more suitable for our connectivity mapping problem. Finally, we will present different possible measures of connectivity, reflecting the degree of connectivity between different regions of the brain. We will illustrate these notions on synthetic and real DTI datasets.

The Split Bregman Method for L1-Regularized Problems

Tom Goldstein and Stanley Osher

SIAM J. Imaging Sci. 2, pp. 323-343 (21 pages) | Cited 4 times

Online Publication Date: April 01, 2009

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The class of L1-regularized optimization problems has received much attention recently because of the introduction of “compressed sensing,” which allows images and signals to be reconstructed from small amounts of data. Despite this recent attention, many L1-regularized problems still remain difficult to solve, or require techniques that are very problem-specific. In this paper, we show that Bregman iteration can be used to solve a wide variety of constrained optimization problems. Using this technique, we propose a “split Bregman” method, which can solve a very broad class of L1-regularized problems. We apply this technique to the Rudin–Osher–Fatemi functional for image denoising and to a compressed sensing problem that arises in magnetic resonance imaging.

Synthetic Aperture Imaging of Multiple Point Targets in Rician Fading Media

Albert C. Fannjiang, Knut Solna, and Pengchong Yan

SIAM J. Imaging Sci. 2, pp. 344-366 (23 pages)

Online Publication Date: April 01, 2009

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This paper presents a study of stability, resolution, and detection for broad-band synthetic aperture (SA) imaging in Rician fading media. The stability condition $BNK^2/(K+1)\gg M$ is derived, where $K$ is the Rician factor, $B$ is the effective number of coherence bands, $N$ is the effective number of array elements, and $M$ is the number of (widely separated) targets. The imaging method is tested numerically with randomly distributed discrete scatterers, and comparisons with the imaging with the full response matrix (RM) are made. The resolution study reveals several interesting effects: First, given the same measurement resources, SA imaging has better resolution performance, although less stable, than RM imaging; second, for both imaging methods, the cross-range resolution measure (i.e., “full width at half maximum”) decreases with the aperture ($N$ fixed) and the probe spacing (the total aperture fixed) while the range resolution increases with both parameters. A statistical scheme is introduced to reduce the uncertainty when the stability regime is not realized.

Classification via Minimum Incremental Coding Length

John Wright, Yi Ma, Yangyu Tao, Zhouchen Lin, and Heung-Yeung Shum

SIAM J. Imaging Sci. 2, pp. 367-395 (29 pages)

Online Publication Date: April 09, 2009

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We present a simple new criterion for classification, based on principles from lossy data compression. The criterion assigns a test sample to the class that uses the minimum number of additional bits to code the test sample, subject to an allowable distortion. We demonstrate the asymptotic optimality of this criterion for Gaussian distributions and analyze its relationships to classical classifiers. The theoretical results clarify the connections between our approach and popular classifiers such as maximum a posteriori (MAP), regularized discriminant analysis (RDA), $k$-nearest neighbor ($k$-NN), and support vector machine (SVM), as well as unsupervised methods based on lossy coding. Our formulation induces several good effects on the resulting classifier. First, minimizing the lossy coding length induces a regularization effect which stabilizes the (implicit) density estimate in a small sample setting. Second, compression provides a uniform means of handling classes of varying dimension. The new criterion and its kernel and local versions perform competitively on synthetic examples, as well as on real imagery data such as handwritten digits and face images. On these problems, the performance of our simple classifier approaches the best reported results, without using domain-specific information. All MATLAB code and classification results are publicly available for peer evaluation at http://perception.csl.uiuc.edu/coding/home.htm.

