Linear Image Reconstruction

Bart Janssenb.j.janssen@tue.nl

13-11, 2007Eindhoven

2

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.

QuickTime™ and aCinepak decompressor

are needed to see this picture.

4

Gaussian Scale Space

s

x

y

Solution of

5

QuickTime™ and aCinepak decompressor

are needed to see this picture.

Singular points of a Gaussian scale space image

QuickTime™ and a decompressor

are needed to see this picture.

6

Reconstruction from Singular Points

Use differential structure in singular pointsas features.

=

7

Image Reconstruction

Given features

Select

8

Image Reconstruction• Kanters et al.:

which is a projection of onto span( )

9

Iff A unbounded then solution A-orthogonal

projection of onto span( )

Minimisation ofCorresponding filters

So

10

Reconstruction from Singular Points

-reconstruction

We should choose a smooth prior:

11

This means

Gram matrix:

Projection:

Reconstruction from Singular Points

12

13

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

14

Bounded Domain Reconstruction

Feature

Equivalence

Reconstruction

15

Completion of space of 2k differentiable functions that vanish on

Sobolev space

Endowed with the inner product

Reconstruction

16

Reconstruction

Reciprocal basis functions

Subspace is spanned by

So

17

Find the image

that satisfy

next compute

Boundary conditions of source image

18

Its right inverse: minus Dirichlet operator

Laplace operator on the bounded domain

19

Green’s function of Dirichlet operator I

Schwarz-Christoffel mapping (inverse) Linear Fractional Transform

20

Green’s function of Dirichlet operator II

21

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

22

Scale space on the bounded domain

Operators:

Scale space image:

Reciprocal filters by application of:

23

Implementation in discrete framework

Discrete sine transform

own inverse and

24

Evaluation - “Top Points”

25

Evaluation - “Laplacian Top Points”

26

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

27

Current & Future Work

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

28

Questions?

Topological Abduction of Europe - Homage to Rene ThomSalvador Dali

29

Filtering interpretation of Parameters I

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

30

Filtering interpretation of Parameters II

31

Iterative Reconstruction

QuickTime™ and a decompressor

are needed to see this picture.