Sherif Makram-Ebeid

Quadratic Structuring Functions in Mathematical Morphology

In this contribution we look at the quadratic structuring functions as alternatives to the often used “flat” structuring functions. The quadratic structuring functions (henceforth abbreviated as QSF’s) are the morphological counterpart of the Gaussian function in linear image processing in the sense that: the class of QSF’s is closed under dilation (i.e. dilating two QSF’s results in a third QSF), the QSF’s are easily dimensionally decomposed, (i.e. any n-dimensional QSF can be obtained through dilation of n one-dimensional QSF’s in n independent directions) and the class of QSF’s contains the unique rotational symmetric structuring function that can be dimensionally decomposed with respect to dilation.

Non-Rigid Image Registration using a Hierarchical Partition of Unity Finite Element Method

We use a hierarchical partition of unity finite element method (H-PUFEM) to represent and analyse the non-rigid deformation fields involved in multidimensional image registration. We make use of the Ritz-Galerkin direct variational method to solve non-rigid image registration problems with various deformation constraints. In this method, we directly seek a set of parameters that minimizes the objective function. We thereby avoid the loss of information that may occur when an Euler-Lagrange formulation is used. Experiments are conducted to demonstrate the advantages of our approach when registering synthetic images having little of or no localizing features. As a special case, conformal mapping problems can be accurately solved in this manner. We also illustrate our approach with an application to cardiac magnetic resonance temporal sequences.

3D harmonic phase tracking with anatomical regularization

This paper presents a novel algorithm that extends HARP to handle 3D tagged MRI images. HARP results were regularized by an original regularization framework defined in an anatomical space of coordinates. In the meantime, myocardium incompressibility was integrated in order to correct the radial strain which is reported to be more challenging to recover. Both the tracking and regularization of LV displacements were done on a volumetric mesh to be computationally efficient. Also, a window-weighted regression method was extended to cardiac motion tracking which helps maintain a low complexity even at finer scales. On healthy volunteers, the tracking accuracy was found to be as accurate as the best candidates of a recent benchmark. Strain accuracy was evaluated on synthetic data, showing low bias and strain errors under 5% (excluding outliers) for longitudinal and circumferential strains, while the second and third quartiles of the radial strain errors are in the (-5%,5%) range. In clinical data, strain dispersion was shown to correlate with the extent of transmural fibrosis. Also, reduced deformation values were found inside infarcted segments.

Scale-space image analysis based on Hermite polynomials theory

The Hermite transform allows to locally approximate an image by a linear combination of polynomials. For a given scale σ and position ξ, the polynomial coefficients are closely related to the differential jet (set of partial derivatives of the blurred image) for the same scale and position. By making use of a classical formula due to Mehler (late 19th century), we establish a linear relationship linking the differential jets at two different scales σ and positions ξ involving Hermite polynomials. Pattern registration and matching applications are suggested. We introduce a Gaussian windowed correlation function K(Υ) for locally matching two images. When taking the mutual translation parameter (Υ) as independent variable, we express the Hermite coefficients of K(Υ) in terms of the Hermite coefficients of the two images being matched. This new result bears similarity with the Wiener-Khinchin theorem which links the Fourier transform of the conventional (flat-windowed) correlation function with the Fourier spectra of the images being correlated. Compared to the conventional correlation function, ours is more suited for matching localized image features. The mathematical tools we propose are shown to have attractive computational features. Numerical simulations using synthetic 1D and 2D test patterns demonstrate the advantages of our proposals for signal and image matching in terms of accuracy and low algorithm complexity.

Vessel geometry modeling and segmentation using convolution surfaces and an implicit medial axis

In the context of vessel tree structures segmentation with implicit deformable models, we propose to exploit convolution surfaces to introduce a novel variational formulation, robust to bifurcations, tangential vessels and aneurysms. Vessels are represented by an implicit function resulting from the convolution of the centerlines of the vessels, modeled as a second implicit function, with localized kernels of continuously-varying scales. The advantages of this coupled representation are twofold. First, it allows for a joint determination of the vessels centerlines and radii, with a single model relevant for segmentation and visualization tasks. Second, it allows us to define a new shape constraint on the implicit function representing the centerlines, to enforce the tubular shape of the segmented objects. The algorithm has been evaluated on the segmentation of the portal veins in 20 CT-scans of the liver from the 3D-IRCADb-01 database, achieving an average recovery of 73% of the trees with fast computational times.

Prior-Based Piecewise-Smooth Segmentation by Template Competitive Deformation Using Partitions of Unity

We propose a new algorithm for two-phase, piecewise-smooth segmentation with shape prior. The image is segmented by a binary template that is deformed by a regular geometric transformation. The choice of the template together with the constraint on the transformation introduce the shape prior. The deformation is guided by the maximization of the likelihood of foreground and background intensity models, so that we can refer to this approach as Competitive Deformation. In each region, the intensity is modelled as a smooth approximation of the original image. We represent the transformation using a Partition of Unity Finite Element Method, which consists in representing each component with polynomial approximations within local patches. A conformity constraint between the patches provides a way to control the globality of the deformation. We show several results on synthetic images, as well as on medical data from different modalities.

Fast solver for some computational imaging problems: A regularized weighted least-squares approach

In this paper we propose to solve a range of computational imaging problems under a unified perspective of a regularized weighted least-squares (RWLS) framework. These problems include data smoothing and completion, edge-preserving filtering, gradient-vector flow estimation, and image registration. Although originally very different, they are special cases of the RWLS model using different data weightings and regularization penalties. Numerically, we propose a preconditioned conjugate gradient scheme which is particularly efficient in solving RWLS problems. We provide a detailed analysis of the system conditioning justifying our choice of the preconditioner that improves the convergence. This numerical solver, which is simple, scalable and parallelizable, is found to outperform most of the existing schemes for these imaging problems in terms of convergence rate.

Computational and physical phantom setups for the second cardiac motion analysis challenge (cMAC2)

This paper describes the data setup of the second cardiac Motion Analysis Challenge (cMac2). The purpose of this challenge is to initiate a public data repository for the benchmark of motion and strain quantification algorithms on 3D ultrasound images. The data currently includes synthetic images that combine ultrasound and biomechanical simulators. We also collected sonomicrometry curves and ultrasound images acquired on a Polyvinyl alcohol phantom.

Motion estimation in 3d echocardiography using smooth field registration

This paper describes an algorithm for motion and deformation quantification of 3D cardiac ultrasound sequences. The algorithm is based on the assumption that the deformation field is smooth inside the myocardium. Thus, we assume that the displacement field can be represented as the convolution of an unknown field with a Gaussian kernel. We apply our algorithm to datasets with reliable ground truth: a set of synthetic sequences with known trajectories and a set of sequences of a mechanical phantom implanted with microsonometry crystals.

New methods and algorithms for the accurate, real-time motion analysis of the left ventricle with MRI-tagging

Abstract Despite its tremendous advantages, magnetic resonance tagging remains unserviceable in clinical conditions due to very tedious image pre-processing tasks. We present here a novel algorithm, fully automatic, real-time and extremely accurate, to perform the tagging pattern extraction. We then address the problem of motion computation and analysis using a novel kinetic deformation class. Very accurate registrations enable to perform automatic ventricle segmentation propagation and a highly sharp motion analysis of the walls. Our work mainly focuses on the CSPAMM protocol, which is briefly recalled. Experiments on acquisitions from 15 healthy volunteers are shown.