Chromatic Derivatives Research Articles

An Adaptive Denoising Algorithm for Improving Frequency Estimation and Tracking

We propose a novel adaptive denoising algorithm which, in presence of high levels of noise, significantly improves super-resolution and noise robustness of standard frequency estimation algorithms, such as (root-)MUSIC and ESPRIT. During the course of its operation, the algorithm dynamically estimates the power spectral density of noise and adapts to it. In addition, the proposed denoising front-end allows signal samples to be non-uniform, enabling the standard frequency estimation algorithms to achieve the same super-resolution, accuracy and noise robustness for non-uniformly sampled signals as for uniformly sampled signals of the same sample density. Extensive numerical tests verify superior denoising performance compared to the standard Cadzow method, especially when the noise present is not white. Our algorithm exploits salient features of numerically robust differential operators known as chromatic derivatives and the associated chromatic approximations which provide a method for digital processing of continuous time signals superior to processing which operates directly on their discrete samples.

Signal recovery algorithm for 2-level amplitude sampling using chromatic signal approximations

A novel, highly accurate signal reconstruction algorithm for signal recovery from 2-level amplitude sampling is proposed. Our method is non-iterative and exploits high fidelity local signal approximation properties of special, numerically robust differential operators known as chromatic derivatives and the associated chromatic signal expansions. Algorithms are proposed to obtain highly accurate timing information from 2-level amplitude sampling followed by signal reconstruction from thus obtained non-uniform time stamps. Numerical simulations with synthetic sinusoidal and standard speech inputs verify signal reconstruction accuracy better than 200 dB and 100 dB, respectively, in terms of the mean square error (MSE). With level crossings corresponding to two amplitude levels that are very closely placed and asymmetric (i.e., either on only the positive or the negative side of the signal), the proposed algorithm can still reconstruct synthetic and speech signals with MSE accuracies better than 175 dB and 100 dB, respectively. Experimental results with fixed-point quantized time stamps are also presented, along with a comprehensive comparison with state of the art methods in literature.

Chromatic Derivatives and Approximations in Practice—Part I: A General Framework

Chromatic derivatives are special, numerically robust differential operators that preserve spectral features of a signal; the associated chromatic approximations accurately capture local features of a signal. For this reason they allow digital processing of continuous time signals often superior to processing of discrete samples of such signals. We introduce a new concept of “matched filter” chromatic approximations, where the underlying basis functions are chosen to match the spectral profile of the signals being approximated. We then derive a collection of formulas and theorems that form a general framework for practical applications of chromatic derivatives and approximations. In the second part of this paper, we use such a general framework in several case studies of such applications that aim to illustrate how chromatic derivatives and approximations can be used in signal processing with an intention of motivating DSP engineers to find applications of these novel concepts in their own practice.

Chromatic Derivatives and Approximations in Practice—Part II: Nonuniform Sampling, Zero-Crossings Reconstruction, and Denoising

Chromatic derivatives are special, numerically robust differential operators that preserve spectral features of a signal; the associated chromatic approximations accurately capture local features of a signal. In the first part of this paper, entitled “Chromatic Derivatives and Approximations in Practice–Part I: A General Framework,” we have derived a collection of formulas and theorems which we use in this paper to demonstrate practical applications of chromatic derivatives and approximations. We present four case studies: a highly accurate (better than 170 dB) reconstruction of a signal from its nonuniformly spaced samples; a highly accurate method for obtaining timing information, such as timing of the zeros of a signal as well as timing of the zeros of its first derivative; a method of reconstruction of a sampled speech signal of 64 000 samples using such timing information only, with a reconstruction error of only about 1% of the rms of the original signal; and finally, a denoising algorithm that significantly outperforms the well-known Cadzow's denoising algorithm. The main purpose of these case studies is to illustrate the potential of chromatic derivatives and expansions and motivate DSP engineers to find applications of these novel concepts in their own practice.

Chromatic derivatives and expansions with weights

Chromatic derivatives and series expansions of bandlimited functions have recently been introduced in signal processing and they have been shown to be useful in practical applications. We extend the notion of chromatic derivative using varying weights. When the kernel function of the integral operator is positive, this extension ensures chromatic expansions around every points. Besides old examples, the modified method is demonstrated via some new ones as Walsh-Fourier transform, and Poisson-wavelet transform. Moreover the chromatic expansion of a function in some $L^p$-space is investigated.

