Multidimensional factorization through helical mapping

This paper proposes a new perspective on the problem of multidimensional spectral factorization, through helical mapping: $d$-dimensional ($d$D) data arrays are vectorized, processed by $1$D cepstral analysis and then remapped onto the original space. Partial differential equations (PDEs) are the basic framework to describe the evolution of physical phenomena. We observe that the minimum phase helical solution asymptotically converges to the $d$D semi-causal solution, and allows to decouple the two solutions arising from PDEs describing physical systems. We prove this equivalence in the theoretical framework of cepstral analysis, and we also illustrate the validity of helical factorization through a $2$D wave propagation example and a $3$D application to helioseismology.

[1]  T. Taxt,et al.  Two-dimensional noise-robust blind deconvolution of ultrasound images , 2001, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[2]  A. Oppenheim,et al.  Nonlinear filtering of multiplied and convolved signals , 1968 .

[3]  Pierre Comon,et al.  Tensors : A brief introduction , 2014, IEEE Signal Processing Magazine.

[4]  Jon F. Claerbout,et al.  Multidimensional recursive filter preconditioning in geophysical estimation problems , 2003 .

[5]  G. Garibotto,et al.  2-D recursive phase filters for the solution of two-dimensional wave equations , 1979 .

[6]  D. Dudgeon,et al.  The computation of two-dimensional cepstra , 1977 .

[7]  Aydin Kizilkaya On the Parameter Estimation of 2-D Moving Average Random Fields , 2007, IEEE Transactions on Circuits and Systems II: Express Briefs.

[8]  Dan E. Dudgeon,et al.  Multidimensional Digital Signal Processing , 1983 .

[9]  R.W. Schafer,et al.  From frequency to quefrency: a history of the cepstrum , 2004, IEEE Signal Processing Magazine.

[10]  M. Ekstrom,et al.  Two-dimensional spectral factorization with applications in recursive digital filtering , 1976 .

[11]  T. Taxt,et al.  Noise robust one-dimensional blind deconvolution of medical ultrasound images , 1999, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[12]  Everton Z. Nadalin,et al.  Unsupervised Processing of Geophysical Signals: A Review of Some Key Aspects of Blind Deconvolution and Blind Source Separation , 2012, IEEE Signal Processing Magazine.

[13]  S. Sarpal,et al.  ARMA modelling based on root cepstral deconvolution , 2002, 2002 14th International Conference on Digital Signal Processing Proceedings. DSP 2002 (Cat. No.02TH8628).

[14]  Afshin Gholamy Salehabady Why Ricker Wavelets Are Successful In Processing Seismic Data: Towards A Theoretical Explanation , 2014 .

[15]  Ju-Hong Lee,et al.  Design Of 2-D Recursive Digital Filters Using Nonsymmetric Half-Plane Allpass Filters , 2007, IEEE Transactions on Signal Processing.

[16]  Kenneth Steiglitz,et al.  Computation of the complex cepstrum by factorization of the z-transform , 1977 .

[17]  Umberto Spagnolini,et al.  Seismic Velocity and Polarization Estimation for Wavefield Separation , 2008, IEEE Transactions on Signal Processing.

[18]  Paul Dalsgaard,et al.  Separation of mixed phase signals by zeros of the z-transform - A reformulation of complex cepstrum based separation by causality , 2010, 2010 IEEE International Conference on Acoustics, Speech and Signal Processing.

[19]  R. Mersereau,et al.  The representation of two-dimensional sequences as one-dimensional sequences , 1974 .

[20]  Jon F. Claerbout,et al.  Calculation of the sun's acoustic impulse response by multi-dimensional spectral factorization , 2000 .

[21]  M. Ekstrom,et al.  Multidimensional spectral factorization and unilateral autoregressive models , 1980 .

[22]  J. Aggarwal,et al.  Design of two-dimensional semicasual recursive filters , 1978 .

[23]  Soo-Chang Pei,et al.  Two-Dimensional Partially Differential Cepstrum and Its Applications on Filter Stabilization and Phase Unwrapping , 2011, IEEE Transactions on Circuits and Systems I: Regular Papers.