Morteza Shahram

Reproducible Research in Computational Harmonic Analysis

Scientific computation is emerging as absolutely central to the scientific method. Unfortunately, its error-prone and currently immature-traditional scientific publication is incapable of finding and rooting out errors in scientific computation-which must be recognized as a crisis. An important recent development and a necessary response to the crisis is reproducible computational research in which researchers publish the article along with the full computational environment that produces the results. In this article, the authors review their approach and how it has evolved over time, discussing the arguments for and against working reproducibly.

ShearLab: A Rational Design of a Digital Parabolic Scaling Algorithm

Multivariate problems are typically governed by anisotropic features such as edges in images. A common bracket of most of the various directional representation systems which have been proposed to deliver sparse approximations of such features is the utilization of parabolic scaling. One prominent example is the shearlet system. Our objective in this paper is threefold: We first develop a digital shearlet theory which is rationally designed in the sense that it is the digitization of the existing shearlet theory for continuous data. This implies that shearlet theory provides a unified treatment of both the continuum and digital realms. Second, we analyze the utilization of pseudo-polar grids and the pseudo-polar Fourier transform for digital implementations of parabolic scaling algorithms. We derive an isometric pseudo-polar Fourier transform by careful weighting of the pseudo-polar grid, allowing exploitation of its adjoint for the inverse transform. This leads to a digital implementation of the shearlet...

On the resolvability of sinusoids with nearby frequencies in the presence of noise

This correspondence develops statistical algorithms and performance limits for resolving sinusoids with nearby frequencies in the presence of noise. We address the problem of distinguishing whether the received signal is a single-frequency sinusoid or a double-frequency sinusoid, with possibly unequal, and unknown, amplitudes and phases. We derive a locally optimal detection strategy that can be applied in a standalone fashion or as a refinement step for existing spectral estimation methods to yield improved performance. We further derive explicit relationships between the minimum detectable difference between the frequencies of two tones for any particular false alarm and detection rate and at a given SNR.

Complex data analysis in high‐resolution SSFP fMRI

In transition‐band steady‐state free precession (SSFP) functional MRI (fMRI), functional contrast originates from a bulk frequency shift induced by a deoxygenated hemoglobin concentration change in the activated brain regions. This frequency shift causes a magnitude and/or phase‐signal change depending on the off‐resonance distribution of a voxel in the balanced‐SSFP (bSSFP) profile. However, in early low‐resolution studies, only the magnitude signal activations were shown. In this paper the task‐correlated phase‐signal change is presented in a high‐resolution (1 × 1 × 1 mm3) study. To include this phase activation in a functional analysis, a new complex domain data analysis method is proposed. The results show statistically significant phase‐signal changes in a large number of voxels comparable to that of the magnitude‐activated voxels. The complex‐data analysis method successfully includes these phase activations in the activation map and thus provides wider coverage compared to magnitude‐data analysis results. Magn Reson Med 57:905–917, 2007.

Development of a digital shearlet transform based on Pseudo-Polar FFT

Shearlab is a Matlab toolbox for digital shearlet transformation of two-D (image) data we developed following a rational design process. The Pseudo-Polar FFT fits very naturally with the continuum theory of the Shearlet transform and allows us to translate Shearlet ideas naturally into a digital framework. However, there are still windows and weights which must be chosen. We developed more than a dozen performance measures quantifying precision of the reconstruction, tightness of the frame, directional and spatial localization and other properties. Such quantitative performance metrics allow us to: (a) tune parameters and objectively improve our implementation; and (b) compare different directional transform implementations. We present and interpret the most important performance measures for our current implementation.

ECG beat classification based on a cross-distance analysis

This paper presents a multi-stage algorithm for QRS complex classification into normal and abnormal categories using an unsupervised sequential beat clustering and a cross-distance analysis algorithm. After the sequential beat clustering, a search algorithm based on relative similarity of created classes is used to detect the main normal class. Then other classes are labeled based on a distance measurement from the main normal class. Evaluated results on the MIT-BIH ECG database exhibits an error rate less than 1% for normal and abnormal discrimination and 0.2% for clustering of 15 types of arrhythmia existing on the MIT-BIH database.

