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**1 - 4**of**4**### Analytic Sensing: Sparse Source Recovery from Boundary Measurements using an Extension of Prony’s Method for the Poisson Equation

, 2011

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### Linear Image Reconstruction by Sobolev Norms on the Bounded Domain

"... Abstract. The reconstruction problem is usually formulated as a variational problem in which one searches for that image that minimizes a so called prior (image model) while insisting on certain image features to be preserved. When the prior can be described by a norm induced by some inner product o ..."

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Abstract. The reconstruction problem is usually formulated as a variational problem in which one searches for that image that minimizes a so called prior (image model) while insisting on certain image features to be preserved. When the prior can be described by a norm induced by some inner product on a Hilbert space the exact solution to the variational problem can be found by orthogonal projection. In previous work we considered the image as compactly supported in L2(R 2) and we used Sobolev norms on the unbounded domain including a smoothing parameter γ> 0 to tune the smoothness of the reconstructed image. Due to the assumption of compact support of the original image, components of the reconstructed image near the image boundary are too much penalized. Therefore, in this work we minimize Sobolev norms only on the actual image domain, yielding much better reconstructions (especially for γ ≫ 0). As an example we apply our method to the reconstruction of singular points that are present in the scale space representation of an image. 1

### Variational Approach to Tomographic Reconstruction

, 2000

"... We formulate the tomographic reconstruction problem in a variational setting. The object to be reconstructed is considered as a continuous density function, unlike in the pixel-based approaches. The measurements are modeled as linear operators (Radon transform), integrating the density function alon ..."

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We formulate the tomographic reconstruction problem in a variational setting. The object to be reconstructed is considered as a continuous density function, unlike in the pixel-based approaches. The measurements are modeled as linear operators (Radon transform), integrating the density function along the ray path. The criterion that we minimize consists of a data term and a regularization term. The data term represents the inconsistency between applying the measurement model to the density function and the real measurements. The regularization term corresponds to the smoothness of the density function. We show that this leads to a solution lying in a finite dimensional vector space which can be expressed as a linear combination of generating functions. The coe#cients of this linear combination are determined from a linear equation set, solvable either directly, or by using an iterative approach. Our experiments show that our new variational method gives results comparable to the classical filtered backprojection for high number of measurements (projection angles and sensor resolution). The new method performs better for medium number of measurements. Furthermore, the variational approach gives usable results even with very few measurements when the filtered back-projection fails. Our method reproduces amplitudes more faithfully and can cope with high noise levels; it can be adapted to various characteristics of the acquisition device. Keywords: tomography, reconstruction, variational, filtered back-projection 1.

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"... In this work, we focus on the problem of reconstructing a volume (scalar 3D field) based on non-uniform point samples and then rendering the volume by exploiting the processing power of GPUs. In the first part involving the reconstruction, we motivate our choice of tensor-product uniform B-splines f ..."

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In this work, we focus on the problem of reconstructing a volume (scalar 3D field) based on non-uniform point samples and then rendering the volume by exploiting the processing power of GPUs. In the first part involving the reconstruction, we motivate our choice of tensor-product uniform B-splines for the discretized representation of the continuous volume. They allow for highly efficient, scalable and accurate reconstruction at multiple scales (resolution levels) at once. By subdividing the volume into blocks and reconstructing them independently, current desktop PCs are able to reconstruct large volumes and multiple CPU cores can be efficiently exploited. We focus on linear and cubic B-splines and on how to eliminate otherwise resulting block discontinuities. Once we have reconstructed the volume at multiple scales, we can derive different Levels of Detail (LoDs) by subdividing the volume into blocks and selecting a suitable scale for each block. We present a fusion scheme which guarantees global C 0 continuity for linear LoDs and C 2 continuity for cubic ones. The challenge here is to minimize visual block interscale discontinuities. A LoD, consisting of a hierarchical spatial subdivision into blocks and an autonomous B-spline coefficient grid for each block, is then rendered via a GPU ray-caster. We achieve interactive frame-rates for qualitative Direct Volume Renderings (DVRs) and real-time frame-rates for iso-surface renderings. ii