Marc C. Robini

Simulated annealing, acceleration techniques, and image restoration

Typically, the linear image restoration problem is an ill-conditioned, underdetermined inverse problem. Here, stabilization is achieved via the introduction of a first-order smoothness constraint which allows the preservation of edges and leads to the minimization of a nonconvex functional. In order to carry through this optimization task, we use stochastic relaxation with annealing. We prefer the Metropolis dynamics to the popular, but computationally much more expensive, Gibbs sampler. Still, Metropolis-type annealing algorithms are also widely reported to exhibit a low convergence rate. Their finite-time behavior is outlined and we investigate some inexpensive acceleration techniques that do not alter their theoretical convergence properties; namely, restriction of the state space to a locally bounded image space and increasing concave transform of the cost functional. Successful experiments about space-variant restoration of simulated synthetic aperture imaging data illustrate the performance of the resulting class of algorithms and show significant benefits in terms of convergence speed.

Comparison of regularization methods for human cardiac diffusion tensor MRI

Diffusion tensor MRI (DT-MRI) is an imaging technique that is gaining importance in clinical applications. However, there is very little work concerning the human heart. When applying DT-MRI to in vivo human hearts, the data have to be acquired rapidly to minimize artefacts due to cardiac and respiratory motion and to improve patient comfort, often at the expense of image quality. This results in diffusion weighted (DW) images corrupted by noise, which can have a significant impact on the shape and orientation of tensors and leads to diffusion tensor (DT) datasets that are not suitable for fibre tracking. This paper compares regularization approaches that operate either on diffusion weighted images or on diffusion tensors. Experiments on synthetic data show that, for high signal-to-noise ratio (SNR), the methods operating on DW images produce the best results; they substantially reduce noise error propagation throughout the diffusion calculations. However, when the SNR is low, Rician Cholesky and Log-Euclidean DT regularization methods handle the bias introduced by Rician noise and ensure symmetry and positive definiteness of the tensors. Results based on a set of sixteen ex vivo human hearts show that the different regularization methods tend to provide equivalent results.

ENHANCEMENT OF CONTRAST REGIONS IN SUBOPTIMAL ULTRASOUND IMAGES WITH APPLICATION TO ECHOCARDIOGRAPHY

In this paper we propose a novel feature-based contrast enhancement approach to enhance the quality of noisy ultrasound (US) images. Our approach uses a phase-based feature detection algorithm, followed by sparse surface interpolation and subsequent nonlinear postprocessing. We first exploited the intensity-invariant property of phase-based acoustic feature detection to select a set of relevant image features in the data. Then, an approximation to the low-frequency components of the sparse set of selected features was obtained using a fast surface interpolation algorithm. Finally, a nonlinear postprocessing step was applied. Results of applying the method to echocardiographic sequences (2-D + T) are presented. The results demonstrate that the method can successfully enhance the intensity of the interesting features in the image. Better balanced contrasted images are obtained, which is important and useful both for manual processing and assessment by a clinician, and for computer analysis of the sequence. An evaluation protocol is proposed in the case of echocardiographic data and quantitative results are presented. We show that the correction is consistent over time and does not introduce any temporal artefacts.

A Stochastic Continuation Approach to Piecewise Constant Reconstruction

We address the problem of reconstructing a piecewise constant 3-D object from a few noisy 2-D line-integral projections. More generally, the theory developed here readily applies to the recovery of an ideal n-D signal (n ges 1) from indirect measurements corrupted by noise. Stabilization of this ill-conditioned inverse problem is achieved with the Potts prior model, which leads to a challenging optimization task. To overcome this difficulty, we introduce a new class of hybrid algorithms that combines simulated annealing with deterministic continuation. We call this class of algorithms stochastic continuation (SC). We first prove that, under mild assumptions, SC inherits the finite-time convergence properties of generalized simulated annealing. Then, we show that SC can be successfully applied to our reconstruction problem. In addition, we look into the concave distortion acceleration method introduced for standard simulated annealing and we derive an explicit formula for choosing the free parameter of the cost function. Numerical experiments using both synthetic data and real radiographic testing data show that SC outperforms standard simulated annealing.

From simulated annealing to stochastic continuation: a new trend in combinatorial optimization

Simulated annealing (SA) is a generic optimization method that is quite popular because of its ease of implementation and its global convergence properties. However, SA is widely reported to converge very slowly, and it is common practice to allow extra freedom in its design at the expense of losing global convergence guarantees. A natural way to increase the flexibility of SA is to allow the objective function and the communication mechanism to be temperature-dependent, the idea being to gradually reveal the complexity of the optimization problem and to increase the mixing rate at low temperatures. We call this general class of annealing processes stochastic continuation (SC). In the first part of this paper, we introduce SC starting from SA, and we derive simple sufficient conditions for the global convergence of SC. Our main result is interesting in two respects: first, the conditions for global convergence are surprisingly weak—in particular, they do not involve the variations of the objective function with temperature—and second, exponential cooling makes it possible to be arbitrarily close to the best possible convergence speed exponent of SA. The second part is devoted to the application of SC to the problem of producing aesthetically pleasing drawings of undirected graphs. We consider the objective function defined by Kamada and Kawai (Inf Process Lett 31(1):7–15, 1989), which measures the quality of a drawing as a weighted sum of squared differences between Euclidean and graph-theoretic inter-vertex distances. Our experiments show that SC outperforms SA with optimal communication setting both in terms of minimizing the objective function and in terms of standard aesthetic criteria.

