Dominique Zosso, PhD

And still, many associated research questions have not been answered to full satisfaction, yet. Mathematically speaking, most image processing tasks are instances of inverse problems, where one looks for an underlying, hidden layer of information given a set of derived measurements. New applications, types of images, and questions to be answered through imaging come up everyday. My research interest is to ride on this highly dynamic wave at the cutting-edge frontier of image understanding problems. I want to contribute new mathematical models and efficient schemes for their computation, to provide new and better ways of getting information out of images, and to create new possibilities of imaging in the first place. Taking images is easy. Understanding them is the challenge.

Research statement

Variational Mode Decomposition

In the late nineties, Huang introduced the Hilbert-Huang transform, also known as Empirical Mode Decomposition. The goal is to recursively decompose a signal into different modes of separate spectral bands, which are unknown beforehand. The HHT/EMD algorithm is widely used today, although there is no exact mathematical model corresponding to this algorithm, and, consequently, the exact properties and limits are widely unknown. Here, we propose an entirely non-recursive variational mode decomposition model, where the modes are extracted concurrently. The model looks for a number of modes and their respective center frequencies, such that the modes reproduce the input signal, while being smooth after demodulation into baseband. In Fourier domain, this corresponds to a narrow-band prior. We show important relations to Wiener filter denoising. Indeed, the proposed method is a generalization of the classic Wiener filter into adaptive, multiple bands.

Unified Retinex

The fundamental assumption in retinex is that the observed image is a multiplication between the illumination and the true underlying reflectance of the object. We define our retinex model in two steps: First, we look for a filtered gradient that is the solution of an optimization problem consisting of two terms: A sparsity prior of the reflectance, such as the TV or H1 norm, and a quadratic fidelity prior of the reflectance gradient with respect to the observed image gradients. In a second step, since this filtered gradient almost certainly is not a consistent image gradient, we then look for a reflectance whose actual gradient comes close. Beyond unifying existing models, we are able to derive entirely novel retinex formulations by using more interesting non-local versions for the sparsity and fidelity prior. Hence we define within a single framework new retinex instances particularly suited for texture-preserving shadow removal, cartoon-texture decomposition, color and hyperspectral image enhancement.

Geodesic Active Fields

In image registration, one looks for the underlying deformation field that best maps one image onto another. This is a classic ill-posed inverse problem, which is usually solved by adding a regularization term. Here, we propose a multiplicative coupling between the registration term and the regularization term, which turns out to be equivalent to embed the deformation field in a weighted minimal surface problem. Then, the deformation field is driven by a minimization flow toward a harmonic map corresponding to the solution of the registration problem.

Fast Geodesic Active Fields

The directly resulting minimizing energy flow has poor numerical properties. Here, we provide an efficient numerical scheme that uses a splitting approach: data and regularity term are optimized over two distinct deformation fields that are constrained to be equal via an augmented Lagrangian approach. Overall, we can show the advantages of the proposed FastGAF method. It compares favorably against Demons, both in terms of registration speed and quality.

Cortical Scale Space

Here, we define a scale-space for cortical mean curvature maps on the sphere, that offers a hierarchical representation of the brain cortical structures, useful in multi-scale registration and analysis algorithms. A spherical feature map is obtained through inflation of the cortical surface of one hemisphere, extracted from structural MR images. Using the Beltrami framework, we embed this spherical mesh in a higher dimensional space and the feature assigned to a mesh vertex becomes an additional component of its coordinates. This enhanced mesh then evolves under Beltrami flow. The collection of all maps produced by this PDE forms a scale-space. Our results suggest that this scale-space provides a generalization of the brain map suitable for use e.g. within a multi-scale registration framework.

Tomography Reconstruction

We present an open-source ITK implementation of a direct Fourier method for tomographic reconstruction, applicable to parallel-beam x-ray images. Direct Fourier reconstruction makes use of the central-slice theorem to build a polar 2D Fourier space from the 1D transformed projections of the scanned object, that is resampled into a Cartesian grid. Inverse 2D Fourier transform eventually yields the reconstructed image. Additionally, we provide a complex wrapper to the BSplineInterpolateImageFunction to overcome ITK’s current lack for image interpolators dealing with complex data types. A sample application is presented and extensively illustrated on the Shepp-Logan head phantom. We show that appropriate input zeropadding and 2D-DFT oversampling rates together with radial cubic b-spline interpolation improve 2D-DFT interpolation quality and are efficient remedies to reduce reconstruction artifacts.