Prolate spheroidal wavelets and multidimensional chromatic series expansions

Chromatic series were originally introduced for bandlimited functions. The nth chromatic derivative of an analytic function is a linear combination of kth ordinary derivatives with 0 ≤ k ≤ n, where the coefficients of linear combination are based on a suitable system of orthogonal polynomials. Chromatic derivative and series expansions of bandlimited functions have been used as a replacement for Taylor's series and they have been shown to be more useful in practical signal processing applications than Taylor series. In this paper we have shown that the theory can be extended to prolate spheroidal wavelet series that than combine chromatic series with sampling series in higher dimensions. The multidimensional case has much reacher structure than in the univariate case and will find more applications in image processing and analysis.

Recovery of archaeological wall paintings using novel multispectral imaging approaches

Abstract New approaches in the application of multispectral imaging to the recovery of archeological wall paintings are presented, based on statistical techniques and on a novel method of image treatment (Chromatic Derivative Imaging – ChromaDI) which offers a way of embedding information coming from four spectral bands into a standard RGB image. The methods are applied to some wall paintings from the Tomb of the Monkey, an Etruscan tomb in the necropolis of Poggio Renzo, near the city of Chiusi (Siena), Italy, dated around 480-470 BC. It is shown that the techniques described are able to highlight and enhance a number of details that cannot be perceived in either any of the original channel images or any single processed output channel.

Chromatic expansions of generalized functions

Chromatic series expansions of bandlimited functions have recently been introduced as an alternative representation of the Whittaker–Shannon–Kotel'nikov sampling series. Chromatic series share similar properties with the Taylor series insofar as the coefficients of the expansions, which are called chromatic derivatives, are based on the ordinary derivatives of the function. The goal of this paper is to extend the notion of chromatic derivatives and series to spaces of generalized functions and obtain chromatic series for integral transforms of generalized functions, such as the Laplace, Fourier, and Hankel transforms.

Chromatic series for functions of slow growth

The theory of chromatic derivatives leads to chromatic series which replace Taylor's series for bandlimited functions. For such functions, these series have a global convergence property not shared by Taylor's series. In this work the theory is extended to bandlimited functions of slow growth. This includes many signals of practical importance such as polynomials, periodic functions and almost periodic functions. This extension also enables us to get improved local convergence results for chromatic series.

Multidimensional chromatic derivatives and series expansions

Chromatic derivatives and series expansions of bandlimited functions have recently been introduced as an alternative representation to the Taylor series, and they have been shown to be more useful in practical signal processing applications than in the Taylor series. Although chromatic series were originally introduced for bandlimited functions, they have now been extended to a larger class of functions. The n-th chromatic derivative of an analytic function is a linear combination of the k-th ordinary derivatives with 0 ≤ k ≤ n, where the coefficients of the linear combination are based on a suitable system of orthogonal polynomials. The goal of this article is to extend chromatic derivatives and series to higher dimensions. This is of interest not only because the associated multivariate orthogonal polynomials have much reacher structure than in the univariate case, but also because we believe that the multidimensional case will find natural applications to fields such as image processing and analysis.

Decorrelating Properties of Chromatic Derivative Signal Representations

This letter is concerned with the decorrelating properties of the recently introduced chromatic derivative signal expansions, which encode signal information by employing suitably orthogonalized differential operators. An upper bound is derived on the condition number of the autocorrelation matrix of chromatic derivative expansion coefficients, extracted at a sampling instant. It is shown that chromatic derivative signal representations may be matched to the power spectrum of the signal under analysis, in order to effect maximum signal decorrelation and in order to ensure a small condition number of the autocorrelation matrix under the chromatic derivative signal representation.

Generalizations of Chromatic Derivatives and Series Expansions

Chromatic series expansions of bandlimited functions provide an alternative representation to the Whittaker-Shannon-Kotel'nikov sampling series. Chromatic series share similar properties with Taylor series insofar as the coefficients of the expansions, which are called chromatic derivatives, are based on the ordinary derivatives of the function. Chromatic derivatives are linear combinations of ordinary derivatives in which the coefficients of the combinations are related to a system of orthonormal polynomials. But unlike Taylor series, chromatic series have more useful applications in signal processing because they represent bandlimited functions more efficiently than Taylor series. The goal of this article is to generalize the notion of chromatic derivatives and then extend chromatic series expansions to a larger class of signals than the class of bandlimited signals. In particular, we extend chromatic series to signals given by integral transforms other than the Fourier transform, such as the Laplace and Hankel transforms.