Multiscale representation for data on the sphere and applications to geopotential data

We develop a wavelet transform on the sphere, based on the spherical HEALPix coordinate system (Hierarchical Equal Area iso-Latitude Pixelization). HEALPix is heavily used for astronomical data processing applications; it is intrinsically multiscale and locally euclidean, hence appealing for building multiscale system. Furthermore, the equal-area pixelization enables us to employ average-interpolating refinement, giving wavelets of local support. HEALPix wavelets have numerous applications in geopotential modeling. A statistical analysis demonstrates wavelet compressibility of the geopotential field and shows that geopotential wavelet coefficients have many of the statistical properties that were previously observed with wavelet coefficients of natural images. The HEALPix wavelet expansion allows to evaluate a gravimetric quantity over a local region far more rapidly than the classic approach based on spherical harmonics. Our software tools are specifically tailored to demonstrate these advantages.

Combining complex signal change in functional MRI

With the development of functional magnetic resonance imaging (fMRI) techniques, data analysis methods based on complex MR data have been proposed. However, the methods have not been popular for fMRI community, in part because the phase activation in conventional GRE fMRI has been suggested to originate from the large veins (1). Recently, novel fMRI methods such as transition-band SSFP fMRI and an alternating balanced SSFP method for neuronal current measurement (2) have been proposed. In these methods, the functional contrasts exist in the complex domain, providing significant and localized complex signal change. Hence, the usefulness of the complex-data analysis methods has become increasingly important for these applications by allowing them to reliably obtain complex activation. As mentioned in his letter, Dr. Rowe has proposed a complex-data analysis method based on the generalized likelihood ratio test (3). Despite the usefulness of the method, the computational complexity of the method, which requires multiple iterations to estimate the parameters, hampers the routine use of the method. This is particularly true for high-resolution studies that we targeted in our study. To overcome this inefficiency, we have proposed a new method based on T 2 statistics combined with generalized linear model (4). Dr. Rowe’s letter expressed concerns about the relationship of our model to his model and some mathematical errors. Here we present our responses to his points:

Local detectors for high-resolution spectral analysis: Algorithms and performance

This paper develops local signal detection strategies for spectral resolution of frequencies of nearby tones. The problem of interest is to decide whether a received noise-corrupted and discrete signal is a single-frequency sinusoid or a double-frequency sinusoid. This paper presents an extension to M. Shahram and P. Milanfar (On the resolvability of sinusoids with nearby frequencies in the presence of noise, IEEE Trans. Signal Process., to appear, available at http://www.soe.ucsc.edu/~milanfar) the case where the noise variance is unknown. A general signal model is considered where the frequencies, amplitudes, phases and also the level of the noise variance is unknown to the detector. We derive a fundamental trade-off between SNR and the minimum detectable difference between the frequencies of two tones, for any desired decision error rate. We also demonstrate that the algorithm, when implemented in a practical scenario, yields significantly better performance compared to the standard subspace-based methods like MUSIC. It is also observed that the performance for the case where the noise variance is unknown, is very close to that when the noise variance is known to the detector. r.

Improved spectral analysis of nearby tones using local detectors

This paper concerns the problem of resolvability power in the frequency domain. The canonical case of interest is to distinguish whether the received noise-corrupted signal is a single-frequency sinusoid or a two-frequency sinusoid, where the amplitudes, phases and frequencies are unknown to the receiver. Using a model-based hypothesis testing approach, we quantify a measure of attainable resolution between sinusoids with nearby frequencies, in the presence of noise. An explicit relationship is derived for the minimum detectable difference between the frequencies of two tones, for any particular false alarm and detection rate, and at a given SNR. An associated algorithm is proposed that produces significantly better performance compared to the standard subspace-based methods like MUSIC and can be effectively used in practice as a postprocessing step for the existing spectral estimation methods.