Srikanth V. Tenneti

Nested Periodic Matrices and Dictionaries: New Signal Representations for Period Estimation

In this paper, we propose a new class of techniques to identify periodicities in data. We target the period estimation directly rather than inferring the period from the signals spectrum. By doing so, we obtain several advantages over the traditional spectrum estimation techniques such as DFT and MUSIC. Apart from estimating the unknown period of a signal, we search for finer periodic structure within the given signal. For instance, it might be possible that the given periodic signal was actually a sum of signals with much smaller periods. For example, adding signals with periods 3, 7, and 11 can give rise to a period 231 signal. We propose methods to identify these “hidden periods” 3, 7, and 11. We first propose a new family of square matrices called Nested Periodic Matrices (NPMs), having several useful properties in the context of periodicity. These include the DFT, Walsh-Hadamard, and Ramanujan periodicity transform matrices as examples. Based on these matrices, we develop high dimensional dictionary representations for periodic signals. Various optimization problems can be formulated to identify the periods of signals from such representations. We propose an approach based on finding the least l2 norm solution to an under-determined linear system. Alternatively, the period identification problem can also be formulated as a sparse vector recovery problem and we show that by a slight modification to the usual l1 norm minimization techniques, we can incorporate a number of new and computationally simple dictionaries.

Ramanujan filter banks for estimation and tracking of periodicities

We propose a new filter-bank structure for the estimation and tracking of periodicities in time series data. These filter-banks are inspired from recent techniques on period estimation using high-dimensional dictionary representations for periodic signals. Apart from inheriting the numerous advantages of the dictionary based techniques over conventional period-estimation methods such as those using the DFT, the filter-banks proposed here expand the domain of problems that can be addressed to a much richer set. For instance, we can now characterize the behavior of signals whose periodic nature changes with time. This includes signals that are periodic only for a short duration and signals such as chirps. For such signals, we use a time vs period plane analogous to the traditional time vs frequency plane. We will show that such filter banks have a fundamental connection to Ramanujan Sums and the Ramanujan Periodicity Transform.

Properties of Ramanujan filter banks

This paper studies a class of filter banks called the Ramanujan filter banks which are based on Ramanujan-sums. It is shown that these filter banks have some important mathematical properties which allow them to reveal localized hidden periodicities in real-time data. These are also compared with traditional comb filters which are sometimes used to identify periodicities. It is shown that non-adaptive comb filters cannot in general reveal periodic components in signals unless they are restricted to be Ramanujan filters. The paper also shows how Ramanujan filter banks can be used to generate time-period plane plots which track the presence of time varying, localized, periodic components.

Dictionary approaches for identifying periodicities in data

We propose several dictionary representations for periodic signals and use them for estimating their periodicity. This includes estimating concurrent multiple periodicities. These are inspired from the recently proposed DFT based Farey dictionary, where period estimation was cast as a sparse vector recovery problem. We show that this can instead be framed as an l2 norm based data-fitting problem with closed form solutions and much faster computations. We also generalize the complex valued Farey dictionary to simpler integer valued dictionaries. We find that dictionaries constructed using the recently proposed Ramanujan Periodicity Transforms provide the best trade-off between complexity and noise immunity.

Reconfigurable processor for energy-scalable computational photography

Computational photography applications, such as lightfield photography [1], enable capture and synthesis of images that could not be captured with a traditional camera. Non-linear filtering techniques like bilateral filtering [2] form a significant part of computational photography. These techniques have a wide range of applications, including High-Dynamic Range (HDR) imaging [3], Low-Light Enhanced (LLE) imaging [4], tone management and video enhancement. The high computational complexity of such multimedia processing applications necessitates fast hardware implementations [5] to enable real-time processing. This paper describes a hardware implementation of a reconfigurable multi-application processor for computational photography.

A Unified Theory of Union of Subspaces Representations for Period Estimation

Several popular period estimation techniques use union-of-subspaces models to represent periodic signals. The main idea behind these techniques is to compare the components of the signal along a set of subspaces representing different periods. Such techniques were shown to offer important advantages over traditional methods, such as those based on DFT. So far, most of these subspace techniques have been developed independent of each other, and there has not been a unified theory analyzing them from a common perspective. In this paper, all such methods are first unified under one general framework. Further, several fundamental aspects of such subspaces are investigated, such as the conditions under which a generic set of subspaces offers unique periodic decompositions, their minimum required dimensions, etc. A number of basic questions in the context of dictionaries spanning periodic signals are also answered. For example, what is the theoretically minimum number of atoms required in any type of dictionary, in order to be able to represent periods 1 ≤ P ≤ Pmax? For each period P , what should be the minimum dimension of the subspace of atoms representing the Pth period itself? Unlike in traditional Fourier dictionaries, it is shown that nonuniform and compact grids are crucial for period estimation. Interestingly, it will be seen that the Euler totient function from number theory plays an important role in providing the answers to all such questions.

Arbitrarily Shaped Periods in Multidimensional Discrete Time Periodicity

Traditionally, most of the analysis of discrete time multidimensional periodicity in DSP is based on defining the period as a parallelepiped. In this work, we study whether this framework can incorporate signals that are repetitions of more general shapes than parallelepipeds. For example, the famous Dutch artist M. C. Escher constructed many interesting shapes such as fishes, birds and animals, which can tile the continuous 2-D plane. Inspired from Eschers tilings, we construct discrete time signals that are repetitions of various kinds of shapes. We look at periodicity in the following way - a given shape repeating itself along fixed directions to tile the entire space. By transcribing this idea into a mathematical framework, we explore its relationship with the traditional analysis of periodicity based on parallelepipeds. Our main result is that given any such signal with an arbitrarily shaped period, we can always find an equivalent parallelepiped shaped period that has the same number of points as the original period.

Detecting tandem repeats in DNA using Ramanujan Filter Bank

Tandem repeats are periodic segments in the DNA. They play an important role in forensics, tracing population evolution, genetic disorders and so on. Locating them in long DNA sequences is the problem addressed in this paper. A new technique is presented, based on the recently proposed Ramanujan Filter Bank (RFB). The RFB was shown to offer several advantages over the traditional period estimation techniques in DSP, such as those based on spectral estimation (STFT etc.). It involves simple integer operations, and detects several new repeats that could not be identified by popular existing techniques.1 Project Website: see [16].

Minimal dictionaries for spanning periodic signals

Recently, several high dimensional dictionary representations were proposed for discrete time periodic signals. These dictionaries could span any periodic signal whose period lies in a given range 1 ≤ P ≤ Pmax. Such dictionaries were used in various ways to estimate unknown periods. In this work, we derive some fundamental properties that any such dictionary must satisfy. For example, we derive bounds on the minimum size of such dictionaries, necessary conditions on their composition, and so on. Our results also demonstrate a natural connection between the well-known Euler Totient function (φ-function) from number theory, and periodicity analysis.1

Ramanujan subspaces and digital signal processing

Ramanujan-sums have in the past been used to extract hidden periods. In a recent paper it was shown that for finite duration (FIR) sequences, the traditional representation is not suitable. Two new types of Ramanujan-sum expansions were proposed for the FIR case, each offering an integer basis, and applications in the extraction of hidden periodicities were developed. Crucial to these developments was the introduction of Ramanujan spaces. The aim of this paper is to develop some properties of these subspaces in the context of signal processing. The design of near orthogonal bases for these spaces is emphasized.