Spatial Processes and Image Analysis

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Spatial Processes and Image Analysis
Yassir Moudden Sandrine Pires CEA/DAPNIA/SEDI-SA
P

• Lectures
• Basic models and tools in signal and image
  processing.
• Multiscale transforms wavelets, ridgelets,
  curvelets, etc.
• Multiresolution analysis and wavelet bases.
• Noise modeling and image restoration.
• Problems and methods in multispectral data
  analysis.

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Basic models and tools in signal and image
processing

• Outline
• Different types of images
• Sampling and quantification
• Fourier transform, Power spectrum
• Linear Filtering, convolution
• Non-linear operators mathematical morphology
• Statistical properties of images

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Signals, Images, etc.

• Quantitative data
• Organized in time or space

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Different types of signals and images (1)

• Continuous or Discrete index.
• Continuous or Quantized values.
• Finite energy, finite power, etc.

Computer processing requires finite energy,
discrete, quantized data.
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Different types of signals and images (2)

• Signals and images grouped in terms of regularity
  properties
• Continuous and global order of differentiability
• Local regularity
• Fractal dimension
• Statistical properties
• Marginal distributions, moments, etc.
• Coherence, correlations and non linear
  dependencies
• Stationarity

Different formal environment for handling indexed
data sets.
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Periodic signals and images

• Fourier series expansion
• where
• Plancherel-Parceval formula

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Signals and images as energy distributions in
time or space

• Energy
• Localization
• Spread
• More detailed characterization higher order
  moments of the energy distribuition.

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and in Fourier space

• Fourier transform
• Parceval
• Localization
• Spread
• Heisenberg Uncertainty Principle

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A few properties of the Fourier transform

• examples

(Poisson Sommation Formula)
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Sampling from continuous to discrete time (1)

• Ideal sampling multiplication by a Dirac comb
  with rate Fs 1/T.
• Properties
• Linear oprator.
• Not shift invariant..
• Shannon-Nyquist sampling theorem
• Given a uniform sampling rate of Fs 1/T, the
  highest frequency
• that can be unambiguously represented is Fs/2.
• Reconstruction (interpolation) formula

where
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Sampling from continuous to discrete time (2)

• Sampling in time
• periodizes in frequency space resulting in
  aliasing.

In higher dimensions, separable sampling schemes
are most commonly used. But there are other non
trivial possibilities.
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Linear operators - Filtering

• Simplest possible operators are linear.
• Shift invariant linear operators convolutive
  systems
• Harmonic signals are eigenvectors of linear
  filters
• with

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Example (1) low pass spatial filter

• Used for smoothing (removal of small details
  prior to large object extraction, bridging small
  gaps in lines) and noise reduction.
• Low-pass (smoothing) spatial filtering
• Neighborhood averaging
• Results in spatial blurring

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Example (2) median filter (non-linear)

• Replace the current pixel value by the median
  pixel value in a given neighborhood.
• Achieves effective noise supression.
• Preserves the sharpness of real boundaries.

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Mathematical morphology

• Two basic non-linear operators
• Dilation
• Erosion
• Several composite operators
• Closing
• Opening
• Conditionnal closing, etc.
• A strucutring element is used in each of these
  operations

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Dilation

• Principle takes the binary image B, places the
  origin of structuring element S over each pixel
  of value 1, and ORs the structuring element S
  into the output image at the corresponding
  position.
• It is typically applied to binary image, but
  there are versions that work on gray scale image.
  
• The basic effect of the operator on a binary
  image is to gradually enlarge the boundaries of
  regions of foreground pixels (i.e. white pixels,
  typically).
• Thus areas of foreground pixels grow in size
  while holes within those regions become smaller.

example dilation using a 3 by 3 square
structuring element for gap bridging.
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Erosion

• Principle takes the binary image B, places the
  origin of structuring element S over each pixel
  of value 1, and ANDs the structuring element S
  into the output image at the corresponding
  position.
• It is typically applied to binary image, but
  there are versions that work on gray scale image.
  
