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A fresh look at the Bayesian bounds of the Weiss–Weinstein family
- IEEE TRANS. SIGNAL PROCESS
, 2008
"... Minimal bounds on the mean square error (MSE) are generally used in order to predict the best achievable performance of an estimator for a given observation model. In this paper, we are interested in the Bayesian bound of the Weiss–Weinstein family. Among this family, we have Bayesian Cramér-Rao bo ..."
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Minimal bounds on the mean square error (MSE) are generally used in order to predict the best achievable performance of an estimator for a given observation model. In this paper, we are interested in the Bayesian bound of the Weiss–Weinstein family. Among this family, we have Bayesian Cramér-Rao bound, the Bobrovsky–MayerWolf–Zakaï bound, the Bayesian Bhattacharyya bound, the Bobrovsky–Zakaï bound, the Reuven–Messer bound, and the Weiss–Weinstein bound. We present a unification of all these minimal bounds based on a rewriting of the minimum mean square error estimator (MMSEE) and on a constrained optimization problem. With this approach, we obtain a useful theoretical framework to derive new Bayesian bounds. For that purpose, we propose two bounds. First, we propose a generalization of the Bayesian Bhattacharyya bound extending the works of Bobrovsky, Mayer–Wolf, and Zakaï. Second, we propose a
Ziv-zakai bounds on image registration
- IEEE Trans. Signal Proc
"... Abstract—Image registration is a fundamental and important task in image processing. The goal essentially is to estimate the transformation that aligns two images. We focus on the general rigid body transformation case. In this paper, we derive the Ziv–Zakai bounds (ZZB) on image registration by ass ..."
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Abstract—Image registration is a fundamental and important task in image processing. The goal essentially is to estimate the transformation that aligns two images. We focus on the general rigid body transformation case. In this paper, we derive the Ziv–Zakai bounds (ZZB) on image registration by assuming an uncertainty model for the rotation and translation errors, and propose to use the ZZB as a benchmark to evaluate the registra-tion ability of an image pair. We also compare the performance of several image registration algorithms with the derived bounds when applied to several datasets. Index Terms—Image registration, parameter estimation, Ziv–Zakai bound. I.
How accurate can block matches be in stereo vision
- SIAM Journal on Imaging Sciences
, 2011
"... Abstract. This article explores the sub-pixel accuracy attainable for the disparity computed from a rectified stereo pair of images with small baseline. In this framework we consider translations as the local deformation model between patches in the images. A mathematical study shows first how discr ..."
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Abstract. This article explores the sub-pixel accuracy attainable for the disparity computed from a rectified stereo pair of images with small baseline. In this framework we consider translations as the local deformation model between patches in the images. A mathematical study shows first how discrete block-matching can be performed with arbitrary precision under Shannon-Whittaker conditions. This study leads to the specification of a block-matching algorithm which is able to refine disparities with sub-pixel accuracy. Moreover, a formula for the variance of the disparity error caused by the noise is introduced and proved. Several simulated and real experiments show a decent agreement between this theoretical error variance and the observed RMSE in stereo pairs with good SNR and low baseline. A practical consequence is that under realistic sampling and noise conditions in optical imaging, the disparity map in stereo-rectified images can be computed for the majority of pixels (but only for those pixels with meaningful matches) with a 1/20 pixel precision. Key words. Block-matching, sub-pixel accuracy, noise error estimate. 1. Introduction. Stereo algorithms aim at reconstructing a 3D model from two or more images of the same scene acquired from different angles. Assuming for a sake of simplicity that the cameras are calibrated, and that the image pair has been
Spatial Confidence Regions for Quantifying and Visualizing Registration Uncertainty
"... For image registration to be applicable in a clinical setting, it is important to know the degree of uncertainty in the returned point-correspondences. In this paper, we propose a data-driven method that allows one to visualize and quantify the registration uncertainty through spatially adaptive co ..."
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For image registration to be applicable in a clinical setting, it is important to know the degree of uncertainty in the returned point-correspondences. In this paper, we propose a data-driven method that allows one to visualize and quantify the registration uncertainty through spatially adaptive confidence regions. The method applies to various parametric deformation models and to any choice of the similarity criterion. We adopt the B-spline model and the negative sum of squared differences for concreteness. At the heart of the proposed method is a novel shrinkage-based estimate of the distribution on deformation parameters. We present some empirical evaluations of the method in 2-D using images of the lung and liver, and the method generalizes to 3-D.