Passive Sensor Imaging Using Cross Correlations of Noisy Signals in a Scattering Medium

Josselin Garnier and George Papanicolaou

SIAM J. Imaging Sci. 2, pp. 396-437 (42 pages) | Cited 4 times

Online Publication Date: April 09, 2009

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It is well known that the travel time or even the full Green's function between two passive sensors can be estimated from the cross correlation of recorded signal amplitudes generated by ambient noise sources. It is also known that the direction of the energy flux from the noise sources affects the estimation of the travel time. Using the stationary phase method, we show here that the travel time can be effectively estimated when the ray joining the two sensors continues into the noise source region. We extend this analysis to passive sensor imaging of reflectors with different ambient noise source configurations by suitably migrating the cross correlations. If in addition there is multiple scattering in the medium, then reflectors can be imaged with passive sensor networks or arrays by migrating suitable fourth-order cross correlations. Fourth-order cross correlations can also be used with auxiliary passive sensors in order to enhance travel time estimation in a scattering medium.

ASIFT: A New Framework for Fully Affine Invariant Image Comparison

Jean-Michel Morel and Guoshen Yu

SIAM J. Imaging Sci. 2, pp. 438-469 (32 pages)

Online Publication Date: April 22, 2009

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If a physical object has a smooth or piecewise smooth boundary, its images obtained by cameras in varying positions undergo smooth apparent deformations. These deformations are locally well approximated by affine transforms of the image plane. In consequence the solid object recognition problem has often been led back to the computation of affine invariant image local features. Such invariant features could be obtained by normalization methods, but no fully affine normalization method exists for the time being. Even scale invariance is dealt with rigorously only by the scale-invariant feature transform (SIFT) method. By simulating zooms out and normalizing translation and rotation, SIFT is invariant to four out of the six parameters of an affine transform. The method proposed in this paper, affine-SIFT (ASIFT), simulates all image views obtainable by varying the two camera axis orientation parameters, namely, the latitude and the longitude angles, left over by the SIFT method. Then it covers the other four parameters by using the SIFT method itself. The resulting method will be mathematically proved to be fully affine invariant. Against any prognosis, simulating all views depending on the two camera orientation parameters is feasible with no dramatic computational load. A two-resolution scheme further reduces the ASIFT complexity to about twice that of SIFT. A new notion, the transition tilt, measuring the amount of distortion from one view to another, is introduced. While an absolute tilt from a frontal to a slanted view exceeding 6 is rare, much higher transition tilts are common when two slanted views of an object are compared (see Figure hightransitiontiltsillustration). The attainable transition tilt is measured for each affine image comparison method. The new method permits one to reliably identify features that have undergone transition tilts of large magnitude, up to 36 and higher. This fact is substantiated by many experiments which show that ASIFT significantly outperforms the state-of-the-art methods SIFT, maximally stable extremal region (MSER), Harris-affine, and Hessian-affine.

Measure-Valued Images, Associated Fractal Transforms, and the Affine Self-Similarity of Images

D. La Torre, E. R. Vrscay, M. Ebrahimi, and M. F. Barnsley

SIAM J. Imaging Sci. 2, pp. 470-507 (38 pages)

Online Publication Date: April 22, 2009

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We construct a complete metric space $(Y,d_Y)$ of measure-valued images, $\mu : X \to {\cal M}(\mathbb{R}_g)$, where $X$ is the base or pixel space and ${\cal M}(\mathbb{R}_g)$ is the set of probability measures supported on the greyscale range $\mathbb{R}_g$. Such a formalism is well suited to nonlocal (NL) image processing, i.e., the manipulation of the value of an image function $u(x)$ based upon values $u(y_k)$ elsewhere in the image. We then show how the space $(Y,d_Y)$ can be employed with a general model of affine self-similarity of images that includes both same-scale as well as cross-scale similarity. We focus on two particular applications: NL-means denoising (same-scale) and multiparent block fractal image coding (cross-scale). In order to accommodate the latter, a method of fractal transforms is formulated over the metric space $(Y,d_Y)$. Under suitable conditions, a transform $M : Y \to Y$ is contractive, implying the existence of a unique fixed point measure-valued function $\bar \mu = M \bar \mu$. We also show that the pointwise moments of this measure satisfy a set of recursion relations that are generalizations of those satisfied by moments of invariant measures of iterated function systems with probabilities.