• The basic effect of the operator on a binary
  image is to gradually eliminate small objects.

origin
0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0
0 0 1 1 0 0 0 1 1 0 0 0 1 1 0 1 1 1 1 1
1 1 1
erode
B
S
B S
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Closing and opening

• Closing is a dilation followed by an erosion
  (with the same structuring element). Closing also
  produces the smoothing of sections of contours
  but fuses narrow breaks, fills gaps in the
  contour and eliminates small holes.
• Opening is an erosion followed by dilation
  (with the same structuring element). Opening
  smoothes the contours of objects, breaks narrow
  isthmuses and eliminates thin protrusions.

Effect of closing using a 3 by 3 square
structuring element
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Closing and opening

• Closing is a dilation followed by an erosion
  (with the same structuring element). Closing also
  produces the smoothing of sections of contours
  but fuses narrow breaks, fills gaps in the
  contour and eliminates small holes.
• Opening is an erosion followed by dilation
  (with the same structuring element). Opening
  smoothes the contours of objects, breaks narrow
  isthmuses and eliminates thin protrusions.

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Statistical signal and image processing

• Another way to build classes of signal and image
  data.
• Doesnt mean the signal or image data are
  stochastic.
• Means that our incomplete prior knowledge of what
  is noise and what is information requires a
  probabilistic framework for bayesian inference or
  maximum likelihood estimation.
• Many algorithms for image denoising, restoration
  etc. are in this general framework MEM, Wiener,
  shrinkage, detection.
• Prior probabilities express our knowledge of
  noise and signal.

NOISE NOT STRUCTURED SIGNAL STRUCTURED
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Statistical properties of signals and images

• A stochastic process/field is completely defined
  by its probability law
• Simplest model considers IID processes,
  isotropic, stationary
• How to account for coherent behaviour of
  neighboring (or not) samples or pixels, in a
  generic way?
• Different priors for differents classes of
  images.

• Gibbs-Markov fields
• New representations of structured image data.

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Gibbs-Markov field models for images

• The probability distribution of the value of
  pixel s does not depend on all the other pixels
  but only on those pixels in the considered
  neighborhood sort range local interactions.

• example Monte-Carlo simulations of an Ising
  model for different values of coupling.

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Gibbs-Markov random fields in segmentation
Segmentation of satelite images of urban areas
using MRF.
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Multiscale transforms wavelets, ridgelets,
curvelets, etc.

• Outline
• The Fourier transform
• Transient world and singularities Gibbs effect,
  regularity
• Time-frequency analysis and the Heisenberg
  principle
• Optimal spatiospectral localization
• Wavelets, the continuous transform coherence,
  sparsity, redundancy
• Cauchy Schwartz inequality
• Approximation theory vanishing moments
• Non-linear operators mathematical morphology
• Markov random fields
• Problems in Astronomical data analysis
• Frames, radon, ridgelets, curvelets
• Parceval plancherel
• 2D wavelets

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Multiresolution analysis and wavelet bases

• Outline
• Multiresolution analysis
• The scaling function and scaling equation
• Examples
• Fast algorithms
• Orthogonal and biorthogonal wavelets
• Building wavelet bases
• Vanishing moments
• Applications in compression, approximation
• Trees
• Wavelet packets
• A trous algorithm
• Pyramidal algorithm

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Image restoration, noise models, detection,
deconvolution

• Outline
• Image fornation model, Inverse problems in image
  processing
• Algorithms for deconvolution Richardson-Lucy,
  CLEAN,
• Wiener filtering, Gaussian filter, Maximumentropy
  methode
• Spike processes
• Application of multiresolution methods
• Shrinkage, Sparsity, bayesesian approaches
• Inpainting
• Again Cauchy-Schwartz
• Pierpaoli
• Complex models accounting for coherent
  behaviour of wavelet coefficients

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Multi-dimensional data analysis

• Outline
• What is multidimensional data
• Where does it come from
• Gaussianity
• Representations ans sparsity
• Projection pusuit
• Principal Component Analysis Karhunen-Loeve
  Basis
• Standard mainstream ICA
• Diversity and separability
• Non gaussianity, Non stationarity
• Linear mixture model
• Applications