Multiresolution Moment Filters: Theory and Applications

We introduce local weighted geometric moments that are computed from an image within a sliding window at multiple scales. When the window function satisfies a two-scale relation, we prove that lower order moments can be computed efficiently at dyadic scales by using a multiresolution wavelet-like algorithm. We show that B-splines are well-suited window functions because, in addition to being refinable, they are positive, symmetric, separable, and very nearly isotropic (Gaussian shape). We present three applications of these multiscale local moments. The first is a feature-extraction method for detecting and characterizing elongated structures in images. The second is a noise-reduction method which can be viewed as a multiscale extension of Savitzky-Golay filtering. The third is a multiscale optical-flow algorithm that uses a local affine model for the motion field, extending the Lucas-Kanade optical-flow method. The results obtained in all cases are promising

Variational Image Reconstruction from Arbitrarily Spaced Samples: A Fast Multiresolution Spline Solution

We propose a novel method for image reconstruction from nonuniform samples with no constraints on their locations. We adopt a variational approach where the reconstruction is formulated as the minimizer of a cost that is a weighted sum of two terms: 1) the sum of squared errors at the specified points and 2) a quadratic functional that penalizes the lack of smoothness. We search for a solution that is a uniform spline and show how it can be determined by solving a large, sparse system of linear equations. We interpret the solution of our approach as an approximation of the analytical solution that involves radial basis functions and demonstrate the computational advantages of our approach. Using the two-scale relation for B-splines, we derive an algebraic relation that links together the linear systems of equations specifying reconstructions at different levels of resolution. We use this relation to develop a fast multigrid algorithm. We demonstrate the effectiveness of our approach on some image reconstruction examples

Myocardial Motion Analysis from B-Mode Echocardiograms

The quantitative assessment of cardiac motion is a fundamental concept to evaluate ventricular malfunction. We present a new optical-flow-based method for estimating heart motion from two-dimensional echocardiographic sequences. To account for typical heart motions, such as contraction/expansion and shear, we analyze the images locally by using a local-affine model for the velocity in space and a linear model in time. The regional motion parameters are estimated in the least-squares sense inside a sliding spatiotemporal B-spline window. Robustness and spatial adaptability is achieved by estimating the model parameters at multiple scales within a coarse-to-fine multiresolution framework. We use a Wavelet-like algorithm for computing B-spline-weighted inner products and moments at dyadic scales to increase computational efficiency. In order to characterize myocardial contractility and to simplify the detection of myocardial dysfunction, the radial component of the velocity with respect to a reference point is color coded and visualized inside a time-varying region of interest. The algorithm was first validated on synthetic data sets that simulate a beating heart with a speckle-like appearance of echocardiograms. The ability to estimate motion from real ultrasound sequences was demonstrated by a rotating phantom experiment. The method was also applied to a set of in vivo echocardiograms from an animal study. Motion estimation results were in good agreement with the expert echocardiographic reading

Full Motion and Flow Field Recovery from Echo Doppler Data

We present a new computational method for reconstructing a vector velocity field from scattered, pulsed-wave ultrasound Doppler data. The main difficulty is that the Doppler measurements are incomplete, for they do only capture the velocity component along the beam direction. We thus propose to combine measurements from different beam directions. However, this is not yet sufficient to make the problem well posed because 1) the angle between the directions is typically small and 2) the data is noisy and nonuniformly sampled. We propose to solve this reconstruction problem in the continuous domain using regularization. The reconstruction is formulated as the minimizer of a cost that is a weighted sum of two terms: 1) the sum of squared difference between the Doppler data and the projected velocities 2) a quadratic regularization functional that imposes some smoothness on the velocity field. We express our solution for this minimization problem in a B-spline basis, obtaining a sparse system of equations that can be solved efficiently. Using synthetic phantom data, we demonstrate the significance of tuning the regularization according to the a priori knowledge about the physical property of the motion. Next, we validate our method using real phantom data for which the ground truth is known. We then present reconstruction results obtained from clinical data that originate from 1) blood flow in carotid bifurcation and 2) cardiac wall motion

Multiscale optical flow computation from the monogenic signal

National audienceWe have developed an algorithm for the estimation of cardiac motion from medical images. The algorithm exploits monogenic signal theory, recently introduced as an N-dimensional generalization of the analytic signal. The displacement is computed locally by assuming the conservation of the monogenic phase over time. A local affine displacement model replaces the standard translation model to account for more complex motions as contraction/expansion and shear. A coarse-to-fine B-spline scheme allows a robust and effective computation of the models parameters and a pyramidal refinement scheme helps handle large motions. Robustness against noise is increased by replacing the standard pointwise computation of the monogenic orientation with a more robust least-squares orientation estimate. This paper reviews the results obtained on simulated cardiac images from different modalities, namely 2D and 3D cardiac ultrasound and tagged magnetic resonance. We also show how the proposed algorithm represents a valuable alternative to state-of-the-art algorithms in the respective fields

