D. Van De Ville

Noise reduction by fuzzy image filtering

A new fuzzy filter is presented for the noise reduction of images corrupted with additive noise. The filter consists of two stages. The first stage computes a fuzzy derivative for eight different directions. The second stage uses these fuzzy derivatives to perform fuzzy smoothing by weighting the contributions of neighboring pixel values. Both stages are based on fuzzy rules which make use of membership functions. The filter can be applied iteratively to effectively reduce heavy noise. In particular, the shape of the membership functions is adapted according to the remaining noise level after each iteration, making use of the distribution of the homogeneity in the image. A statistical model for the noise distribution can be incorporated to relate the homogeneity to the adaptation scheme of the membership functions. Experimental results are obtained to show the feasibility of the proposed approach. These results are also compared to other filters by numerical measures and visual inspection.

Multiresolution Monogenic Signal Analysis Using the Riesz–Laplace Wavelet Transform

The monogenic signal is the natural 2D counterpart of the 1D analytic signal. We propose to transpose the concept to the wavelet domain by considering a complexified version of the Riesz transform which has the remarkable property of mapping a real-valued (primary) wavelet basis of L2(R2) into a complex one. The Riesz operator is also steerable in the sense that it give access to the Hilbert transform of the signal along any orientation. Having set those foundations, we specify a primary polyharmonic spline wavelet basis of L2(R2) that involves a single Mexican-hat-like mother wavelet (Laplacian of a B-spline). The important point is that our primary wavelets are quasi-isotropic: they behave like multiscale versions of the fractional Laplace operator from which they are derived, which ensures steerability. We propose to pair these real-valued basis functions with their complex Riesz counterparts to specify a multiresolution monogenic signal analysis. This yields a representation where each wavelet index is associated with a local orientation, an amplitude and a phase. We give a corresponding wavelet-domain method for estimating the underlying instantaneous frequency. We also provide a mechanism for improving the shift and rotation-invariance of the wavelet decomposition and show how to implement the transform efficiently using perfect-reconstruction filterbanks. We illustrate the specific feature-extraction capabilities of the representation and present novel examples of wavelet-domain processing; in particular, a robust, tensor-based analysis of directional image patterns, the demodulation of interferograms, and the reconstruction of digital holograms.

Isotropic polyharmonic B-splines: scaling functions and wavelets

In this paper, we use polyharmonic B-splines to build multidimensional wavelet bases. These functions are nonseparable, multidimensional basis functions that are localized versions of radial basis functions. We show that Rabuts elementary polyharmonic B-splines do not converge to a Gaussian as the order parameter increases, as opposed to their separable B-spline counterparts. Therefore, we introduce a more isotropic localization operator that guarantees this convergence, resulting into the isotropic polyharmonic B-splines. Next, we focus on the two-dimensional quincunx subsampling scheme. This configuration is of particular interest for image processing because it yields a finer scale progression than the standard dyadic approach. However, up until now, the design of appropriate filters for the quincunx scheme has mainly been done using the McClellan transform. In our approach, we start from the scaling functions, which are the polyharmonic B-splines and, as such, explicitly known, and we derive a family of polyharmonic spline wavelets corresponding to different flavors of the semi-orthogonal wavelet transform; e.g., orthonormal, B-spline, and dual. The filters are automatically specified by the scaling relations satisfied by these functions. We prove that the isotropic polyharmonic B-spline wavelet converges to a combination of four Gabor atoms, which are well separated in the frequency domain. We also show that these wavelets are nearly isotropic and that they behave as an iterated Laplacian operator at low frequencies. We describe an efficient fast Fourier transform-based implementation of the discrete wavelet transform based on polyharmonic B-splines.

Fuzzy Filters for Image Processing

1. Fuzzy Filters for Noise Removal.- 2. Fuzzy Filters for Noise Reduction in Images.- 3. Real-time Image Noise Cancellation Based on Fuzzy Similarity.- 4. Fuzzy Rule-Based Color Filtering Using Statistical Indices.- 5. Fuzzy Based Image Segmentation.- 6. Fuzzy Thresholding and Histogram Analysis.- 7. Color Image Segmentation by Analysis of 3D Histogram with Fuzzy Morphological Filters.- 8. Fast and Robust Fuzzy Edge Detection.- 9. Fuzzy Data Fusion for Multiple Cue Image and Video Segmentation.- 10. Fuzzy Image Enhancement in the Framework of Logarithmic Models.- 11. Observer-Dependent Image Enhancement.- 12. Fuzzy Techniques in Digital Image Processing and Shape Analysis.- 13. Adaptive Fuzzy Filters and Their Application to Online Maneuvering Target Tracking.- 14. Lossy Image Compression and Reconstruction Based on Fuzzy Relational Equation.- 15. Avoidance of Highlights through ILFOs in Automated Visual Inspection.- Appendix Color Images.

Model-Based 2.5-D Deconvolution for Extended Depth of Field in Brightfield Microscopy

Due to the limited depth of field of brightfield microscopes, it is usually impossible to image thick specimens entirely in focus. By optically sectioning the specimen, the in-focus information at the specimens surface can be acquired over a range of images. Commonly based on a high-pass criterion, extended-depth-of-field methods aim at combining the in-focus information from these images into a single image of the texture on the specimens surface. The topography provided by such methods is usually limited to a map of selected in-focus pixel positions and is inherently discretized along the axial direction, which limits its use for quantitative evaluation. In this paper, we propose a method that jointly estimates the texture and topography of a specimen from a series of brightfield optical sections; it is based on an image formation model that is described by the convolution of a thick specimen model with the microscopes point spread function. The problem is stated as a least-squares minimization where the texture and topography are updated alternately. This method also acts as a deconvolution when the in-focus PSF has a blurring effect, or when the true in-focus position falls in between two optical sections. Comparisons to state-of-the-art algorithms and experimental results demonstrate the potential of the proposed approach.