A Graph-Based Approach for Automatic Cardiac Tractography

A new automatic algorithm for assessing fiber‐bundle organization in the human heart using diffusion‐tensor magnetic resonance imaging is presented. The proposed approach distinguishes from the locally “greedy” paradigm, which uses voxel‐wise seed initialization intrinsic to conventional tracking algorithms. It formulates the fiber tracking problem as the global problem of computing paths in a boolean‐weighted undirected graph, where each voxel is a vertex and each pair of neighboring voxels is connected with an edge. This leads to a global optimization task that can be solved by iterated conditional modes‐like algorithms or Metropolis‐type annealing. A new deterministic optimization strategy, namely iterated conditional modes with α‐relaxation using (t2)‐ and (t4)‐moves, is also proposed; it has similar performance to annealing but offers a substantial computational gain. This approach offers some important benefits. The global nature of our tractography method reduces sensitivity to noise and modeling errors. The discrete framework allows an optimal balance between the density of fiber bundles and the amount of available data. Besides, seed points are no longer needed; fibers are predicted in one shot for the whole diffusion‐tensor magnetic resonance imaging volume, in a completely automatic way. Magn Reson Med, 2010.

Structure-adaptive sparse denoising for diffusion-tensor MRI

Diffusion tensor magnetic resonance imaging (DT-MRI) is becoming a prospective imaging technique in clinical applications because of its potential for in vivo and non-invasive characterization of tissue organization. However, the acquisition of diffusion-weighted images (DWIs) is often corrupted by noise and artifacts, and the intensity of diffusion-weighted signals is weaker than that of classical magnetic resonance signals. In this paper, we propose a new denoising method for DT-MRI, called structure-adaptive sparse denoising (SASD), which exploits self-similarity in DWIs. We define a similarity measure based on the local mean and on a modified structure-similarity index to find sets of similar patches that are arranged into three-dimensional arrays, and we propose a simple and efficient structure-adaptive window pursuit method to achieve sparse representation of these arrays. The noise component of the resulting structure-adaptive arrays is attenuated by Wiener shrinkage in a transform domain defined by two-dimensional principal component decomposition and Haar transformation. Experiments on both synthetic and real cardiac DT-MRI data show that the proposed SASD algorithm outperforms state-of-the-art methods for denoising images with structural redundancy. Moreover, SASD achieves a good trade-off between image contrast and image smoothness, and our experiments on synthetic data demonstrate that it produces more accurate tensor fields from which biologically relevant metrics can then be computed.

Optimization by Stochastic Continuation

Simulated annealing (SA) and deterministic continuation are well-known generic approaches to global optimization. Deterministic continuation is computationally attractive but produces suboptimal solutions, whereas SA is asymptotically optimal but converges very slowly. In this paper, we introduce a new class of hybrid algorithms which combines the theoretical advantages of SA with the practical advantages of deterministic continuation. We call this class of algorithms stochastic continuation (SC). In a nutshell, SC is a variation of SA in which both the energy function and the communication mechanism are allowed to be time-dependent. We first prove that SC inherits the convergence properties of generalized SA under weak assumptions. Then, we show that SC can be successfully applied to optimization issues raised by the Bayesian approach to signal reconstruction. The considered class of energy functions arises in maximum a posteriori estimation with a Markov random field prior. The associated minimization task is NP-hard and beyond the scope of popular methods such as loopy belief propagation, tree-reweighted message passing, and graph cuts and its extensions. We perform numerical experiments in the context of three-dimensional reconstruction from a very limited number of projections; our results show that SC can substantially outperform both deterministic continuation and SA.

Stochastic nonlinear image restoration using the wavelet transform

The dominant methodology for image restoration is to stabilize the problem by including a roughness penalty in addition to faithfulness to the data. Among various choices, concave stabilizers stand out for their boundary detection capabilities, but the resulting cost function to be minimized is generally multimodal. Although simulated annealing is theoretically optimal to take up this challenge, standard stochastic algorithms suffer from two drawbacks: i) practical convergence difficulties are encountered with second-order prior models and ii) it remains computationally demanding to favor the formation of smooth contour lines by taking the discontinuity field explicitly into account. This work shows that both weaknesses can be overcome in a multiresolution framework by means of the 2-D discrete wavelet transform (DWT). We first propose to improve convergence toward global minima by single-site updating on the wavelet domain. For this purpose, a new restricted DWT space is introduced and a theoretically sound updating mechanism is constructed on this subspace. Next, we suggest to incorporate the smoothness of the discontinuity field via an additional penalty term defined on the high frequency subbands. The resulting increase in complexity is small and the approach requires the specification of a unique extra parameter for which an explicit selection formula is derived.

Two-dimensional ultrasonic flaw detection based on the wavelet packet transform

An important issue in ultrasonic nondestructive evaluation is the detection of flaw echoes in the presence of coherent background noise associated with the microstructure of materials. Many signal processing techniques have proven to be useful for this purpose, but fully 2-D flaw detection techniques remain desirable. In this paper, we describe a novel automatic flaw detection method based on the wavelet packet transform, which is particularly well adapted to B-scan image analysis. After a brief review of the essential elements of the theory of wavelets and wavelet packets, a detailed description of the method is provided. The detection process operates on a set of spatially oriented frequency channels, i.e., detail images, obtained from successive wavelet packet decompositions of the initial B-scan. A statistical selection procedure based on the modeling of the detail image histograms retains the useful information-bearing frequency channels. The flaw information is then extracted from these selected channels by means of a specific thresholding scheme. Some experimental detection results in B-scan images of austenitic stainless steel samples comprising artificial flaws are presented.