Air Systems Division

Definition of anisotropic denoising operators viasectional curvature

Stanley DurrlemanSeptember 19, 2006

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The problem

coast

crest

Purpose : Denoising homogeneous areas… …without smoothing the signal at the interfaces

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The problem

Autoregressive model :

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The problem

Autoregressive model :

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The problem

Autoregressive model :

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The problem

Autoregressive model :

Burg algorithm enables :-better estimation in case of short sample signals-fewer interference peaks-recursive computation : real time algorithm-estimation of the spectral density function :

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The problem

Example : record of turbulent atmospheric clutter Images du CR 1magnitude angle

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What’s in the image proceesing toolbox ?

- Statistical models of noiseBayesian models, Markov fields… :

- good model of noise- how to take the geometry into account ?

- Geometrical models : Linear filters (Gaussian,…) : do not preserve the discontinuitiesNon-linear filters :- Curvature motion & morphologic filters (AMSS, mean curvature motion, median filter) :

- noise = level set of small areas- specific for gray-level images

- Geometric filters : (Kimmel, Sochen, Barbaresco) : - model data as a sub-manifold- depend on the way data are parametrized (mean curvature flow)- model of noise ?

Our goal : define anisotropic operators that can denoise data… of any dimension (gray-level images, radar signal…) independently of the data parametrization and restore piecewise constant data

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Outline

Noise characterization via sectional curvature

De-noising algorithms

Results

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I. Noise characterization through sectional curvature

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I. Noise & Sectional Curvature

1. Question : what is noise ??

statistics : Bayesian filters, maximum likelihood…

geometry : which tool ? Gradient ? Curvature !

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I. Noise & Sectional Curvature

2- Basic idea : the surface Gaussian Curvature

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I. Noise & Sectional Curvature

2- Basic idea : the surface Gaussian Curvature

Examples :

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I. Noise & Sectional Curvature

Noise and curvatureAxiom : pixel of noise = pixel of big curvatureAxiom : pixel of noise = pixel of big curvature

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How to denoise ?

By minimizing the following energy :

I. Noise & Sectional Curvature

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I. Noise & Sectional Curvature

3 – Modeling

A generic ‘image’ :

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I. Noise & Sectional Curvature

3 – Modeling

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I. Noise & Sectional Curvature

3 – ModelingCurvature of a metric :

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I. Noise & Sectional Curvature

3 – ModelisationCurvature of a metric :

That is the surface Gaussian curvature !That is the surface Gaussian curvature !

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I. Noise & Sectional Curvature

Summary :1/ One defines :

h metric on the data space e metric on the acquisition space

=> a ‘mixed’ metric : g

2/ One computes the sectional curvature: K

3/ One defines the energy : E

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II. De-noising algorithms

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II. De-noising algorithms

Purpose :Minimizing :

2 methods :- Partial Differential Equation- Stochastic algorithm

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1. Descent gradient scheme :1/ initialise with the given noisy image2/ Evolve towards a minimum of :

using the gradient :

Hence, the evolution equation :

implemented with a finite difference scheme.

II. De-noising algorithms

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II. De-noising algorithms

Case of gray-level images :

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II. De-noising algorithms

2. Stochastic method :

- One picks randomly a pixel in the (noisy) image.

- One adds a small random Gaussian variable to the pixel’s value.

- If the energy decreases : one keeps the change Else : the change is rejected.

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III. Results

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III. Results

1 – Gray-level images (1)

Metric :

Curvature :

Flow :

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III. Results

t = 0 t = 1 t = 10 t = 100

Flow equation

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III. Results

Transversal view :

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III. Results

Stochastic

Stoch. + PDE

Stochastic algorithm :Stochastic algorithm :

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III. Results

2 – Gray level images (2) Adaptative metric :

‘Dilate’ geodesics in D far from the minimum

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III. Results

original PDE Stoch. Algo. Adaptative metric

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III. Results

median

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III. Results

RSO Image

original amss PDE Stoch

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III. Results

2. Radar signal

Reminder : Parametrization thanks to complex auto-regressive analysiscomplex auto-regressive analysis

8 complex magnitudes

7 reflection coefficients

Doppler spectrum

Burg Algo.

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III. Results

Data : Reflection coefficients

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III. Results

Simulated data :Image of CR 1 :

magnitude angle

azimut 16

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III. Results

Simulated data :Image of CR 1 :

magnitude angle

azimut 16

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III. Results

After de-noising:

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III. Results

After de-noising :

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III. Results

Real data : Images du CR 1

magnitude angle

azimut 19

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III. Results

After de-noising :

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Thank you for your kind attention

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