Signal Processing Algorithms Hans G. Feichtinger (Univ. of Vienna) NuHAG

Previous Work * Mathematical methods for image processing (interdisciplinary FSP 1994-2000) * Gabor Analysis (Book, 1998) * Algorithms for irregular sampling (e.g., geophysics)

Establish new parallel basic algorithms for * scattered data approximation in 2D/3D * Gabor analysis for images (denoising, space variant filtering)

Objectives of Planned Work

Signal Processing Algorithms Hans G. Feichtinger (Univ. of Vienna)

Scattered Data (irregular sampling) Problem

Signal model: smooth function f (e.g., band-limited)

Task: Recovery of f from sampling values f(ti)

Methods: linear recovery using iterations: f(t) =if(ti) ei (t)

Numerical aspects: fast iterative (CG-based) algorithms and well structured (e.g., Toeplitz) system matrix.

• image restoration (lost pixel problem)

• geophysical data approximation

• nearest neighborhood approximation

Signal Processing Algorithms Hans G. Feichtinger (Univ. of Vienna)

Signal Processing Algorithms (Scattered Data) Hans G. Feichtinger (Univ. of Vienna)

Signal Processing Algorithms Hans G. Feichtinger (Univ. of Vienna)

Background within NUHAG

* variety of iterative algorithms (CG);

* guaranteed rates of convergence;

* established robustness (e.g., jitter error);

* good locality possible (T. Werther);

* adaptive weights improve condition;

* no a priori information of f is required (function spaces);

Scattered Data or Irregular Sampling Problem (1st step):

2D-Voronoi method = nearest neighborhood interpolation

Fourier-based method applied to color images

Signal Processing Algorithms Hans G. Feichtinger (Univ. of Vienna)

Signal Processing Algorithms (Scattered Data) Hans G. Feichtinger (Univ. of Vienna)

Irregular sampling Reconstruction

Signal Processing Algorithms Hans G. Feichtinger (Univ. of Vienna)

Explicit and hidden Parallelism

A) Evident opportunities

* local iteration versus data exchange* real time applications* time / space variant smoothness* time variant Gabor based filters

B) Hidden parallelism and new problems

* frequent FFT2* establishing system (Toeplitz) matrix* parallel variants of POCS

Signal Processing Algorithms (Scattered Data) Hans G. Feichtinger (Univ. of Vienna)

A possible application: move restoration

Signal Processing Algorithms (Scattered Data) Hans G. Feichtinger (Univ. of Vienna)

Reconstruction with nearest neighbourhood

Signal Processing Algorithms (Scattered Data) Hans G. Feichtinger (Univ. of Vienna)

Reconstruction with adaptive filteringrespecting directional information

Signal Processing Algorithms Hans G. Feichtinger (Univ. of Vienna)

Foundations of Gabor AnalysisTwo (mutually dual) equivalent fares both involving a STFT

(for some window g): STFTgf(t,r)=[FT(Tt g*f)](r)

(eliminate redundancy by sampling over some TF-lattice) A) Recover signal f from sampled STFT

B) Gabor´s “Atomic Approach“: Expand a given signal as series of time-frequency shifted atoms

Problem: good locality requires non-orthogonality of system

Joint Solution: “dual“ Gabor-atoms (for given g and lattice).

Operations based on Gabor Analysis – Signal denoising (*)

– time-variant filtering

– texture analysis (image segmentation)

– foveation

(focus of attention)

– musical transcription

– image compression (*)

Signal Processing Algorithms Hans G. Feichtinger (Univ. of Vienna)

Signal Processing Algorithms (Gabor Analysis) Hans G. Feichtinger (Univ. of Vienna)

The Time-Frequency-representation of a sound

signal showing the temporal frequency variation

time

freq

ency

Signal Processing Algorithms (Gabor Analysis) Hans G. Feichtinger (Univ. of Vienna)

Signal Processing Algorithms (Gabor Analysis) Hans G. Feichtinger (Univ. of Vienna)

Signal Processing Algorithms (Gabor Analysis) Hans G. Feichtinger (Univ. of Vienna)