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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!
23-June-2006TSI, ENST
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)
23-June-2006TSI, ENST
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,
23-June-2006TSI, ENST
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
23-June-2006TSI, ENST
SIAM06 Imaging Science
TVLo
gan-
Shep
p Ph
anto
m
512 samples on 22 lines
Mini. Energy
Perfect reconstruction with:
23-June-2006TSI, ENST
SIAM06 Imaging Science
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).
23-June-2006TSI, ENST
SIAM06 Imaging Science
(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
23-June-2006TSI, ENST
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
23-June-2006TSI, ENST
SIAM06 Imaging Science
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.
23-June-2006TSI, ENST
SIAM06 Imaging Science3. “Curvature depending functional for image
segmentation with occlusions and transparencies”R. March, ITALY
2.1D sketch of Nitzberg-Mumford
23-June-2006TSI, ENST
SIAM06 Imaging Science
Complete occluded edges by linking end points
Regularity on image discontinuities
23-June-2006TSI, ENST
No algorithmTo study: links with dead leaves model from Matheron, Gousseau
SIAM06 Imaging Science
23-June-2006TSI, ENST
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
23-June-2006TSI, ENST
SIAM06 Imaging Science
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/
23-June-2006TSI, ENST
257 x 257257 x 25712 bits/channel12 bits/channel
3D geometry3D geometrycompletely regular samplingcompletely regular sampling
RGB colors encode XYZ positions
SIAM06 Imaging Science
http://research.microsoft.com/~hoppe/
23-June-2006TSI, ENST
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
http://research.microsoft.com/~hoppe/
23-June-2006TSI, ENST
SIAM06 Imaging Science
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
http://research.microsoft.com/~hoppe/
23-June-2006TSI, ENST
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
23-June-2006TSI, ENST
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 .
23-June-2006TSI, ENST
SIAM06 Imaging Science7. ‘’Morphological Diversity and Source Separation’’, J. L. Starck,
CEA, France
23-June-2006TSI, ENST
SIAM06 Imaging Science8. ‘’ 3D Directional Filter Banks and Surfacelets ‘’, M. Do, Urbana-
Champaign, USA
23-June-2006TSI, ENST
SIAM06 Imaging Science
Pyramidal construction through orthogonal 2D wedge-shape planes.
23-June-2006TSI, ENST
SIAM06 Imaging Science
Trick on decimation sequence & dimension to have pyramid=wedge*wedge
23-June-2006TSI, ENST
SIAM06 Imaging Science
Model of interest: singularities live on smooth surfaces‘’surface-like’’ basis function.
Software package: c++ & Matlab
23-June-2006TSI, ENST
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?
23-June-2006TSI, ENST
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?
23-June-2006TSI, ENST
SIAM06 Imaging Science11. ‘’Shape Representation based on Integral Kernels: Application to
Image Matching and Segmentation’’, Prados-Vese, UCLA, USA
23-June-2006TSI, ENST
SIAM06 Imaging Science12. ‘’A Multi-scale Image Representation using Hierarchical (BV,L2)
Decomposition’’, Tadmor-Vese
23-June-2006TSI, ENST
For image segmentation (ROF)
Proposed multiscale image representation
SIAM06 Imaging Science
23-June-2006TSI, ENST
SIAM06 Imaging Science
Hierarchical decomposition
Convergence of Decomposition
(weak)