Chromatic Derivatives and Local Approximations

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> We present a detailed motivation for the notions of chromatic derivatives and chromatic expansions. Chromatic derivatives are special, numerically robust linear differential operators; chromatic expansions are the associated local expansions, which possess the best features of both the Taylor and the Nyquist expansions. We give a simplified treatment of some of the basic properties of chromatic derivatives and chromatic expansions which are relevant for applications. We also consider some signal spaces with a scalar product defined by a CesÀro-type sum of products of chromatic derivatives, as well as an approximation of such a scalar product which is relevant for signal processing. We also introduce a new kind of local approximations based on trigonometric functions. </para>

Local Approximations Based on Orthogonal Differential Operators

Let M be a symmetric positive definite moment functional and let \(\{P_n^{\cal M}(\omega)\}_{n\in {\Bbb N}}\) be the family of orthonormal polynomials that corresponds to M. We introduce a family of linear differential operators \({\cal K}^n =(-i)^nP_n^{\cal M}(i\frac{d}{dt})\), called the chromatic derivatives associated with M, which are orthonormal with respect to a suitably defined scalar product. We consider a Taylor type expansion of an analytic function f(t), with the values f(n) (t0) of the derivatives replaced by the values \({\cal K}^n[f](t_0)\) of these orthonormal operators, and with monomials (t − t0)n/n! replaced by an orthonormal family of "special functions" of the form \((−1)^n{\cal K}^n[m](t-t_0)\), where \(m(t) = \sum_{n=0}^{\infty} (−1)^n{\cal M}(\omega^{2n}) t^{2n}/(2n)!\). Such expansions are called the chromatic expansions. Our main results relate the convergence of the chromatic expansions to the asymptotic behavior of the coefficients appearing in the three term recurrence satisfied by the corresponding family of orthogonal polynomials PMn(ω). Like the truncations of the Taylor expansion, the truncations of a chromatic expansion at t = t0 of an analytic function f(t) approximate f(t) locally, in a neighborhood of t0. However, unlike the values of f(n)(t0), the values of the chromatic derivatives Kn[f](t0) can be obtained in a noise robust way from sufficiently dense samples of f(t). The chromatic expansions have properties which make them useful in fields involving empirically sampled data, such as signal processing.

A sampling expansion for nonbandlimited signals in chromatic derivatives

In this paper, we construct infinite-band filterbanks for perfect reconstruction (PR) using Hermite polynomials and Hermite functions. The analysis filters are linear combinations of derivative operators based on these polynomials-the so-called chromatic derivative filters. Together with the synthesis filterbanks, they give PR for a large class of signals that may have infinite bandwidth. Several other related filterbanks are discussed as well. The error in reconstruction for finite-channel chromatic derivative filterbanks is calculated. Examples are given to demonstrate the use of these chromatic filterbanks.

Finite-channel chromatic derivative filter banks

Two previous contributions discussed the theory of perfect-reconstruction (PR) chromatic derivative filter banks comprising an infinite number of channels. This paper extends the theory to the case of finite channels. A novel time domain procedure is delineated for designing the synthesis filters that achieve PR in this case.

Chromatic derivative filter banks

A new contribution to generalized sampling of bandlimited signals based on the so-called chromatic derivative operators was recently introduced by Ignjatovic (see Kromos Technology, Los Altos, CA. [Online] Tech. Rep. 1, 2001). Chromatic derivatives are linear combinations of the ordinary derivatives, where the coefficients of the combination are derived from orthogonal polynomial theory. This article describes the connection between these operators and the well-established ideas of perfect reconstruction and biorthogonality in analog filter banks.

Optical aberration functions: chromatic aberrations and derivatives with respect to refractive indices for symmetrical systems

Formulas are presented for the computation of derivatives of the optical aberration functions S, T, V, W, and K with respect to the relative refractive indices, leading to the chromatic aberrations, for symmetrical optical systems. The formulas are suitable for use in computer programs for automatic computation of optical aberration coefficients.