Practical Box Splines for Reconstruction on the Body Centered Cubic Lattice

We introduce a family of box splines for efficient, accurate, and smooth reconstruction of volumetric data sampled on the body-centered cubic (BCC) lattice, which is the favorable volumetric sampling pattern due to its optimal spectral sphere packing property. First, we construct a box spline based on the four principal directions of the BCC lattice that allows for a linear C0 reconstruction. Then, the design is extended for higher degrees of continuity. We derive the explicit piecewise polynomial representations of the C0 and C2 box splines that are useful for practical reconstruction applications. We further demonstrate that approximation in the shift-invariant space - generated by BCC-lattice shifts of these box splines - is twice as efficient as using the tensor-product B-spline solutions on the Cartesian lattice (with comparable smoothness and approximation order and with the same sampling density). Practical evidence is provided demonstrating that the BCC lattice not only is generally a more accurate sampling pattern, but also allows for extremely efficient reconstructions that outperform tensor-product Cartesian reconstructions.

Hex-splines: a novel spline family for hexagonal lattices

This paper proposes a new family of bivariate, nonseparable splines, called hex-splines, especially designed for hexagonal lattices. The starting point of the construction is the indicator function of the Voronoi cell, which is used to define in a natural way the first-order hex-spline. Higher order hex-splines are obtained by successive convolutions. A mathematical analysis of this new bivariate spline family is presented. In particular, we derive a closed form for a hex-spline of arbitrary order. We also discuss important properties, such as their Fourier transform and the fact they form a Riesz basis. We also highlight the approximation order. For conventional rectangular lattices, hex-splines revert to classical separable tensor-product B-splines. Finally, some prototypical applications and experimental results demonstrate the usefulness of hex-splines for handling hexagonally sampled data.

Machine Learning with Brain Graphs: Predictive Modeling Approaches for Functional Imaging in Systems Neuroscience

The observation and description of the living brain has attracted a lot of research over the past centuries. Many noninvasive imaging modalities have been developed, such as topographical techniques based on the electromagnetic field potential [i.e., electroencephalography (EEG) and magnetoencephalography (MEG)], and tomography approaches including positron emission tomography and magnetic resonance imaging (MRI). Here we will focus on functional MRI (fMRI) since it is widely deployed for clinical and cognitive neurosciences today, and it can reveal brain function due to neurovascular coupling (see ?From Brain Images to fMRI Time Series?). It has led to a much better understanding of brain function, including the description of brain areas with very specialized functions such as face recognition. These neuroscientific insights have been made possible by important methodological advances in MR physics, signal processing, and mathematical modeling.

Wavelet Steerability and the Higher-Order Riesz Transform

Our main goal in this paper is to set the foundations of a general continuous-domain framework for designing steerable, reversible signal transformations (a.k.a. frames) in multiple dimensions (d ¿ 2). To that end, we introduce a self-reversible, Nth-order extension of the Riesz transform. We prove that this generalized transform has the following remarkable properties: shift-invariance, scale-invariance, inner-product preservation, and steerability. The pleasing consequence is that the transform maps any primary wavelet frame (or basis) of L 2( \BBR d) into another ¿steerable¿ wavelet frame, while preserving the frame bounds. The concept provides a functional counterpart to Simoncellis steerable pyramid whose construction was primarily based on filterbank design. The proposed mechanism allows for the specification of wavelets with any order of steerability in any number of dimensions; it also yields a perfect reconstruction filterbank algorithm. We illustrate the method with the design of a novel family of multidimensional Riesz-Laplace wavelets that essentially behave like the N th-order partial derivatives of an isotropic Gaussian kernel.

Dynamic PET Reconstruction Using Wavelet Regularization With Adapted Basis Functions

Tomographic reconstruction from positron emission tomography (PET) data is an ill-posed problem that requires regularization. An attractive approach is to impose an lscr1-regularization constraint, which favors sparse solutions in the wavelet domain. This can be achieved quite efficiently thanks to the iterative algorithm developed by Daubechies et al., 2004. In this paper, we apply this technique and extend it for the reconstruction of dynamic (spatio-temporal) PET data. Moreover, instead of using classical wavelets in the temporal dimension, we introduce exponential-spline wavelets (E-spline wavelets) that are specially tailored to model time activity curves (TACs) in PET. We show that the exponential-spline wavelets naturally arise from the compartmental description of the dynamics of the tracer distribution. We address the issue of the selection of the ldquooptimalrdquo E-spline parameters (poles and zeros) and we investigate their effect on reconstruction quality. We demonstrate the usefulness of spatio-temporal regularization and the superior performance of E-spline wavelets over conventional Battle-Lemarie wavelets in a series of experiments: the 1-D fitting of TACs, and the tomographic reconstruction of both simulated and clinical data. We find that the E-spline wavelets outperform the conventional wavelets in terms of the reconstructed signal-to-noise ratio (SNR) and the sparsity of the wavelet coefficients. Based on our simulations, we conclude that replacing the conventional wavelets with E-spline wavelets leads to equal reconstruction quality for a 40% reduction of detected coincidences, meaning an improved image quality for the same number of counts or equivalently a reduced exposure to the patient for the same image quality.