Denosing Using Wavelets and Projections onto the L1-Ball

Both wavelet denoising and denosing methods using the concept of sparsity are based on soft-thresholding. In sparsity based denoising methods, it is assumed that the original signal is sparse in some transform domains such as the wavelet domain and the wavelet subsignals of the noisy signal are projected onto L1-balls to reduce noise. In this lecture note, it is shown that the size of the L1-ball or equivalently the soft threshold value can be determined using linear algebra. The key step is an orthogonal projection onto the epigraph set of the L1-norm cost function.Comment: Submitted to Signal Processing Magazin

Phase and TV Based Convex Sets for Blind Deconvolution of Microscopic Images

In this article, two closed and convex sets for blind deconvolution problem are proposed. Most blurring functions in microscopy are symmetric with respect to the origin. Therefore, they do not modify the phase of the Fourier transform (FT) of the original image. As a result blurred image and the original image have the same FT phase. Therefore, the set of images with a prescribed FT phase can be used as a constraint set in blind deconvolution problems. Another convex set that can be used during the image reconstruction process is the epigraph set of Total Variation (TV) function. This set does not need a prescribed upper bound on the total variation of the image. The upper bound is automatically adjusted according to the current image of the restoration process. Both of these two closed and convex sets can be used as a part of any blind deconvolution algorithm. Simulation examples are presented.Comment: Submitted to IEEE Selected Topics in Signal Processin

Cosine Similarity Measure According to a Convex Cost Function

In this paper, we describe a new vector similarity measure associated with a convex cost function. Given two vectors, we determine the surface normals of the convex function at the vectors. The angle between the two surface normals is the similarity measure. Convex cost function can be the negative entropy function, total variation (TV) function and filtered variation function. The convex cost function need not be differentiable everywhere. In general, we need to compute the gradient of the cost function to compute the surface normals. If the gradient does not exist at a given vector, it is possible to use the subgradients and the normal producing the smallest angle between the two vectors is used to compute the similarity measure

Period estimation of an almost periodic signal using persistent homology with application to respiratory rate measurement

Time-frequency techniques have difficulties in yielding efficient online algorithms for almost periodic signals. We describe a new topological method to find the period of signals that have an almost periodic waveform. Proposed method is applied to signals received from a pyro-electric infrared sensor array for the online estimation of the respiratory rate (RR) of a person. Timevarying analog signals captured from the sensors exhibit an almost periodic behavior due to repetitive nature of breathing activity. Sensor signals are transformed into two-dimensional point clouds with a technique that allows preserving the period information. Features, which represent the harmonic structures in the sensor signals, are detected by applying persistent homology and the RR is estimated based on the persistence barcode of the first Betti number. Experiments have been carried out to show that our method makes reliable estimates of the RR. © 2017 IEEE

Pulse shape design using iterative projections

In this paper, the pulse shape design for various communication systems including PAM, FSK, and PSK is considered. The pulse is designed by imposing constraints on the time and frequency domains constraints on the autocorrelation function of the pulse shape. Intersymbol interference, finite duration and spectral mask restrictions are a few examples leading to convex sets in L 2. The autocorrelation function of the pulse is obtained by performing iterative projections onto convex sets. After this step, the minimum phase or maximum phase pulse producing the autocorrelation function is obtained by cepstral deconvolution