We present a framework for the design of wavelet transforms tailored to data defined on multislice graphs (i.e., multiplex or dynamic graphs). Graphs with multiple types of interactions are ubiquitous in real life, motivating the extension of wavelets to these complex domains. Our framework generalizes the recently proposed spectral graph wavelet transform (SGWT) [D. Hammond, P. Vandergheynst, and R. Gribonval, “Wavelets on Graphs via Spectral Graph Theory,” Appl. Comput. Harmon. Anal., vol. 30, pp. 129-150, Mar. 2011], which is designed in the spectral (frequency) domain of an arbitrary finite weighted graph. We extend the SGWT to form a tight frame, which conserves energy in the wavelet domain, and define the relationship between conventional and spectral graph wavelets. We then propose a design for multislice graphs that is based on the higher-order singular value decomposition (HOSVD), a powerful tool from multilinear algebra. In particular, the multiple adjacency matrices are stacked to form a tensor and the HOSVD decomposition provides information about its third-order structure, analogous to that provided by matrix factorizations. We obtain a set of “eigennetworks” and from these graph wavelets, which exploit the variability across the graphs. We demonstrate the feasibility of our method 1) by capturing different orientations of a gray-scale image and 2) by decomposing brain signals from functional magnetic resonance imaging. We show its effectiveness to identify variability across graph edges and provide meaningful decompositions.

/pdf/tight-wavelet-frames-on-multislice-graphs-4mvptds941.pdf

Tight Wavelet Frames on Multislice Graphs

In this paper, we present a novel framework for quantifying physiological stress at a distance via thermal imaging. The method captures stress-induced neurophysiological responses on the perinasal area that manifest as transient perspiration. We have developed two algorithms to extract the perspiratory signals from the thermophysiological imagery. One is based on morphology and is computationally efficient, while the other is based on spatial isotropic wavelets and is flexible; both require the support of a reliable facial tracker. We validated the two algorithms against the clinical standard in a controlled lab experiment where orienting responses were invoked on n=18 subjects via auditory stimuli. Then, we used the validated algorithms to quantify stress of surgeons (n=24) as they were performing suturing drills during inanimate laparoscopic training. This is a field application where the new methodology shines. It allows nonobtrusive monitoring of individuals who are naturally challenged with a task that is localized in space and requires directional attention. Both algorithms associate high stress levels with novice surgeons, while low stress levels are associated with experienced surgeons, raising the possibility for an affective measure (stress) to assist in efficacy determination. It is a clear indication of the methodology's promise and potential.

/pdf/perinasal-imaging-of-physiological-stress-and-its-affective-4jxr4vjfkb.pdf

Perinasal Imaging of Physiological Stress and Its Affective Potential

We present a functional framework for the design of tight steerable wavelet frames in any number of dimensions. The 2-D version of the method can be viewed as a generalization of Simoncelli's steerable pyramid that gives access to a larger palette of steerable wavelets via a suitable parametrization. The backbone of our construction is a primal isotropic wavelet frame that provides the multiresolution decomposition of the signal. The steerable wavelets are obtained by applying a one-to-many mapping (N th-order generalized Riesz transform) to the primal ones. The shaping of the steerable wavelets is controlled by an M × M unitary matrix (where M is the number of wavelet channels) that can be selected arbitrarily; this allows for a much wider range of solutions than the traditional equiangular configuration (steerable pyramid). We give a complete functional description of these generalized wavelet transforms and derive their steering equations. We describe some concrete examples of transforms, including some built around a Mallat-type multiresolution analysis of L2(Rd), and provide a fast Fourier transform-based decomposition algorithm. We also propose a principal-component-based method for signal adapted wavelet design. Finally, we present some illustrative examples together with a comparison of the denoising performance of various brands of steerable transforms. The results are in favor of an optimized wavelet design (equalized principal component analysis), which consistently performs best.

/pdf/steerable-pyramids-and-tight-wavelet-frames-in-l-2-bbr-d-7oi35wqo6p.pdf

Steerable Pyramids and Tight Wavelet Frames in $L_{2}({\BBR}^{d})$

We introduce a complete parameterization of the family of two-dimensional steerable wavelets that are polar-separable in the Fourier domain under the constraint of self-reversibility. These wavelets are constructed by multiorder generalized Riesz transformation of a primary isotropic bandpass pyramid. The backbone of the transform (pyramid) is characterized by a radial frequency profile function $h(\omega)$, while the directional wavelet components at each scale are encoded by an $M \times (2N+1)$ shaping matrix ${\bf U}$, where $M$ is the number of wavelet channels and $N$ the order of the Riesz transform. We provide general conditions on $h(\omega)$ and ${\bf U}$ for the underlying wavelet system to form a tight frame of $L_2(\mathbb{R}^2)$ (with a redundancy factor $4/3M$). The proposed framework ensures that the wavelets are steerable and provides new degrees of freedom (shaping matrix ${\bf U}$) that can be exploited for designing specific wavelet systems. It encompasses many known transforms as part...

http://bigwww.epfl.ch/publications/unser1301.pdf

A Unifying Parametric Framework for 2D Steerable Wavelet Transforms

In this paper we investigate Isotropic Multiresolution Analysis (IMRA), isotropic refinable functions, and wavelets. The main results are the characterization of IMRAs in terms of the Lax–Wiener Theorem, and the characterization of isotropic refinable functions in terms of the support of their Fourier transform. As an immediate consequence of these results, there are no compactly supported (in the space domain) isotropic refinable functions in many dimensions. Next we study the approximation properties of IMRAs. Finally, we discuss the application of IMRA wavelets to 2D and 3D-texture segmentation in natural and biomedical images.

/pdf/the-geometry-and-the-analytic-properties-of-isotropic-4r52rsd6uq.pdf

The geometry and the analytic properties of isotropic multiresolution analysis

We analyze localized textural consistencies in high-resolution X-ray (computed tomography) CT scans of coronary arteries to identify the appearance of diagnostically relevant changes in tissue. For the efficient and accurate processing of CT volume data, we use fast wavelet algorithms associated with three-dimensional isotropic multiresolution wavelets that implement a redundant, frame-based image encoding without directional preference. Our algorithm identifies textural consistencies by correlating coefficients in the wavelet representation.

/pdf/texture-based-tissue-characterization-for-high-resolution-ct-1jmn34gfkx.pdf

Juan R. Romero

Papers

The geometry and the analytic properties of isotropic multiresolution analysis

Texture-based tissue characterization for high-resolution CT scans of coronary arteries