Linear Image Reconstruction Bart Janssen b.j.janssen@tue.nl 13-11, 2007 Eindhoven.

Linear Image Reconstruction

Bart Janssenb.j.janssen@tue.nl

13-11, 2007Eindhoven

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Outline

• Introduction• Linear Image Reconstruction• Bounded Domain• Future Work

3Gala looking into the Mediterranean Sea

Salvador Dali

•Objects exist at certain ranges of scale.•It is not known a priory at what scale to look.

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Gaussian Scale Space

s

x

y

Solution of

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Singular points of a Gaussian scale space image

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Reconstruction from Singular Points

Use differential structure in singular pointsas features.

=

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Image Reconstruction

Given features

Select

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Image Reconstruction• Kanters et al.:

which is a projection of onto span( )

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Iff A unbounded then solution A-orthogonal

projection of onto span( )

Minimisation ofCorresponding filters

So

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Reconstruction from Singular Points

-reconstruction

We should choose a smooth prior:

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This means

Gram matrix:

Projection:

Reconstruction from Singular Points

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Bounded Domain• Features are penalized while outside the image• Control of boundary is needed for Image Editing (and other applications)

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Bounded Domain Reconstruction

Feature

Equivalence

Reconstruction

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Completion of space of 2k differentiable functions that vanish on

Sobolev space

Endowed with the inner product

Reconstruction

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Reconstruction

Reciprocal basis functions

Subspace is spanned by

So

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Find the image

that satisfy

next compute

Boundary conditions of source image

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Its right inverse: minus Dirichlet operator

Laplace operator on the bounded domain

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Green’s function of Dirichlet operator I

Schwarz-Christoffel mapping (inverse) Linear Fractional Transform

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Green’s function of Dirichlet operator II

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Green’s function of Dirichlet operator III

Spectral Decomposition : extends to compact, self-adjoint operator onSo normalized eigenfunctions + eigenvalues of

Eigenfunctions of Dirichlet operator coincide(eigenvalues are inverted) since

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Scale space on the bounded domain

Operators:

Scale space image:

Reciprocal filters by application of:

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Implementation in discrete framework

Discrete sine transform

own inverse and

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Evaluation - “Top Points”

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Evaluation - “Laplacian Top Points”

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Conclusions

• On the bounded domain the solution can still be obtained by orthogonal projection

• Efficient implementation possible (Fast Sine Transform)

• Better reconstructions for• The method extends readily to Neumann boundary conditions

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Current & Future Work

•Approximation•Select resolution/scale•Best Numerical Method?•Force absence of toppoints ?

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Questions?

Topological Abduction of Europe - Homage to Rene ThomSalvador Dali

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Filtering interpretation of Parameters I

Operator equivalent to filtering by low-pass Butterworth filter of order and cut-off frequency

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Filtering interpretation of Parameters II

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Iterative Reconstruction

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