January 17, 2019
January-August 2012
Thematic Program on Inverse Problems and Imaging
August 13-17, 2012
Workshop on Microlocal Methods in Medical Imaging

Peter Gibson (York), Allan Greenleaf (Rochester)
Luigi Rodino (Torino), M. W. Wong (York)
Hongmei Zhu (York)
Speaker Abstracts


Ryuichi Ashino, Osaka Kyoiku University
Image separation by wavelet analysis

Blind image separation is the separation of a set of images, called source images, from a set of mixed images, called observed images, without information about the source images nor the mixing process. Besides methods based on independent component analysis, several methods based on time-frequency analysis have been proposed. In this talk, a new method using wavelet analysis is proposed.

Allan Greenleaf, University of Rochester
Basic Microlocal Analysis

Microlocal analysis allows one to precisely describe the singularities of functions and distributions (generalized functions), and analyze how operators transform these singularities. The locations of singularities are expressed in terms of the cotangent bundle, which includes both spatial location and momentum (phase space) direction. Since many of the material parameters one is interested in medicine have discontinuities across interfaces, microlocal analysis lends itself to applications in medical imaging. This two-lecture mini-course will present some of the basics of microlocal analysis,
including a discussion of elliptic partial differential operators, their fundamental solutions and hypoellipticity; the notions of support, singular support and wave-front set; the calculus of pseudodifferential operators;and construction of parametrices for elliptic pseudodifferential operators.

Alexander Katsevich, University of Central Florida
Recent advances in tomographic image reconstruction from interior x-ray data

Using the Gelfand-Graev formula, the interior problem of tomography reduces to invertion of the finite Hilbert transform (FHT) from incomplete data. In this talk we study several aspects of inverting the FHT when the data are incompelte. Using the Cauchy transform and an approach based on the Riemann-Hilbert problem we derive a differential operator that commutes with the FHT. Our second result is the characterization of the null-space of the FHT in the case of incomplete data. Also we derive the asymptotics of the singular values of the FHT in three different cases of incomplete data.

Michael Lamoureux, University of Calgary
Gabor methods for imaging

The Gabor transform of a signal creates a time-frequency representation of physical data which can be directly manipulated through multiplication by symbols, analogous to the action of pseudodifferential operation and their representation through symbols in the time-frequency domain.

This two-lecture minicourse describes the basics of Gabor transforms and Gabor multipliers, windowing considerations, representation of linear operators including differential operators, and the use of Gabor multipliers in nonstationary filtering, deconvolution, and numerical wave propagation. Some particular imaging applications are described.

Cheng Liu, Kinectrics Inc.
An Adaptive S Transform with Applications in Studying Brain Functions

Discovering how the brain functions has been proved valuable for understanding the brain's behavioral control as well as guiding treatment of mental diseases. In response to stimuli, the brain generates a mix of brain waves that are dynamic and frequency-specified. Thus, time-frequency analysis has been widely used in analyzing brain signals. However, due to huge variation of the characteristics of brain signals, analysis measures providing the signal-invariant resolution cannot well reveal dynamic structure of various brain signals.

We introduce an adaptive S transform (AST), a new multi-resolution time-frequency representation whose resolution is adaptively adjusted to its analyzed signal. The proposed representation is built on the S transform with additional parameters to control its resolution. Given any specific signal, we implement a numerical procedure that automatically determines optimal parameters so that the resulting representation has the signal energy highly concentrated at the involved frequencies and time duration. It hence provides a time-frequency analysis tool offering good resolution to describe behaviors of various signals. We then use the AST to derive a number of measures for analyzing brain time series recorded by electroencephalography and magnetoencephalography. These measures include the AST-based power spectrum for revealing the characteristics of functional activity at a single brain area, and the AST-based coherence and phase-locking statistic for investigating the functional connectivity between multiple brain areas. Numerical simulations are presented to demonstrate performances of the AST and the corresponding brain time series measures. Finally, we apply the proposed AST-based analysis tools to investigate functional activity of motor cortices when subjects perform the multi-source interference task, a behavioral experiment involving tasks at multiple levels of difficulty.

