elsa angelini - telecom paris

SIAM Imaging Science 2006Overview

Elsa AngeliniDepartment TSI, ENST

Jérôme DarbonLRDE, EPITA

23-June-2006TSI, ENST

SIAM06 Imaging Science3 days of conferenceConcurrent sessionsThemes: acquisition and formation , Image restoration and reconstruction , Image modeling and analysis , Image understanding , Biomedical imaging , Inverse problems in imaging sciences , Statistical aspects in image processing , Image segmentation, inpainting and registration , Mathematics of visualization , PDEs in image processing ,Novel imaging methods ,Applications

Abstract publication2 plenary talks per day1 poster session

Very intensive!


SIAM06 Imaging Science1. “Imaging by Random Sensing”, E. Candes, CalTec

USA

Problem of image acquisition with few samples ⇨Efficient image acquisition to reconstruct a N pixel image from k<N measurements.Example of indirect acquisition (Fourier sampling) with MRI.Analysis of structures in the image in terms of sparsity & compressibility

Ex. of image compression with WT.Compressibility has bearing on data acquisition

Show ex. on PET phantom with TV used to select samples to reconstruct. Perfect reconstruction!Example of imaging fuel cells, with very expensive measurements (Fourier acquisitions)


SIAM06 Imaging ScienceShannon/Nyquist sampling theory: the number of samples needed to capture a signal is dictated by its bandwidth.Alternative theory of "compressive sampling“:

uses nonlinear recovery algorithms (based on convex optimization)Can reconstruct super-resolved signals and images can be reconstructed from what appears to be highly incomplete datadata compression implicitly incorporated into data acquisition process,


SIAM06 Imaging ScienceTheorem: need k>S Log(N) to use k Fourier samples to ~perfectly reconstruct a signal of support size S over N samples, with L1 minimization.Compressive sampling: Theory based on incoherence between representation and measurements.Sensing matrix (matrix of measurements):

Robust compressive sampling for Noisy measurements ⇨no blow up in reconstruction errors.

Applications to new optical devices: www.l1-magic.org


SIAM06 Imaging Science

TVLo

gan-

Shep

p Ph

anto

m

512 samples on 22 lines

Mini. Energy

Perfect reconstruction with:


SIAM06 Imaging ScienceProof



Given these 80 observed samples, the set of length-256 signals that have samples that match our observations is an affine subspace of dimension 256-80=176. From the candidate signals in this set, we choose the one whose DFT has minimum L1 norm; that is, the sum of the magnitudes of the Fourier transform is the smallest. In doing this, we are able to recover the signal exactly!

In general, if there are B sinusoids in the signal, we will be able to recover using L1 minimization from on the order of B log N samples (see the "Robust Uncertainty Principles..." paper for a full exposition).



(1) Randomly choose a K-dimensional subspace(2) project the signal onto it (3) If f is B sparse in a known ortho-basis & K ≈ B

log N, ⇒ f can be recovered without error by solving an

L1 minimization problem. More general types of measurements &types of

sparsity.106 pixels & Perfectly sparse WT (25 103 coef Db-8 ). 100,000 "random measurements", subspace.

Image whose WT has the smallest L1 norm & has the same projection onto this subspace

Extension to projection

spaces


SIAM06 Imaging Science2. “Overview of high-order PDEs in Image Processing”,

A. L. Bertozzi, UCLAImage InpantingNavier-Stoke and 2D fluid dynamicNo maximum principles, Difficult to design numerical schemes, especially to preserve smoothnessNeed implicit time steps (problem very stiff)

Low-curvature image

simplifier



Intrinsic term of thin-plate splineIntroduce anisotropic Willmore flow (4th order) into level setsVolume preservation & non-increasing areasNumerical schemes carefully designedMumford-Shah surfaces

Very slow speeds of convergence. Discretization very delicateGood results.

To study: Direct minimization of the energies instead of Euler-Lagrange. Could reduce the order of the functional.


SIAM06 Imaging Science3. “Curvature depending functional for image

segmentation with occlusions and transparencies”R. March, ITALY

2.1D sketch of Nitzberg-Mumford


SIAM06 Imaging Science2.1D sketch

p=2



Complete occluded edges by linking end points

Regularity on image discontinuities


No algorithmTo study: links with dead leaves model from Matheron, Gousseau



SIAM06 Imaging Science5. “Geometry Image Description”, Hoppe, Microsoft

Corp. USA

General cut: 2g cuts for g-genus surfacesIllustration on Budda shapesMulti-chart approach: each chart gives 2 piecewise regular meshesSpherical geometry:

no cutting to map objects on sphereSphere is cut to create a square image

Applications:Morphing, animation, geometry amplification, compression with spherical wavelets



Remesh an arbitrary surface onto a completely regular structure, called a geometry image:

• captures geometry as a simple 2D array of quantized points.

