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Sparse PDF Volumes for Consistent Multi-Resolution Volume Rendering
Authors:
Ronell Sicat, KAUST
Jens Kruger, University of Duisburg-Essen
Torsten Moller, University of Vienna
Markus Hadwiger, KAUST
Presented by:
Subhashis Hazarika,
Ohio State University
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Challenges
• Substitutes each voxel by a weighted average of its neighbourhood, this changes the distribution values in the volume. Standard approaches use a single value (mean) to represent the voxel footprint distribution.
• Application of transfer functions becomes incompatible and results in inconsistent image across resolution levels. (inconsistency artifacts).
• Ideally an accurate representation of voxel footprint would provide a consistent multi-resolution volume rendering.
– Histogram storage overhead.
– Application of transfer function becomes expensive.
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Proposed Idea
• A compact sparse pdf representation for voxels in 4D (joint space X range domain of the volume).
• Optimize the sparse pdf volume data structure for parallel rendering in GPU.
• A novel approach for computing a sparse 4D pdf approximation via a greedy pursuit algorithm.
• An out-of-core framework for efficient parallel computation of sparse pdf volumes for large scalar volume data.
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Process Overview
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Basic Model
• Xp random variable for voxels associated with position p across different resolution levels.
• fp(r) pdf at position p, r is the intensity range of the volume data.
• t(r) transfer function in the domain of the range of the volume, r.
• Goal of the paper is to: – Store fp(r) effectively and apply t(r)
– Challenge : • Storage overhead
• How to evaluate eq 1.
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Joint 4D space x range domain
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Hierarchy of 4D Gaussian Mixtures
• All the Gaussians at level m have the same standard deviation. – Easy of using convolutions
– Don’t have to store s.d for all Gaussians.
• d
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Hierarchy Computation
• Initial Gaussian Mixture: – Start at level l0 and Gaussian Mixture vo
– Standard deviation:
– Weight:
• Subsequent computation: – Compute m from preceding level m.
– Low pass filter vm to avoid artifacts • By updating spatial s.d and the coefficient ci.
– Our goal is to represent m with fewer Gaussians than vm
– km=km..
– This is done by sparse approximation to m.
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Sparse PDF Volume Computation
• Sparse Approximation Theory:
– H dictionary of atoms (basis functions)
– c is the coefficient vector that determines the linear combination that should best approximate v, given H.
– H in our case consists of translates of Gaussians.
– Target signal v to approximate is a chosen vm after low-pass filter.
– Inorder to obtain sparse representation, c should have as few non-zero elements as possible.
• An NP-hard problem.
• Pursuit Algorithm: greedy iterative method of finding sparse c. – In each iteration the atom from H that best approximates the target function g(x) is
picked by projecting the g(x) into the dictionary.
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Dictionary Projection as Convolution
• Consider 1D function g(x) that we want to approximate.
• h() dictionary of atoms, where u selects the atom
• We will project g(x) onto h(x) (i.e finding inner product of the two functions)
• All dictionary atoms are translates of the same kernel h(x), where h is symmetric around zero. Therefore in terms of kernels h(x).
• This converts the eq.9 to convolution form:
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Dictionary Projection as Convolution
• In order to determine the atom that best approximates g(x) we have to determine which atom results in the largest inner product.
• In terms of convolution:
• Observation: in order to find the dictionary element that best approximates g(x) we simply have to find the max of the function
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Gaussian Dictionaries & Mixtures
• Gaussian Dictionaries:
• Gaussian Mixture: the g(x) function that we approximate is given by k Gaussians with identical s.d.
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Pursuit Algorithm
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Projection in 4D using mode finding
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Sparse PDF Volume Data Structure
• Original volume is subdivided into bricks.
• At l0, stored in usual way, with one scalar per voxel.
• For the other levels, lm, m>0: – 1st sort the set of mixture component …………… based on spatial position p.
– For each voxel with position p we count how many tuples have the same p(p=pi)
– This count is stored in a coefficient count block.
– The pi value is dropped from the tuple and the r and c values are stored in coefficient info array.
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Sparse PDF Volume Data Structure
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Run Time Classification
• Applying the transfer function to the Gaussian mixture.:
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Performance & Scalability
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Results
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Thank You
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