Generalized Newton-Type Methods for Energy Formulations in Image Processing

Leah Bar and Guillermo Sapiro

SIAM J. Imaging Sci. 2, pp. 508-531 (24 pages)

Online Publication Date: April 22, 2009

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Many problems in image processing are addressed via the minimization of a cost functional. The most prominently used optimization technique is gradient-descent, often used due to its simplicity and applicability where other techniques, e.g., those coming from discrete optimization, cannot be applied. Yet, gradient-descent suffers from slow convergence, and often to just local minima which highly depend on the initialization and the condition number of the functional Hessian. Newton-type methods, on the other hand, are known to have a faster, quadratic convergence. In its classical form, the Newton method relies on the $L^2$-type norm to define the descent direction. In this paper, we generalize and reformulate this very important optimization method by introducing Newton-type methods based on more general norms. Such norms are introduced both in the descent computation (Newton step) and in the corresponding stabilizing trust-region. This generalization opens up new possibilities in the extraction of the Newton step, including benefits such as mathematical stability and the incorporation of smoothness constraints. We first present the derivation of the modified Newton step in the calculus of variation framework needed for image processing. Then, we demonstrate the method with two common objective functionals: variational image deblurring and geometric active contours for image segmentation. We show that in addition to the fast convergence, norms adapted to the problem at hand yield different and superior results.

Total Variation Regularization for Image Denoising, III. Examples.

William K. Allard

SIAM J. Imaging Sci. 2, pp. 532-568 (37 pages)

Online Publication Date: May 06, 2009

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Let $\mathcal{F}(\mathbb{R}^{2})$ be the family of bounded nonnegative Lebesgue measurable functions on $\mathbb{R}^{2}$. Suppose $s\in\mathcal{F}(\mathbb{R}^{2})$ and $\gamma:\mathbb{R}\rightarrow[0,\infty)$ is zero at zero, positive away from zero, and convex. For $f\in\mathcal{F}(\Omega)$ let $F(f)=\int_\Omega\gamma(f(x)-s(x))\,d\mathcal{L}^{2}x$; here $\mathcal{L}^{2}$ is Lebesgue measure on $\mathbb{R}^2$. In the denoising literature $F$ would be called a fidelity in that it measures how much $f$ differs from $s$ which could be a noisy grayscale image. Suppose $0<\epsilon<\infty$ and let ${\bf m}^{loc}_{\epsilon}(F)$ be the set of those $f\in\mathcal{F}(\mathbb{R}^{2})$ such that ${\bf TV}(f)<\infty$ and $\epsilon{\bf TV}(f)+F(f)\leq\epsilon{\bf TV}(g)+F(g)$ for $g\in{\bf k}(f)$; here ${\bf TV}(f)$ is the total variation of $f$ and ${\bf k}(f)$ is the set of $g\in\mathcal{F}(\mathbb{R}^{2})$ such that $g=f$ off some compact subset of $\mathbb{R}^{2}$. A member of ${\bf m}^{loc}_{\epsilon}(F)$ is called a total variation regularization of $s$ (with smoothing parameter $\epsilon$). Rudin, Osher, and Fatemi in [Phys. D, 60 (1992), pp. 259–268] and Chan and Esedoḡlu in [SIAM J. Appl. Math., 65 (2005), pp. 1817–1837] have studied total variation regularizations of $s$ where $\gamma(y)=y^2/2$ and $\gamma(y)=y$, $y\in\mathbb{R}$, respectively. It turns out that the character of a member of ${\bf m}^{loc}_{\epsilon}(F)$ changes quite a bit as $\gamma$ changes. In [SIAM J. Imaging Sci., 1 (2008), pp. 400–417] the family ${\bf m}^{loc}_{\epsilon}(F)$ was described in complete detail when $s$ is the indicator function of a compact convex subset of $\mathbb{R}^{2}$. Our main purpose in this paper is to describe, in complete detail, ${\bf m}^{loc}_{\epsilon}(F)$ when $s$ is the indicator function of either $S=([0,1]\times[0,-1])\cup([-1,0]\times[0,1])$ or $S=\{x\in\mathbb{R}^{2}:|x-{\bf c}_+|\leq 1\}\cup\{x\in\mathbb{R}^{2}:|x-{\bf c}_-|\leq 1\}$, where, for some $l\in[1,\infty)$, ${\bf c}_\pm=(\pm l,0)$. We believe these examples reveal a great deal about the nature of total variation regularizations. For example, it is said that total variation denoising preserves edges; while this is certainly true in many cases and in comparison with other denoising methods, the examples given in sections 2squares and 2circles provide evidence to the contrary. In addition, one can test computational schemes for total variation regularization against these examples. We will also establish what we believe to be a number of interesting properties of ${\bf m}^{loc}_{\epsilon}(F)$.
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A Fast Algorithm for Edge-Preserving Variational Multichannel Image Restoration