• Surface normals and colors stored in similar 2D arrays

• No texture coordinates

To create a geometry image, we cut an arbitrary mesh along a network of edge paths, and parameterize the resulting single chart onto a square.

Geometry images can be encoded using traditional image compression algorithms, such as wavelet-based coders. http://research.microsoft.com/~hoppe/


257 x 257257 x 25712 bits/channel12 bits/channel

3D geometry3D geometrycompletely regular samplingcompletely regular sampling

RGB colors encode XYZ positions


http://research.microsoft.com/~hoppe/


SIAM06 Imaging ScienceGeometry images have the potential to simplify the rendering pipeline, eliminating the "gather" operations associated with vertex indices and texture coordinates.

cutcut

topology of a disk

ParameterizatoinParameterizatoinSample on a Sample on a square square

regular gridregular grid

storestore

renderrender




Genus: Largest number of non-intersecting simple closed curves that can be drawn on the surface without separating it. ~ number of holes in a surface.

- Genus-g surface 2g generator loops minimum- Additional cuts can optimize the parameterization via mapping on a regular grid

aaaa’’

aa

aa’’

Optimal geometric strech

Non-Optimal



geometry image geometry image 2572572 2 x 12b/chx 12b/ch

normalnormal--map image map image 5125122 2 x 8b/chx 8b/ch

Rendering with attributeshttp://research.microsoft.com/~hoppe/



SIAM06 Imaging Science‘6. ’Exact Solutions of Some Variational Image Analysis Models’’,

Kevin R. Vixie , Los Alamos Ntl. Lab. USA

If the observed image represents a convex shape, then L1+ λ TV is equivalent to an opening/closing with a ball or radius λNeed to study theoretical common grounds with work from Alter, Caselles and Chambolle .


SIAM06 Imaging Science7. ‘’Morphological Diversity and Source Separation’’, J. L. Starck,

CEA, France



Sparse … … representation



Mixture model

Problem


SIAM06 Imaging ScienceSolution


SIAM06 Imaging Science8. ‘’ 3D Directional Filter Banks and Surfacelets ‘’, M. Do, Urbana-

Champaign, USA



Recursive 2D implementation?



Pyramidal construction through orthogonal 2D wedge-shape planes.



H1

H2



Trick on decimation sequence & dimension to have pyramid=wedge*wedge



Model of interest: singularities live on smooth surfaces‘’surface-like’’ basis function.

Software package: c++ & Matlab


SIAM06 Imaging Science9. ‘’ Optimization Involving L∞-Norm for Image Restoration’’ , P. Weiss, L.

Blanc-Feraud, G. Aubert, INRIA/Université de Nice, Sophia Antipolis, FRANCE

Study of models L∞- +TV suited for image restoration, degraded by a quantification process. Model not strictly convex -> many solutions.Unicity dervived with minimal surface constraintNo convincing results so far.

To study: Is TV the appropriate constraint?


SIAM06 Imaging Science10. ‘’A Majorization-Minimization Algorithm for Total Variation Image

Deconvolution’’, M. T. Figueiredo, J. Dias, Instituto Superior Tecnico, PORTUGAL

TV minimization with a data term modelling a convolution operator. Miminization with surrogate method pour effectuer la minimisation. Impressive results

To study: convergence is guaranteed, but is it towards a global minimum? Probably. Speed of convergence?


SIAM06 Imaging Science11. ‘’Shape Representation based on Integral Kernels: Application to

Image Matching and Segmentation’’, Prados-Vese, UCLA, USA


SIAM06 Imaging Science12. ‘’A Multi-scale Image Representation using Hierarchical (BV,L2)

Decomposition’’, Tadmor-Vese


SIAM06 Imaging ScienceSpace interpolation BV ⇒ L2

For image processing


For image segmentation (ROF)

Proposed multiscale image representation




Hierarchical decomposition

Convergence of Decomposition

(weak)


SIAM06 Imaging ScienceConvergence of Decomposition

(strong)

Initialisation



Numerical Implementation

elsa angelini - telecom paris

Documents