Jalal M. Fadili

Colored noise and computational inference in neurophysiological (fMRI) time series analysis: resampling methods in time and wavelet domains.

Even in the absence of an experimental effect, functional magnetic resonance imaging (fMRI) time series generally demonstrate serial dependence. This colored noise or endogenous autocorrelation typically has disproportionate spectral power at low frequencies, i.e., its spectrum is  f–1 ‐like. Various pre‐whitening and pre‐coloring strategies have been proposed to make valid inference on standardised test statistics estimated by time series regression in this context of residually autocorrelated errors. Here we introduce a new method based on random permutation after orthogonal transformation of the observed time series to the wavelet domain. This scheme exploits the general whitening or decorrelating property of the discrete wavelet transform and is implemented using a Daubechies wavelet with four vanishing moments to ensure exchangeability of wavelet coefficients within each scale of decomposition. For  f–1 ‐like or fractal noises, e.g., realisations of fractional Brownian motion (fBm) parameterised by Hurst exponent 0 < H < 1, this resampling algorithm exactly preserves wavelet‐based estimates of the second order stochastic properties of the (possibly nonstationary) time series. Performance of the method is assessed empirically using  f–1 ‐like noise simulated by multiple physical relaxation processes, and experimental fMRI data. Nominal type 1 error control in brain activation mapping is demonstrated by analysis of 13 images acquired under null or resting conditions. Compared to autoregressive pre‐whitening methods for computational inference, a key advantage of wavelet resampling seems to be its robustness in activation mapping of experimental fMRI data acquired at 3 Tesla field strength. We conclude that wavelet resampling may be a generally useful method for inference on naturally complex time series. Hum. Brain Mapping 12:61–78, 2001.

Morphological Component Analysis: An Adaptive Thresholding Strategy

In a recent paper, a method called morphological component analysis (MCA) has been proposed to separate the texture from the natural part in images. MCA relies on an iterative thresholding algorithm, using a threshold which decreases linearly towards zero along the iterations. This paper shows how the MCA convergence can be drastically improved using the mutual incoherence of the dictionaries associated to the different components. This modified MCA algorithm is then compared to basis pursuit, and experiments show that MCA and BP solutions are similar in terms of sparsity, as measured by the lscr1 norm, but MCA is much faster and gives us the possibility of handling large scale data sets.

Overcoming Confounds of Stimulus Blocking: An Event-Related fMRI Design of Semantic Processing

The way in which meaning is represented and processed in the brain is a key issue in cognitive neuroscience, which can be usefully addressed by functional imaging techniques. In contrast to previous imaging studies of semantic knowledge, which have primarily used blocked designs, in this study we use an event-related fMRI (erfMRI) design, which has the advantage of enabling events to be presented pseudorandomly, thus reducing strategic processes and enabling more direct comparison with psychological behavioral studies. We used a semantic categorization task in which events were words representing either artifact or natural kinds concepts. Significant areas of activation for semantic processing included inferior frontal lobe bilaterally (BA 47) and left temporal regions, both inferior (BA 36 and 20) and middle (BA 21). These are areas that have been identified in previous neuroimaging studies of semantic knowledge. However, there were no significant differences between artifact and natural kinds concepts. These results are consistent with our previous imaging studies using blocked designs and suggest that conceptual knowledge is represented in a unitary, distributed neural system undifferentiated by domain of knowledge. These findings demonstrate that event-related designs can generate activations that are similar to those seen in blocked designs investigating semantics and, moreover, offer a greater capacity for interpretation free from the confounds of block effects.

Effect of slice orientation on reproducibility of fMRI motor activation at 3 Tesla

The effect of slice orientation on reproducibility and sensitivity of 3T fMRI activation using a motor task has been investigated in six normal volunteers. Four slice orientations were used; axial, oblique axial, coronal and sagittal. We applied analysis of variance (ANOVA) to suprathreshold voxel statistics to quantify variability in activation between orientations and between subjects. We also assessed signal detection accuracy in voxels across the whole brain by using a finite mixture model to fit receiver operating characteristic (ROC) curves to the data. Preliminary findings suggest that suprathreshold cluster characteristics demonstrate high motor reproducibility across subjects and orientations, although a significant difference between slice orientations in number of activated voxels was demonstrated in left motor cortex but not cerebellum. Subtle inter-orientation differences are highlighted in the ROC analyses, which are not obvious by ANOVA; the oblique axial slice orientation offers the highest signal detection accuracy, whereas coronal slices give the lowest.