Abdol-Reza Mansouri, Queen's University
Geometric Approaches in Image Diffusion

Image diffusion partial differential equations have been applied to medical and non-medical images in applications such as denoising, sharpening, and interpolating missing data, with great success. In this talk, we will review two recent geometrical approaches -- Beltrami and Hypoelliptic -- that have been proposed for deriving image diffusion equations, and we will present theoretical and experimental results on a class of geometrically inspired diffusion equations that we have recently proposed. The diffusion equations we obtain are derived by changing the Riemannian metric on the space of images from L^2 to Sobolev, and lead to flows which could not be obtained under the standard L^2 metric (Joint work with J. Calder (Michigan) and A. Yezzi (Georgia Tech)).

Ross Mitchell, The Mayo Clinic - Arizona
Microlocal Analysis of Medical Images: Applications in Cancer Imagenomics

Our expanding knowledge of the genetic basis and molecular mechanisms of cancer is beginning to revolutionize the practice of oncology. In fact, personalized medicine, using molecular biomarkers to classify tumors and direct treatment decisions profiles, is becoming the new standard of care. Unfortunately, genomic testing is invasive, costly, and time consuming. In addition, different regions of the same tumor, or the primary and metastatic tumors, can have widely variable genetic signatures. Therefore, a single biopsy in only one area of the tumor may provide an incomplete assessment. Global assessment of tumors for genomic analysis would be preferred but is limited due to high cost and practical limitations in obtaining multiple biopsies.
Noninvasive global tumor assessment, however, is possible with imaging such as Computed Tomography (CT), Magnetic Resonance (MR) or Positron Emission Tomography (PET). The emerging field of "imagenomics" is focused on identifying imaging traits that correlate with genetics. If successfully validated, and proven to have suitable sensitivity and specificity, the use of imagenomics tests could complement conventional surgical biopsies. For example, this could be important in the context of large heterogeneous lesions, multiple lesions, surgically inaccessible lesions, and settings where disease progression needs to be monitored frequently over time.
The Medical Imaging Informatics research group at the Mayo Clinic is developing and applying microlocal analyses of medical images to improve cancer imagenomics. This presentation will discuss some of our recent efforts and results in this area.

Juri Rappoport, Russian Academy of Sciences
Lebedev transforms and some ophthalmic imaging applications

Lebedev transforms and some ophthalmic imaging applications are discussed. The properties of Kontorovitch-Lebedev integral transforms, Lebedev-Skalskaya integral tranforms and modified Bessel functions are elaborated. The approximation and computation of the kernels of transforms are given. The effective applications for some mixed boundary value problems are described. The quantitative and qualitative changes in the transplant endothelium after keratoplasty are analysed.

Luigi Riba, Universita di Torino and York University
Continuous Inversion Formulas for Multi-Dimensional Stockwell Transforms

Stockwell transforms as hybrids of Gabor transforms and wavelet transforms have been studied extensively. Starting from the well known results on the two-diensional Stockwell transform we introduce in this paper multi-dimensional Stockwell transforms that include multi-dimensional Gabor transforms as special cases. Furthermore we give continuous inversion formulas for multi-dimensional Stockwell transforms.

Ervin Sejdic, University of Pittsburgh
Introduction to Time-Frequency Analysis and Its Biomedical Applications

Time-frequency analysis is of great interest when time or frequency based techniques provide insufficient information about signals. Time-frequency representations depict variations of the spectral characteristics of signals as a function of time, which is ideally suited for nonstationary signals, especially, non-stationary biomedical signals. Many biomedical signals (e.g. heart sounds, swallowing accelerometry signals) are multicomponent, one-dimensional signals. The time-frequency analysis of these signals provides a two-dimensional representation of signals' components, which is appropriate for a diagnostic analysis. The resolution, that is, the quality of a representation, depends on a specific time-frequency distribution. Therefore, this talk provides an introduction to time-frequency analysis through an overview of recent contributions.

