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Image Reconstruction and Image Priors Vadim Soloviev, Josias Elisee, Tim Rudge, Simon Arridge Munich April 24, 2009 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A A Slide 2 P4 University College London Computer Science (UCL) UCL has an annual turnover of 500M, academic and research staff totaling 4,000, and over 3,000 PhD research students. The department of Computer Science has over 50 academic staff with specialist groups involved in Imaging Science, Computer Graphics, BioInformatics.Intelligent Systems Networking, and Software Systems Engineering. CMIC In 2005, the Centre for Medical Imaging (CMIC) was formed jointly between Computer Science and the department of Medical Physics & BioEngineering to create a world class grouping combining excellence in medical imaging sciences withinnovative computational methodology, finding application in biomedical research and in healthcare. The research of the group focuses on detailed structural and functional analysis in neurosciences, imaging to guide interventions, image analysis in drug discovery, imaging in cardiology and imaging in oncology with a strong emphasis on e-science technologies. The Centre has very close links with the Faculty of Clinical Sciences, the Faculty of Life Sciences and associated Clinical Institutes, in particular the Institute of Neurology,the Institute of Child Health and the Centre for Neuroimaging Techniques (CNT), Main tasks attributed to the organisation: The main tasks for P4 UCL are WP4 with some input into WP3, WP6 and WP7. We will contribute mathematical and computational techniques for the development of forward and inverse modeling in optical tomography in the diffuse regime particularly using priors, and to the simulation of new imaging devices and the analysis of clinical data. Slide 3 Objective 4.1 : To develop FMT inversion utilizing XCT image priors without strong anatomy function correlations. Progress: Developed structured priors orientating the reconstructed FMT images to have level sets parallel to those of XCT image Developed information theoretic priors orientating the reconstructed FMT images to have maximum joint entropy with the XCT image Initial tests on simulated 3D images of mouse from a realistic atlas Significant Results Reconstructions in 3D depend only linearly on total number of pixels in reconstructed image and independent of the number of pixels in data. Deviations from Annex 1 None Failure to meet critical objectives Not Applicable Use of resources No deviation from work Slide 4 Structural priors Slide 5 Choice of Prior Consider where and D a symmetric tensor By variation where Lagged_Diffusivity Gauss-Newton Method Slide 6 Choice of Prior (2) Examples: 1 st order Tikhonov Total Variation Now choose Where is normal to level set of another image x ref xx ref Slide 7 Target object: cylinder with embedded inhomogeneities Radius: 25 mm, height: 50 mm Background: a =0.01 mm -1, s =1 mm -1 Red: Inclusions with increased absorption Blue: Inclusions with increased scattering Measurements: 80 source locations, 80 detector locations, arranged in 5 rings at elevations -20, -10, 0, +10, +20 Data: log amplitude and phase for source modulated at 100MHz Multiplicative Gaussian noise 0.5% FEM mesh: 83142 nodes, 444278 4-noded tetrahedra Reconstruction grid: 80x80x80 Cross sections through target for planes z=16, z=60 and y=40 aa ss Example: Cylinder with inclusions Slide 8 Reconstruction: Nonlinear conjugate gradients (50 iterations) with line search Prior: TV with hyperparameter = 10 -4 and smoothing parameter = 0.1 aa ss Reconstruction with flat TV prior Iso-surfacesCross sections Slide 9 aa ss Reconstruction with TV prior using correct structural information Edge prior Iso-surfacesCross sections Reconstruction aa ss Iso-surfacesCross sections Slide 10 aa ss TV prior using undifferentiated structural information Edge prior Iso-surfacesCross sections Reconstruction aa ss Iso-surfacesCross sections Slide 11 aa ss TV prior using partial structural information Edge prior Iso-surfacesCross sections Reconstruction aa ss Iso-surfacesCross sections Slide 12 Results using 3D Edge-Weighted Priors Slide 13 Results using 3D Edge-Weighted Priors (2) Slide 14 Information Theoretic Priors Slide 15 Marginal and Joint Entropies Slide 16 Target Distributions and Reference Images Slide 17 Reconstructions aa ss Slide 18 Objective 4.2 : To incorporate XCT image segmentation into the FMT Progress: Developed segmentation of XCT based on anisotropic diffusion (Perona-Malik algorithm) Developed hexahedral adaptive mesh generation from XCT images Incorporated mesh reduction methods using public-domain software ISO2MESH Developed Boundary Element (BEM) and hybrid Boundary-Finite Element (BEM-FEM) methods Significant Results Reconstructions using FEM only for internal organs are much faster than using a complete FEM mesh Deviations from Annex 1 None Failure to meet critical objectives Not Applicable Use of resources No deviation from work Slide 19 Segmentation Requirements construction of meshes for numerical modelling construction of priors as required in Objective 4.1 post-reconstruction object labelling and analysis Slide 20 Anisotropic Diffusion Based Segmentation Slide 21 Mesh Generation Slide 22 BEM and BEM-FEM approach Slide 23 BEM-FEM results Slide 24 Objective 4.3 : To calculate spatially varying optical attenuation in tissues in-vivo. Progress: Developed non-linear reconstruction method for attenuation making use of Louiville transformation from diffusion to Schrodinger equation. Significant Results Reconstruction of attenuation from steady-state data is dependent on good estimates of spatially varying scatter. Deviations from Annex 1 None Failure to meet critical objectives Not Applicable Use of resources No deviation from work Slide 25 Excitation Slide 26 Fluorescence Slide 27 Reconstruction Slide 28 Objective 4.4 :To develop FMT inversion based on simultaneous XCT segmentation and classification Progress: Developed combined reconstruction/segmentation method combining Gauss-Newton image reconstruction with fuzzy-kmeans image classification. Developed fully hierarchical Bayesian framework Significant Results Classification error less than 5% for simulated noisy data. Deviations from Annex 1 None Failure to meet critical objectives Not Applicable Use of resources No deviation from work Slide 29 x x,C x yCyCy Data Noise Statistics Image Image Statistics Class Statistics Reconstruction Step Estimation Step Prior Update Step Combined Reconstruction Classification Slide 30 Heirarchical Bayesian Method Slide 31 3D Animation Slide 32 Deliverables 4.1 Inversion algorithms Deliverable 4.1 was created as an inversion code. Two versions were developed : compiled C++ code using UCL Toast Libraries and OpenGL graphics Matlab program using Mex version of TOAST libraries and Matlab Graphics Individual installations on partner systems will be provided at the next project meeting. Slide 33 Conclusions FEM and BEM based solvers Linear and non-linear reconstruction Large Data Sets using Matrix-Free approach Structural Priors incorporating image information, not dependent on segmentation Statistical Priors based on information theory Matlab based code available on web http://web4.cs.ucl.ac.uk/research/vis/toast/index.html