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

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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)
Objectives of Planned Work
Establish new parallel basic algorithms for
scattered data approximation in 2D/3D Gabor
analysis for images (denoising, space variant
filtering)
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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) S i f(ti) ei (t) Numerical
aspects fast iterative (CG-based) algorithms and
well structured (e.g., Toeplitz) system matrix.
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Signal Processing Algorithms Hans G.
Feichtinger (Univ. of Vienna)

• image restoration (lost pixel problem)
• geophysical data approximation
• nearest neighborhood approximation

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

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Signal Processing Algorithms Hans G.
Feichtinger (Univ. of Vienna)
Scattered Data or Irregular Sampling Problem
(1st step)
2D-Voronoi method nearest neighborhood
interpolation
Fourier-based method applied to color images
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Signal Processing Algorithms (Scattered Data)
Hans G. Feichtinger (Univ. of Vienna)
Irregular sampling
Reconstruction
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Signal Processing Algorithms Hans G.
Feichtinger (Univ. of Vienna)
Explicit and hidden Parallelism
A) Evident opportunities
frequent FFT2 establishing system (Toeplitz)
matrix parallel variants of POCS
B) Hidden parallelism and new problems
local iteration versus data exchange real
time applications time / space variant
smoothness time variant Gabor based filters
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Signal Processing Algorithms (Scattered Data)
Hans G. Feichtinger (Univ. of Vienna)
A possible application move restoration
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Signal Processing Algorithms (Scattered Data)
Hans G. Feichtinger (Univ. of Vienna)
Reconstruction with nearest neighbourhood
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Signal Processing Algorithms (Scattered Data)
Hans G. Feichtinger (Univ. of Vienna)
Reconstruction with adaptive filtering respecting
directional information
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Signal Processing Algorithms Hans G.
Feichtinger (Univ. of Vienna)
Foundations of Gabor Analysis

Two (mutually dual) equivalent fares both
involving a STFT (for some window g)
STFTgf(t,r)FT(Tt gf)(r)
A) Recover signal f from sampled STFT
(eliminate redundancy by sampling over some
TF-lattice)
B) Gabors 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).
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Signal Processing Algorithms Hans G.
Feichtinger (Univ. of Vienna)

• Operations based on Gabor Analysis
• Signal denoising ()
• time-variant filtering
• texture analysis (image segmentation)
• foveation
• (focus of attention)
• musical transcription
• image compression ()

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Signal Processing Algorithms (Gabor Analysis)
Hans G. Feichtinger (Univ. of Vienna)
The Time-Frequency-representation of a
sound signal showing the temporal frequency
variation
freqency
time
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Signal Processing Algorithms (Gabor Analysis)
Hans G. Feichtinger (Univ. of Vienna)
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Signal Processing Algorithms (Gabor Analysis)
Hans G. Feichtinger (Univ. of Vienna)
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Signal Processing Algorithms (Gabor Analysis)
Hans G. Feichtinger (Univ. of Vienna)