Hau-Tieng Wu, Princeton University
Instantaneous frequency, shape functions, Synchrosqueezing transform and some applications

Although one can formulate an intuitive notion of instantaneous frequency, generalizing "frequency" as we understand it in e.g. the Fourier transform, a rigorous mathematical definition is lacking. In this talk, we consider a class of functions composed of waveforms that repeat nearly periodically, and for which the instantaneous frequency can be given a rigorous meaning. In other words, we consider the problem of the following form: given a function $$f(t)=\sum_{k=1}^K A_k(t)s_k(\phi_k(t)),\mbox{ with } A_k(t),\phi'_k(t)>0 ~ \forall t,$$ and $s_k$ are $2\pi$ periodic, compute $s_k(t)$, $A_k(t)$ and $\phi'_k(t)$ or describe their properties from $f$. We introduce the Synchrosqueezing transforms, which can be used to determine the instantaneous frequency of functions in this class, even if the waveform is not harmonic. The properties of the Synchrosqueezing transform, like robustness to many kinds of noises, the ability to detect the dynamics of the system will also be discussed. Finally we provide examples in sleep depth detection, ventilator weaning and seasonality detection.

Hau-Tieng Wu, Princeton University
Vector Diffusion Maps, Connection Laplacian and Their Applications

We introduce vector diffusion maps (VDM), a new mathematical framework for organizing and analyzing massive high dimensional data sets, images and shapes. VDM is a mathematical and algorithmic generalization of diffusion maps and other non-linear dimensionality reduction methods, such as LLE, ISOMAP and Laplacian eigenmaps. While existing methods are either directly or indirectly related to the heat kernel for functions over the data, VDM is based on the heat kernel for 1-forms and vector fields. VDM provides tools for organizing complex data sets, embedding them in a low dimensional space, and interpolating and regressing vector fields over the data. In particular, it equips the data with a metric, which we refer to as the {\em vector diffusion distance}. In the manifold learning setup, where the data set is distributed on (or near) a low dimensional manifold $\text{M}^d$ embedded in $\mathbb{R}^p$, we prove the relation between VDM and the connection-Laplacian operator for 1-forms over the manifold. The algorithm is directly applied to the cryo-EM problem and the result will be demonstrated. We will also discuss its application in determining the orient ability of a given manifold.

Hau-Tieng Wu, Princeton University
Two-Dimensional tomography from noisy projections taken at unknown random directions

Computerized Tomography (CT) is a standard method for obtaining internal structure of objects from their projection images. While CT reconstruction requires the knowledge of the imaging directions, there are some situations in which the imaging directions are unknown, for example, when imaging a moving object. It is therefore desirable to design a reconstruction method from projection images taken at unknown directions. Another difficulty arises from the fact that the projections are often contaminated by noise, practically limiting all current methods, including the recently proposed diffusion map approach. In this paper, we introduce two denoising steps that allow reconstructions at much lower signal-to-noise ratios (SNR) when combined with the diffusion map framework. In the first denoising step we use principal component analysis (PCA) together with classical Wiener filtering to derive an asymptotically optimal linear filter. In the second step, we denoise the graph of similarities between the filtered projections using a network analysis measure such as the Jaccard index. Using this combination of PCA, Wiener filtering, graph denoising and diffusion map, we are able to reconstruct the 2-D Shepp-Logan phantom from simulative noisy projections at SNRs well below their currently reported threshold values. We also report the results of a numerical experiment corresponding to an abdominal CT. Although the focus of this paper is the 2-D CT reconstruction problem, we believe that the combination of PCA, Wiener filtering, graph denoising and diffusion maps is potentially useful in other signal processing and image analysis applications.

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