Junfeng Yang, Wotao Yin, Yin Zhang, and Yilun Wang

SIAM J. Imaging Sci. 2, pp. 569-592 (24 pages) | Cited 3 times

Online Publication Date: May 06, 2009

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Variational models with $\ell_1$-norm based regularization, in particular total variation (TV) and its variants, have long been known to offer superior image restoration quality, but processing speed remained a bottleneck, preventing their widespread use in the practice of color image processing. In this paper, by extending the grayscale image deblurring algorithm proposed in [Y. Wang, J. Yang, W. Yin, and Y. Zhang, SIAM J. Imaging Sci., 1 (2008), pp. 248–272], we construct a simple and efficient algorithm for multichannel image deblurring and denoising, applicable to both within-channel and cross-channel blurs in the presence of additive Gaussian noise. The algorithm restores an image by minimizing an energy function consisting of an $\ell_2$-norm fidelity term and a regularization term that can be either TV, weighted TV, or regularization functions based on higher-order derivatives. Specifically, we use a multichannel extension of the classic TV regularizer (MTV) and derive our algorithm from an extended half-quadratic transform of Geman and Yang [IEEE Trans. Image Process., 4 (1995), pp. 932–946]. For three-channel color images, the per-iteration computation of this algorithm is dominated by six fast Fourier transforms. The convergence results in [Y. Wang, J. Yang, W. Yin, and Y. Zhang, SIAM J. Imaging Sci., 1 (2008), pp. 248–272] for single-channel images, including global convergence with a strong $q$-linear rate and finite convergence for some quantities, are extended to this algorithm. We present numerical results including images recovered from various types of blurs, comparisons between our results and those obtained from the deblurring functions in MATLAB's Image Processing Toolbox, as well as images recovered by our algorithm using weighted MTV and higher-order regularization. Our numerical results indicate that the processing speed, as attained by the proposed algorithm, of variational models with TV-like regularization can be made comparable to that of less sophisticated but widely used methods for color image restoration.

Reconstruction of Single-Grain Orientation Distribution Functions for Crystalline Materials

Per Christian Hansen, Henning Osholm Sørensen, Zsuzsanna Sükösd, and Henning Friis Poulsen

SIAM J. Imaging Sci. 2, pp. 593-613 (21 pages)

Online Publication Date: May 06, 2009

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A fundamental imaging problem in microstructural analysis of metals is the reconstruction of local crystallographic orientations from X-ray diffraction measurements. This work develops a fast, accurate, and robust method for the computation of the three-dimensional orientation distribution function for individual grains of the material in consideration. We study two iterative large-scale reconstruction algorithms, the algebraic reconstruction technique (ART) and conjugate gradients for least squares (CGLS), and demonstrate that right preconditioning is necessary in both algorithms to provide satisfactory reconstructions. Our right preconditioner is not a traditional one that accelerates convergence; its purpose is to modify the smoothness properties of the reconstruction. We also show that a new stopping criterion, based on the information available in the residual vector, provides a robust choice of the number of iterations for these preconditioned methods.

