Optical Deflectometric Tomography (ODT) provides an accurate characterization of transparent materials whose complex surfaces present a real challenge for manufacture and control. In ODT, the refractive index map (RIM) of a transparent object is reconstructed by measuring light deflection under multiple orientations. We show that this imaging modality can be made "compressive", i.e., a correct RIM reconstruction is achievable with far less observations than required by traditional Filtered Back Projection (FBP) methods. Assuming a cartoon-shape RIM model, this reconstruction is driven by minimizing the map Total-Variation under a fidelity constraint with the available observations. Moreover, two other realistic assumptions are added to improve the stability of our approach: the map positivity and a frontier condition. Numerically, our method relies on an accurate ODT sensing model and on a primal-dual minimization scheme, including easily the sensing operator and the proposed RIM constraints. We conclude this paper by demonstrating the power of our method on synthetic and experimental data under various compressive scenarios. In particular, the compressiveness of the stabilized ODT problem is demonstrated by observing a typical gain of 20 dB compared to FBP at only 5 of 360 incident light angles for moderately noisy sensing.
In common Absorption Tomography (AT) we deal with the reconstruction of the absorption index distribution from intensity measurements. As these measurements are directly related to the absorption index, the AT sensing model does not include the frequency @math . In this domain, several works have exploited sparsity based methods. Most recent works in AT have focused on promoting a small TV norm @cite_27 @cite_33 . @cite_8 use a Lagrangian formulation for the tomographic reconstruction problem, promoting a small TV norm under a Kullback-Leiber data divergence and a positivity constraint. They aim at reconstructing a breast phantom from 60 projections with Poisson distributed noise. For this, they use the primal-dual optimization algorithm proposed by @cite_2 . The method results in high quality reconstruction compared to FBP but with a convergence result that is highly dependent on the Lagrangian parameter choosen.
@cite_27 @cite_2 @cite_33 @cite_8
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The reconstruction of photoacoustic images typically neglects the effect of acoustic absorption on the measured time domain signals. Here, a method to compensate for acoustic absorption in photoacoustic tomography is described. The approach is based on time-reversal image reconstruction and an absorbing equation of state which separately accounts for acoustic absorption and dispersion following a frequency power law. Absorption compensation in the inverse problem is achieved by reversing the absorption proportionality coefficient in sign but leaving the equivalent dispersion parameter unchanged. The reconstruction is regularized by filtering the absorption and dispersion terms in the spatial frequency domain using a Tukey window. This maintains the correct frequency dependence of these parameters within the filter pass band. The method is valid in one, two and three dimensions, and for arbitrary power law absorption parameters. The approach is verified through several numerical experiments. The reconstruction of a carbon fibre phantom and the vasculature in the abdomen of a mouse are also presented. When absorption compensation is included, a general improvement in the image magnitude and resolution is seen, particularly for deeper features. Iterative image reconstruction with sparsity-exploiting methods, such as total variation (TV) minimization, investigated in compressive sensing claim potentially large reductions in sampling requirements. Quantifying this claim for computed tomography (CT) is nontrivial, because both full sampling in the discrete-to-discrete imaging model and the reduction in sampling admitted by sparsity-exploiting methods are ill-defined. The present article proposes definitions of full sampling by introducing four sufficient-sampling conditions (SSCs). The SSCs are based on the condition number of the system matrix of a linear imaging model and address invertibility and stability. In the example application of breast CT, the SSCs are used as reference points of full sampling for quantifying the undersampling admitted by reconstruction through TV-minimization. In numerical simulations, factors affecting admissible undersampling are studied. Differences between few-view and few-detector bin reconstruction as well as a relation between object sparsity and admitted undersampling are quantified. We study the compressed sensing reconstruction problem for a broad class of random, band-diagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by KrzakalaEtAl , message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of non-zero coordinates. We use an approximate message passing (AMP) algorithm and analyze it through the state evolution method. We give a rigorous proof that this approach is successful as soon as the undersampling rate @math exceeds the (upper) R 'enyi information dimension of the signal, @math . More precisely, for a sequence of signals of diverging dimension @math whose empirical distribution converges to @math , reconstruction is with high probability successful from @math measurements taken according to a band diagonal matrix. For sparse signals, i.e., sequences of dimension @math and @math non-zero entries, this implies reconstruction from @math measurements. For discrete' signals, i.e., signals whose coordinates take a fixed finite set of values, this implies reconstruction from @math measurements. The result is robust with respect to noise, does not apply uniquely to random signals, but requires the knowledge of the empirical distribution of the signal @math . We consider the problem of the reconstruction of a continuous-time function f(x) spl isin H from the samples of the responses of m linear shift-invariant systems sampled at 1 m the reconstruction rate. We extend Papoulis' generalized sampling theory in two important respects. First, our class of admissible input signals (typ. H=L sub 2 ) is considerably larger than the subspace of band-limited functions. Second, we use a more general specification of the reconstruction subspace V( spl psi ), so that the output of the system can take the form of a band-limited function, a spline, or a wavelet expansion. Since we have enlarged the class of admissible input functions, we have to give up Shannon and Papoulis' principle of an exact reconstruction. Instead, we seek an approximation f spl isin V( spl psi ) that is consistent in the sense that it produces exactly the same measurements as the input of the system. This leads to a generalization of Papoulis' sampling theorem and a practical reconstruction algorithm that takes the form of a multivariate filter. In particular, we show that the corresponding system acts as a projector from H onto V( spl psi ). We then propose two complementary polyphase and modulation domain interpretations of our solution. The polyphase representation leads to a simple understanding of our reconstruction algorithm in terms of a perfect reconstruction filter bank. The modulation analysis, on the other hand, is useful in providing the connection with Papoulis' earlier results for the band-limited case. Finally, we illustrate the general applicability of our theory by presenting new examples of interlaced and derivative sampling using splines.