A new approach for regularization of inverse problems in image processing

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Inverse problems : variational formulation Generalized Diffusion regularization Application Conclusion

A new approach for regularization of inverse

problems in image processing

I. Souopgui1,2, E. Kamgnia2,F.-X. Le Dimet1, A. Vidard1

(1) INRIA / LJK Grenoble (2) University of Yaounde I

10th_{African Conference on Research in Computer Science and Applied} Mathematics - CARI 2010

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1 Inverse problems : variational formulation

Definition

A priori knowledges

Regularization : Vector fields

2 _{From classical regularization to Generalized Diffusion}

Regularization as smoothing operators Case of gradient penalization

3 Application to geophysical fluid motion estimation

Motion estimation problem

From regularization to pseudo covariance operator Experimental result

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Inverse problems : variational formulation Generalized Diffusion regularization Application Conclusion Definition A priori knowledges Regularization

Ingredients

Physical system M: V → Y v 7→ y=M(v)

y∈ Y, the system state

v∈ V, the control variable

M, model mapping V toY Observation yo y o_{∈ O}_{, observed state} Observation system H: Y → O y 7→ H(y) Hobservation operator mappingY to O

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Definition

Giving observed stateyo,

Inverse problem (unconstrained)

Findv∗= MinArg(J(v)), v∈ V where

J(v) =Jo(v) =

1

2kH(M(v))−y

o_k2

O (1)

under adequate conditions, the solutionv∗ is given by the

Euler-Lagrange Equation∇J(v∗) = 0

Problems

ill-posedness⇒ use a priori knowledges;

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A priori knowledges

For a priori knowledgeA, set J=Jo +JA

whereJA is defined to force the solution to satisfyA

Use of a priori informations

Background vb _{and background error covariance}_B

Jb=

1

2αbkv−v

b_k2

B−1 (2)

Regularity of the solution : Φ-smooth (minimum gradient) Jr =

1

2αrkΦ(v)k

2 ₍₃₎

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Definition A priori knowledges Regularization

Vector fields regularization

first order regularization : first order derivatives ofv

Φ(1) ∂vi ∂xj 1≤i,j≤n, gradient penalization : J∇(v) = 1 2α∇ Z Ω n X i=1 k∇vik2dx

second order regularization : second order derivatives ofv Φ(2) ∂2vi ∂xj∂xk 1≤i,j,k≤n, Suter regularization : Jsuter(v) = 1 2 Z Ω α∇divk∇div(v)k 2 +α∇curlk∇curl(v)k 2 dx

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Notations and definition

Let :

v(x) be an incomplete/inconsistent control variable, with

x∈Ω the physical space

Φ(v) regularization operator as defined previously

ϕ(x) a scalar positive trust function given the quality of v atx

small value meaning bad/lack/inconsistent control variable large value for good quality control variable

we define restored control variableu∗= MinArg(ε(u)),u∈ V

ε(u) = 1 2

Z

Ω

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ε(u) = 1 2 Z Ω kΦ(u(x))k2+ϕ(x)k2u(x)−v(x)kdx εis minimized by settingu to be :

close to v when ϕis large (v has adequate properties)

Φ−regular when ϕis small (otherwise)

Under adequate conditions MinArg(ε) is given by the Euler-Lagrange condition

∇uε(u) = 0 (5)

Gateaux derivatives development leads to

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Gradient penalization : mathematical expression

J∇(v) = 1 2α∇ Z Ω n X i=1 k∇vik2dx

Applied as smoothing operator, we get

Φ∗∇◦Φ∇=−∆,with boundary conditions : ∇ui ⊥ν on∂Ω

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Numerical implementation

Generalized diffusion implementation

Classical implementation : given ∇ε, use descent-type algorithms.

Problem : solve the Euler-Lagrange equation

∆ui−ϕ(x)(ui(x)−vi(x)) = 0, 1≤i ≤n (8)

considersui as a function of time and solve the equivalent problem

∂

∂tui(x,t) = ∆ui(x,t)−ϕ(x)(ui(x,t)−vi(x))), 1≤i ≤n (9)

known as the generalized diffusion equations.

As diffusion operator, it can directly be used in background covariance [see Weaver et al.]

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optical flow of Horn and Shunck : luminance conservation

df

dt = 0 (10)

f(x,t) noted f is the luminance function.

For geophysical fluid images, the mass conservation equation is more adequate [Fitzpatrick 1985]

df

dt +f(∇ ·v) = 0 (11)

v(x) is the velocity at x

given the luminance functionf(x,0) =f0(x) at time 0, solution to

equations (10) or (11) definesf(x,t) as function of the static

velocity fieldv(x)

M: V → F

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motion estimation is reduced to the inverse problem : δv∗= MinArg(J(δv)) with J(δv) = 1 2kM(v b₊_δ_v₎₋_fo_k2 F + 1 2kδvk 2 V (12) Aperture problem :

only motion along the normal to iso-contours can be inferred⇒

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Trust function for motion estimation

Proposition: define trust functionϕ

to have large values on discontinuities (contours) for motion component along the normal to the contour, and small values in homogeneous areas.

Example : setϕto be the contours mapc1 orc2 defined as

c1(x,f) = k∇_xf(x)k2

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Twin experiments

Direct Image sequences assimilation [Titaud et al 2009]⇒true initial state (velocity fields)

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Error analysis : Tikhonov regularization

Evolution of diagnostic functions with respect to the weighting parameter α

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Error analysis : gradient penalization

Evolution of diagnostic functions with respect to the weighting parameter α∇

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Error analysis : Generalised diffusion

Evolution of diagnostic functions with respect to the weighting parameter αGD

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Error analysis : Comparison - cost function

Evolution of the observation cost function with minimization iterations

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Error analysis : Comparison - velocity error

Evolution of the velocity error with minimization iterations

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Error analysis : Comparison - vorticity error

Evolution of the vorticity error with minimization iterations

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Error analysis : Comparison - angular error

Evolution of the angular error with minimization iterations

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Analysis : vector field

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Analysis : vector field

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Analysis : vector field

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Conclusion

Inverse problems :

ill-posed ⇒ use regularization

ill-conditioned⇒ use preconditioner

proposed : promising approach for regularization of inverse problems.