Research Areas


Broadly speaking, our research interests lie in the diverse area of mathematical imaging and inverse problems. Our long-term research objective is directed towards developing and validating efficient numerical methodologies for solving real-world, ill-posed inverse problems in the field of medical imaging. Some of our short- and medium-termresearch objectives are outlined below.

Recent Activities

1-Efficient numerical methodologies aimed at Computer Assisted Surgery (CAS) of the breast

Breast cancer is the most common malignant disease among Canadian women. Breast Magnetic Resonance Imaging (MRI) is frequently performed prior to breast conserving surgery in order to assess the location and extent of the lesion. Ideally, the surgeon should be able to use the pre-surgery image information during surgery to guide the excision. This requires the prone pre-surgical MR image to be aligned or co-registered to conform to the patient's supine position on the operating table. The registration step can be modelled as an inverse problem that is ill-posed, i.e., has no unique solution.
Recently, we have developed a computationally efficient registration scheme based on Thin-Plate-Splines to align pre-surgical images to the operating room position using MR-visible surface markers placed before imaging. While the results are promising, the next natural step is to eliminate the requirement of placing the surface markers in the registration scheme. The future breast CAS technology will demand novel efficient hybrid surface-intensity based registration algorithms that employ the surface information of the operating room along with the intensity information of the pre-surgical images. The highly deformable nature of the breast tissue is one of the main challenges of solving this ill-posed inverse problem.

2-Coupled approaches for analysis of Dynamic Contrast Enhanced (DCE) Magnetic Resonance Imaging (MRI)

DCE imaging has emerged as a powerful screening tool for identifying carcinoma in high-risk populations. Accurate registration of DCE images is valuable for proper identification of the lesions. Recently, we have developed a simple coupled PDE approach to joint registration and intensity correction of the DCE images. In general, the contrast uptake curves are results of a physical process. Motion correction of images coupled with this process is a significant step in identification of malignancies. This requires defining regularization expressions that directly incorporate the underlying physical process of this inverse problem.

3-Image registration schemes in presence of discontinuities

In image registration, researchers generally rely on transformations to describe the alignment process relating two images. However, in the clinical setting, there are many situations where the “true” deformation may have discontinuities. For example, one would expect the transformation between the images of a patient before and after tumor resection would contain a singularity or a hole. Previously, the problems of rips/tears have been addressed using the Total Variation (TV) regularization, but this typically only works well for extremely small discontinuities. Furthermore, these approaches do not at all address the clinical problems where singularities or holes are present in the true deformation. Given these limitations of current approaches to image registration, an open problem is to define an approach that would enable deformable registration in the presence of various types of large-scale discontinuities. One potential approach is to develop a registration technique that implicitly allows topological changes in the deformation field, much like the use of level sets in image segmentation allows implicitly for topological changes in region boundaries.

Past Contributions

1-Landmark-based image registration of the breast

Breast magnetic resonance imaging (MRI) is frequently performed prior to breast conserving surgery in order to assess the location and extent of the lesion. It is, however, difficult for the surgeon to use the MR images to guide the surgery. While pre-surgical MRI is typically performed in the prone position, surgery is performed in the supine position. The breast undergoes significant distortion as a result. To overcome this restriction, supine breast MRI has been recently proposed. In most clinical MR scanners the arm of the patient has to be placed parallel to the body, whereas the arm is placed in an outstretched position during surgery. The highly deformable nature of the breast tissue and the computational time constraint are the main challenges in solving this inverse problem. We have examined a thin-plate spline registration scheme to match these two configurations using the positional information of surface markers [ESM+14] [SEH+12a] [ESPM12].

2-Intensity based approaches for medical image registration

We developed a number of new methodologies for automated motion correction of medical images using their intensity information. Specifically, we have been working on registration algorithms aimed at motion correction of dynamic contrast-enhanced magnetic resonance images (DCE-MRI). In [EM09b] we suggested an extension of the well-known Demons image registration algorithm. Our extension allows intensity changes in images due to contrast enhancement as opposed to the Demons algorithm. This extension provides a simple registration algorithm for DCE-MRI. Furthermore, in [Ebr09a] we proposed a coupled PDE-based approach to optimize the combined image registration and intensity correction expression in an optimize-then-discretize framework. More recently in [ELM13] we have presented an efficient multi-level Gauss-Newton approach in a discretize-then-optimize paradigm. The simulated data for validation in this article was prepared by Anthony Lausch (MSc Student, co-author). Lausch’s work has also led to [LEM11]. In addition, we assisted Melissa Hill (PhD student at Sunnybrook) in registration of contrast-enhanced digital mammography images. The work was a collaboration with a group of experts in digital mammography that led to [MEL13] in Medical Physics.

3-Image zooming and super-resolution

Recovery of a high-resolution image from a single image is called image zooming and from a set of distorted images is known as super-resolution. Image zooming and super-resolution are the other ill-posed inverse problems that we have been investigating during the course of our research. In “Solving the inverse problem of image zooming using ‘self-examples’” [EV07c] we presented a recipe to address image zooming problem based on the non-local (NL) self-similarities of an image across scales. This publication received a great deal of attention and citations. The method of “self-examples” combined the ideas of fractal-zoom with the newly developed NL-means denoising algorithm which is demonstrated to perform as a leading classical denoising method. The effectiveness of our method lies in the cross-scale regularity properties of natural images which has been considered in [EV08a]. Furthermore, to address the multi-frame super-resolution problem, we presented an algorithm in [EV08b] that does not require sub-pixel motion estimation as opposed to the common existing super-resolution algorithms. It is believed that a combined reconstruction and motion estimation is a key tool to address the superresolution problem rather than treating the two sub-problems independently. In [EVM09] [EM09c], we proposed a coupled multi-frame super-resolution approach with a non-parametric motion model. A coupled partial differential equation (PDE)-based approach was used to solve the optimization problem. Two variants of this work were also presented in [Ebr10b] [Ebr08c].

4-Inverse problems involving self-similarity

Self-similarity is the basis of fractal image coding which was originally developed by Michael Barnsley, the influential author of Fractals Everywhere. We revisited the concept of fractal-based methods and derived a necessary and sufficient condition for the contractivity of the fractal transform operator [Ebr09]. In [LVEB09] prepared in collaboration with Barnsley and coworkers we proposed more generalized fractal transform operators in the context of measure valued images. To offer flexibility in inclusion of a-priori image constraints at the decoding stage we proposed “Fractal image coding as projections onto convex sets (POCS)” [EV06a] and “Regularization Schemes involving Self-similarity in Imaging Inverse Problems” [EV07b]. These papers provide tools to address the ill-posed inverse problem of missing fractal codes. We summarized the activity in the area of self-similarity in imaging for the past 20 years in two invited presentations [Ebr09b] [EV08c].

Selected Publications

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