Cramér-Rao Lower Bound for Point Based Image Registration with Heteroscedastic Error Model for Application in Single Molecule Microscopy
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ESTIMATING RESPIRATORY MOTION FROM CT IMAGES VIA DEFORMABLE MODELS AND PRIORS
, 2007
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Modifications in Normalized Cross Correlation Expression for Template Matching Applications
"... Abstract- This paper analyzes the performance of sum of squared differences (SSD), sum of absolute differences (SAD), normalized cross correlation (NCC), zero mean normalized cross correlation (ZNCC) and several other proposed modified expressions of NCC. Experimental results on real images demonstr ..."
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Abstract- This paper analyzes the performance of sum of squared differences (SSD), sum of absolute differences (SAD), normalized cross correlation (NCC), zero mean normalized cross correlation (ZNCC) and several other proposed modified expressions of NCC. Experimental results on real images demonstrate that some of the proposed modified expressions of NCC are more efficient than conventional NCC for template matching. Three of the modified expressions of NCC perform similar to ZNCC however they are computationally less intensive. Modified expressions of NCC were also studied under different values of additive white Gaussian noise. Some of them perform better than ZNCC in terms of successfully found points and computation time for noisy images. the search area (reference image). Figure 1 shows one such reference image. Template and region of interest extracted from the reference image are marked as region A and region B respectively. The proposed correlation based template matching algorithm consists of sliding the template over the region of interest and calculating ‘correlation measure ’ at each position, estimating the degree of similarity, between the template and the region of interest. The maximum correlation position or minimum distortion is taken to represent the position of the template in the region of interest. Selection of region of interest reduces the time required for computation.
SIAM J. IMAGING SCIENCES © xxxx Society for Industrial and Applied Mathematics Vol. xx, pp. x x–x Analysis of Image Registration with Tangent Distance
"... Abstract. The computation of the geometric transformation between a reference and a target image, known as image registration or alignment, corresponds to the projection of the target image onto the transformation manifold of the reference image (the set of images generated by its geometric trans-fo ..."
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Abstract. The computation of the geometric transformation between a reference and a target image, known as image registration or alignment, corresponds to the projection of the target image onto the transformation manifold of the reference image (the set of images generated by its geometric trans-formations). It often takes a nontrivial form such that exact computation of projections on the manifold is difficult. The tangent distance method is an effective alignment algorithm that exploits a linear approximation of the transformation manifold of the reference image. As theoretical studies about the tangent distance algorithm have been largely overlooked, we present in this work a detailed performance analysis of this useful algorithm, which can eventually help the selection of algorithm parameters. We consider a popular image registration setting using a multiscale pyramid of lowpass filtered versions of the (possibly noisy) reference and target images, which is particularly useful for recovering large transformations. We first show that the alignment error has a nonmonotonic varia-tion with the filter size, due to the opposing effects of filtering on manifold nonlinearity and image noise. We then study the convergence of the multiscale tangent distance method to the optimal solution. We finally examine the performance of the tangent distance method in image classification applications. Our theoretical findings are confirmed by experiments on image transformation models involving translations, rotations and scalings. Our study is the first detailed study of the tangent distance algorithm that leads to a better understanding of its efficacy and to the proper selection of design parameters. Key words. Image registration, tangent distance, image analysis, hierarchical registration methods, perfor-mance analysis. 1. Introduction. The
Analysis of Image Registration with Tangent Distance
, 2014
"... The computation of the geometric transformation between a reference and a target image, known as image registration or alignment, corresponds to the projection of the target image onto the transformation manifold of the reference image (the set of images generated by its geometric transformations). ..."
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The computation of the geometric transformation between a reference and a target image, known as image registration or alignment, corresponds to the projection of the target image onto the transformation manifold of the reference image (the set of images generated by its geometric transformations). It often takes a nontrivial form such that exact computation of projections on the manifold is difficult. The tangent distance method is an effective alignment algorithm that exploits a linear approximation of the transformation manifold of the reference image. As theoretical studies about the tangent distance algorithm have been largely overlooked, we present in this work a detailed performance analysis of this useful algorithm, which can eventually help the selection of algorithm parameters. We consider a popular image registration setting using a multiscale pyramid of lowpass filtered versions of the (possibly noisy) reference and target images, which is particularly useful for recovering large transformations. We first show that the alignment error has a nonmonotonic variation with the filter size, due to the opposing effects of filtering on manifold nonlinearity and image noise. We then study the convergence of the multiscale tangent distance method to the optimal solution. We finally examine the performance of the tangent distance method in image classification applications. Our theoretical findings are confirmed by experiments on image transformation models involving translations, rotations and scalings. Our study is the first detailed study of the tangent distance algorithm that leads to a better understanding of its efficacy and to the proper selection of design parameters.