Spatio-Temporal Nonrigid Registration for Ultrasound Cardiac Motion Estimation

We propose a new spatio-temporal elastic registration algorithm for motion reconstruction from a series of images. The specific application is to estimate displacement fields from two-dimensional ultrasound sequences of the heart. The basic idea is to find a spatio-temporal deformation field that effectively compensates for the motion by minimizing a difference with respect to a reference frame. The key feature of our method is the use of a semi-local spatio-temporal parametric model for the deformation using splines, and the reformulation of the registration task as a global optimization problem. The scale of the spline model controls the smoothness of the displacement field. Our algorithm uses a multiresolution optimization strategy to obtain a higher speed and robustness. We evaluated the accuracy of our algorithm using a synthetic sequence generated with an ultrasound simulation package, together with a realistic cardiac motion model. We compared our new global multiframe approach with a previous method based on pairwise registration of consecutive frames to demonstrate the benefits of introducing temporal consistency. Finally, we applied the algorithm to the regional analysis of the left ventricle. Displacement and strain parameters were evaluated showing significant differences between the normal and pathological segments, thereby illustrating the clinical applicability of our method

Motion Analysis of Echocardiograms Using a Local-Affine, Spatio-Temporal Model

We present a new method for estimating heart motion from two-dimensional (2D) echocardiographic sequences. It is inspired by the Lucas-Kanade algorithm for optical flow which estimates motion parameters over a sliding window. However, instead of assuming that the motion is constant within the analysis window, we consider a model that is locally affine and can account for typical heart motions such as dilation/contraction and shear. Another refinement is spatial adaptivity which is achieved by estimating displacement vectors at multiple scales and selecting the most promising fit. The affine parameters are estimated in the least squares sense using a separable spatial (resp., spatio-temporal) B-spline window. This particular choice is motivated by the fact that the B-splines are nearly isotropic (Gaussian-like) and that they satisfy a two-scale equation. We use this latter property to derive a wavelet-like algorithm that leads to a fast computation of B-spline-weighted inner products and moments at dyadic scales, which speeds up our method considerably. We test the algorithm on synthetic and real ultrasound sequences and show that it compares favorably with other methods, such as Lucas-Kanade and Horn-Schunk

Multiresolution Moment Filters

We define multi-scale moments that are estimated locally by analyzing the image through a sliding window at multiple scales. When the analysis window satisfies a two-scale relation, we prove that these moments can be computed very efficiently using a multiresolution wavelet-like algorithm. We also show that B-spline windows are best suited for this kind of analysis because, in addition to being refinable, they are positive, symmetric and very nearly isotropic. We present two applications of our method. The first is a feature extraction method for detecting strands of DNA in noisy cryoelectron-micrographs. The second is an extension of the Lucas-Kanade optical flow algorithm that assumes a local affine model of the motion field. The results obtained in both cases are very promising

Multigrid Image Reconstruction from Arbitrarily Spaced Samples

We propose a novel multiresolution-multigrid based signal reconstruction method from arbitrarily spaced samples. The signal is reconstructed on a uniform grid using B-splines basis functions. The computation of spline weights is formulated as a variational problem. Specifically, we minimize a cost that is a weighted sum of two terms: (i) the sum of squared errors at the specified points; (ii) a quadratic functional that penalizes the lack of smoothness. The problem is equivalent to solving a very large system of linear equations, with the dimension equal to the number of grid points. We develop a computationally efficient multiresolution-multigrid scheme for solving the system. We demonstrate the method with image reconstruction from contour points

Bimodal Myocardial Motion Analysis from B-Mode and Tissue Doppler Ultrasound

We present a new method for estimating heart motion from two-dimensional echocardiographic sequences by exploiting two ultrasound modalities: B-mode and tissue Doppler. The algorithmestimates a two-dimensional velocity field locally by using a spatial affine velocity model inside a sliding window. Within each window, we minimize a local cost function that is composed of two quadratic terms: an optical flow constraint that involves the B-mode data and a constraint that enforces the agreement of the velocity field with the directional tissue Doppler measurements. The relative influence of the two differentmodalities to the resulting solution is controlled by an adjustable weighting parameter. Robustness is achieved by a coarse-to-fine multi-scale approach. The method was tested on synthetic ultrasound data and validated by a rotating phantom experiment. First applications to clinical echocardiograms give promising results