1. Spatial Processes and Image Analysis
Yassir Moudden & Sandrine Pires
CEA/DAPNIA/SEDI-SAP
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.

2. 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

3. Signals, Images, etc.
Quantitative data
Organized in time or space

4. 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.

5. 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.

6. Periodic signals and images
Fourier series expansion :
where :
Plancherel-Parceval formula :

7. Signals and images as energy distributions in time or space …
Localization :
Spread :
More detailed characterization : higher order moments of the energy distribuition.

8. … and in Fourier space : Fourier transform : Parceval : Localization :
Spread :
Heisenberg Uncertainty Principle :

9. A few properties of the Fourier transform
examples :
(Poisson Sommation Formula)

10. 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

11. 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.

12. Linear operators - Filtering
Simplest possible operators are linear.
Shift invariant linear operators = convolutive systems :
Harmonic signals are eigenvectors of linear filters :
with

13. 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

14. 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.

15. 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:

16. 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.

17. 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 1
erode B S
B S

18. 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

19. 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.

20. Statistical signal and image processing
Another way to build classes of signal and image data.
Doesn’t 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

21. 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.

22. 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.

23. Gibbs-Markov random fields in segmentation
Segmentation of satelite images of urban areas using MRF.

24. 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

25. 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

26. 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

27. 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