Linear Image Reconstruction Bart Janssen 13-11, 2007 Eindhoven

2 Outline Introduction Linear Image Reconstruction Bounded Domain Future Work

3 Gala 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.

4 Gaussian Scale Space s x y Solution of

5 Singular points of a Gaussian scale space image

6 Reconstruction from Singular Points Use differential structure in singular points as 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 of Corresponding 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 Subspaceis 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 on So 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 Thom Salvador 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