Estimation of Translation, Rotation, and Scaling between Noisy Images Using the Fourier–Mellin Transform

Jérémie Bigot, Fabrice Gamboa, and Myriam Vimond

SIAM J. Imaging Sci. 2, pp. 614-645 (32 pages)

Online Publication Date: May 14, 2009

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In this paper we focus on extended Euclidean registration of a set of noisy images. We provide an appropriate statistical model for this kind of registration problem, and a new criterion based on Fourier-type transforms is proposed to estimate the translation, rotation, and scaling parameters to align a set of images. This criterion is a two-step procedure which does not require the use of a reference template onto which all the images are aligned. Our approach is based on $M$-estimation, and we prove the consistency of the resulting estimators. A small-scale simulation study and real examples are used to illustrate the numerical performances of our procedure.
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On the Use of Start-Stop Approximation for Spaceborne SAR Imaging

S. V. Tsynkov

SIAM J. Imaging Sci. 2, pp. 646-669 (24 pages)

Online Publication Date: May 14, 2009

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The start-stop approximation is a standard tool for processing radar data in synthetic aperture imaging. It assumes that the antenna is motionless when a pulse is emitted and the scattered signal received, after which the antenna moves to its next sending/receiving position along the flight track. However, when the antenna is mounted on a satellite, as opposed to an airplane, its relatively high speed raises at least two questions. The first one is whether the image may be affected by the actual displacement of the antenna during the pulse round-trip time between the orbit and the Earth's surface. This displacement, in fact, can be rather large. Nonetheless, by analyzing the corresponding generalized ambiguity function of a synthetic aperture radar (SAR) sensor we show that in practice this issue can be disregarded. The second question is related to the Doppler frequency shift, which, again, is larger for spaceborne radars than for airborne radars. In the early SAR studies, this frequency shift provided a venue for understanding the azimuthal resolution of a radar. However, in a more rigorous analysis based on the generalized ambiguity function, the Doppler effect is typically left out of consideration. We show that for the image to stay largely unaffected by Doppler, the frequency shift must be included in the definition of a matched filter. Otherwise, there will be a geometric shift (translation) of the entire imaged scene from its true position, and there may also be a slight deterioration of the image sharpness (contrast).

Scale-Space Analysis of Discrete Filtering over Arbitrary Triangulated Surfaces

Chunlin Wu, Jiansong Deng, Falai Chen, and Xuecheng Tai

SIAM J. Imaging Sci. 2, pp. 670-709 (40 pages)

Online Publication Date: May 22, 2009

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Discrete filtering of information over triangulated surfaces has proved very useful in computer graphics applications. This technique is based on diffusion equations and has been extensively applied to image processing, harmonic map regularization and texture generating, etc. [C. L. Bajaj and G. Xu, ACM Trans. Graph., 22 (2003), pp. 4–32], [C. Wu, J. Deng, and F. Chen, IEEE Trans. Vis. Comput. Graph., 14 (2008), pp. 666–679]. However, little has been done on analysis (especially quantitative analysis) of the behavior of these filtering procedures. Since in applications mesh surfaces can be of arbitrary topology and the filtering can be nonlinear and even anisotropic, the analysis of the quantitative behavior is a very difficult issue. In this paper, we first present the discrete linear, nonlinear, and anisotropic filtering schemes via discretizing diffusion equations with appropriately defined differential operators on triangulated surfaces, and then use concepts of discrete scale-spaces to describe these filtering procedures and analyze their properties respectively. Scale-space properties such as existence and uniqueness, continuous dependence on initial value, discrete semigroup property, grey level shift invariance and conservation of total grey level, information reduction (also known as topology simplification), and constant limit behavior have been proved. In particular, the information reduction property is analyzed by eigenvalue and eigenvector analysis of matrices. Different from the direct observation of the local filtering to the diffusion equations and other interpretation methods based on wholly global quantities such as energy and entropy, this viewpoint helps us understand the filtering both globally (information reduction as image components shrink) and locally (how the image component contributes to its shrink rate). With careful consideration of the correspondence between eigenvalues and eigenvectors and their features, differences between linear and nonlinear filtering, as well as between isotropic and anisotropic filtering, are discussed. We also get some stability results of the filtering schemes. Several examples are provided to illustrate the properties.