Stein block thresholding for wavelet-based image deconvolution

In this paper, we propose a fast image deconvolution algorithm that combines adaptive block thresholding and Vaguelet-Wavelet Decomposition. The approach consists in first denoising the observed image using a wavelet-domain Stein block thresholding, and then inverting the convolution operator in the Fourier domain. Our main theoretical result investigates the minimax rates over Besov smoothness spaces, and shows that our block estimator can achieve the optimal minimax rate, or is at least nearlyminimax in the least favorable situation. The resulting algorithm is simple to implement and fast. Its computational complexity is dominated by that of the FFT in the Fourier-domain inversion step. We report a simulation study to support our theoretical findings. The practical performance of our block vaguelet-wavelet deconvolution compares very favorably to existing competitors on a large set of test images.

Sparse Image and Signal Processing: The Ridgelet and Curvelet Transforms

INTRODUCTION The ridgelet and curvelet transforms generalize the wavelet transform. First, they incorporate angular alignment information, and then, in addition, the length of the alignment is covered. As with all of these transforms, multiple scales are supported. The motivation for these transforms is to build up an image from edge-related building blocks. Furthermore, as in previous chapters, the efficiency of computing these transforms is an important practical aspect. In this chapter, we consider the ridgelet transform and a number of algorithms for its implementation. Then we proceed to the curvelet transform and algorithms for it. BACKGROUND AND EXAMPLE Wavelets rely on a dictionary of roughly isotropic elements occurring at all scales and locations. They do not describe well highly anisotropic elements and contain only a fixed number of directional elements, independent of scale. Despite the fact that they have had wide impact in image processing, they fail to efficiently represent objects with highly anisotropic elements such as lines or curvilinear structures (e.g., edges). The reason is that wavelets are nongeometrical and do not exploit the regularity of the edge curve. Following this reasoning, new constructions have been proposed such as ridgelets (Candes and Donoho 1999) and curvelets (Candes and Donoho 2001, 2002; Starck et al. 2002). Ridgelets and curvelets are special members of the family of multiscale orientation-selective transforms, which have recently led to a flurry of research activity in the field of computational and applied harmonic analysis.

3D Sparse Representations

In this chapter, we review a variety of 3-D sparse representations developed in recent years and adapted to different kinds of 3-D signals. In particular, we describe 3-D wavelets, ridgelets, beamlets, and curvelets. We also present very recent 3-D sparse representations on the 3-D ball that has been adapted to 3-D signals naturally observed in spherical coordinates. Illustrative examples are provided for the different transforms.

Poisson noise removal with pyramidal multi-scale transforms

In this paper, we introduce a method to stabilize the variance of decimated transforms using one or two variance stabilizing transforms (VST). These VSTs are applied to the 3-D Meyer wavelet pyramidal transform which is the core of the first generation 3D curvelets. This allows us to extend these 3-D curvelets to handle Poisson noise, that we apply to the denoising of a simulated cosmological volume.

Poisson Noise Removal in Spherical Multichannel Images: Application to Fermi data

The aim of this chapter is to present a multi-scale representation for spherical data with Poisson noise called Multi-Scale Variance Stabilizing Transform on the Sphere (MS-VSTS) [14], combining the MS-VST [25] with various multi-scale transforms on the sphere (wavelets and curvelets) [22, 2, 3]. Section 1.2 presents some multi-scale transforms on the sphere. Section 1.3 introduces a new multi-scale representation for data with Poisson noise called MS-VSTS. Section 1.4 applies this representation to Poisson noise removal on Fermi data. Section 1.5 presents applications to missing data interpolation and source extraction. Section 1.6 extends the method to multichannel data.