Shape and Motion Estimation from Near-Field Echo-Based Sensor Data

Matthew Ferrara and Gregory Arnold

SIAM J. Imaging Sci. 2, pp. 710-729 (20 pages)

Online Publication Date: May 22, 2009

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This paper presents a new approach for reconstructing both shape and motion from data collected by echo-based ranging sensors. The approach is based on geometric invariant theory and exploits object-image relations for near-field (spherical-wavefront) range data. These object-image equations relate the data to a unique matrix of Euclidean invariants that completely describe the object shape. The object-image relations can be used to determine the shape of a scene viewed from unknown vantage points. Specifically, the object-image equations form a linear system of equations whose solution determines the relevant shape parameters for a configuration of features within the scene. Once the shape parameters are estimated, a single shape exemplar from the point in shape space can be used to determine the relative motion (up to an arbitrary rotation) between the sensor and the object. One advantage of this motion-estimation approach is that the geometric-invariant-based strategy allows us to uniquely solve the optimization problem without the need to introduce coordinate-system-dependent “nuisance” parameters. The theorems stated in this paper hold for any range-measurement sensor scenario. As an example of the utility of the given theorems, the object-image relations are used to augment noisy GPS measurements in a circular synthetic aperture radar geometry.

Nested Iterative Algorithms for Convex Constrained Image Recovery Problems

Caroline Chaux, Jean-Christophe Pesquet, and Nelly Pustelnik

SIAM J. Imaging Sci. 2, pp. 730-762 (33 pages)

Online Publication Date: June 04, 2009

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The objective of this paper is to develop methods for solving image recovery problems subject to constraints on the solution. More precisely, we will be interested in problems which can be formulated as the minimization over a closed convex constraint set of the sum of two convex functions $f$ and $g$, where $f$ may be nonsmooth and $g$ is differentiable with a Lipschitz-continuous gradient. To reach this goal, we derive two types of algorithms that combine forward-backward and Douglas–Rachford iterations. The weak convergence of the proposed algorithms is proved. In the case when the Lipschitz-continuity property of the gradient of $g$ is not satisfied, we also show that, under some assumptions, it remains possible to apply these methods to the considered optimization problem by making use of a quadratic extension technique. The effectiveness of the algorithms is demonstrated for two wavelet-based image restoration problems involving a signal-dependent Gaussian noise and a Poisson noise, respectively.

Field Fluctuations, Imaging with Backscattered Waves, a Generalized Energy Theorem, and the Optical Theorem

Roel Snieder, Francisco J. Sánchez-Sesma, and Kees Wapenaar

SIAM J. Imaging Sci. 2, pp. 763-776 (14 pages) | Cited 4 times

Online Publication Date: June 04, 2009

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We show the connection between four aspects of wave propagation: the autocorrelation of field fluctuations, imaging with backscattered waves, a theorem for energy flow, and the generalized optical theorem. The autocorrelation of field fluctuations can be used to extract the imaginary component of the Green's function at the source. The Green's function usually is singular at the source, but the imaginary component is not. The imaginary component of the Green's function at the source can thus be retrieved from the autocorrelation of field fluctuations, and can be used to image the medium using backscattered fields. We also show for general linear systems, which may be open or closed and may be dissipative, that the imaginary component of the Green's function at the source accounts for the loss of generalized energy by dissipation and/or propagation of the fields away from the source. Finally we show that the expressions for the extraction of the Green's function for scalar waves has the same mathematical structure as the generalized optical theorem. The theory presented here is shown to be applicable to damped acoustic waves, quantum mechanics, and diffusion.
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