Energy minimization methods are a classical tool in a multitude of computer vision applications. While they are interpretable and well-studied, their regularity assumptions are difficult to design by hand. Deep learning techniques on the other hand are purely data-driven, often provide excellent results, but are very difficult to constrain to predefined physical or safety-critical models. A possible combination between the two approaches is to design a parametric energy and train the free parameters in such a way that minimizers of the energy correspond to desired solution on a set of training examples. Unfortunately, such formulations typically lead to bi-level optimization problems, on which common optimization algorithms are difficult to scale to modern requirements in data processing and efficiency. In this work, we present a new strategy to optimize these bi-level problems. We investigate surrogate single-level problems that majorize the target problems and can be implemented with existing tools, leading to efficient algorithms without collapse of the energy function. This framework of strategies enables new avenues to the training of parameterized energy minimization models from large data.
The straightforward way of optimizing bi-level problems is to consider @cite_49 @cite_2 @cite_57 . These methods directly differentiate the higher-level loss function with respect to the minimizing argument and descend in the direction of this gradient. An incomplete list of examples in image processing is @cite_29 @cite_23 @cite_7 @cite_94 @cite_100 @cite_76 @cite_64 @cite_86 @cite_58 . This strategy requires both the higher- and lower-level problems to be smooth and the minimizing map to be invertible. This is usually facilitated by implicit differentiation, as discussed in @cite_69 @cite_60 @cite_94 @cite_23 . In more generality, the problem of directly minimizing @math without assuming that smoothness in @math leads to optimization problems with equilibrium constraints (MPECs), see @cite_80 for a discussion in terms of machine learning or @cite_85 @cite_41 @cite_40 and @cite_57 . This approach also applies to the optimization layers of @cite_79 , which lend themselves well to a reformulation as a bi-level optimization problem.
@cite_64 @cite_41 @cite_29 @cite_85 @cite_2 @cite_58 @cite_69 @cite_60 @cite_23 @cite_49 @cite_80 @cite_7 @cite_57 @cite_79 @cite_40 @cite_86 @cite_76 @cite_94 @cite_100
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Maximization of mutual information of voxel intensities has been demonstrated to be a very powerful criterion for three-dimensional medical image registration, allowing robust and accurate fully automated affine registration of multimodal images in a variety of applications, without the need for segmentation or other preprocessing of the images. In this paper, we investigate the performance of various optimization methods and multiresolution strategies for maximization of mutual information, aiming at increasing registration speed when matching large high-resolution images. We show that mutual information is a continuous function of the affine registration parameters when appropriate interpolation is used and we derive analytic expressions of its derivatives that allow numerically exact evaluation of its gradient. Various multiresolution gradient- and non-gradient-based optimization strategies, such as Powell, simplex, steepest-descent, conjugate-gradient, quasi-Newton and Levenberg—Marquardt methods, are evaluated for registration of computed tomography (CT) and magnetic resonance images of the brain. Speed-ups of a factor of 3 on average compared to Powell's method at full resolution are achieved with similar precision and without a loss of robustness with the simplex, conjugate-gradient and Levenberg—Marquardt method using a two-level multiresolution scheme. Large data sets such as 2562 × 128 MR and 5122 × 48 CT images can be registered with subvoxel precision in < 5 min CPU time on current workstations. One central issue in practically deploying network coding is the adaptive and economic allocation of network resource. We cast this as an optimization, where the net-utility-the difference between a utility derived from the attainable multicast throughput and the total cost of resource provisioning-is maximized. By employing the MAX of flows characterization of the admissible rate region for multicasting, this paper gives a novel reformulation of the optimization problem, which has a separable structure. The Lagrangian relaxation method is applied to decompose the problem into subproblems involving one destination each. Our specific formulation of the primal problem results in two key properties. First, the resulting subproblem after decomposition amounts to the problem of finding a shortest path from the source to each destination. Second, assuming the net-utility function is strictly concave, our proposed method enables a near-optimal primal variable to be uniquely recovered from a near-optimal dual variable. A numerical robustness analysis of the primal recovery method is also conducted. For ill-conditioned problems that arise, for instance, when the cost functions are linear, we propose to use the proximal method, which solves a sequence of well-conditioned problems obtained from the original problem by adding quadratic regularization terms. Furthermore, the simulation results confirm the numerical robustness of the proposed algorithms. Finally, the proximal method and the dual subgradient method can be naturally extended to provide an effective solution for applications with multiple multicast sessions Minimization with orthogonality constraints (e.g., (X^ X = I )) and or spherical constraints (e.g., ( x _2 = 1 )) has wide applications in polynomial optimization, combinatorial optimization, eigenvalue problems, sparse PCA, p-harmonic flows, 1-bit compressive sensing, matrix rank minimization, etc. These problems are difficult because the constraints are not only non-convex but numerically expensive to preserve during iterations. To deal with these difficulties, we apply the Cayley transform—a Crank-Nicolson-like update scheme—to preserve the constraints and based on it, develop curvilinear search algorithms with lower flops compared to those based on projections and geodesics. The efficiency of the proposed algorithms is demonstrated on a variety of test problems. In particular, for the maxcut problem, it exactly solves a decomposition formulation for the SDP relaxation. For polynomial optimization, nearest correlation matrix estimation and extreme eigenvalue problems, the proposed algorithms run very fast and return solutions no worse than those from their state-of-the-art algorithms. For the quadratic assignment problem, a gap 0.842 to the best known solution on the largest problem “tai256c” in QAPLIB can be reached in 5 min on a typical laptop. Many database management systems support whole-image matching. However, users may only remember certain subregions of the images. In this paper, we develop Padding and Reduction Algorithms to support subimage queries of arbitrary size based on local color information. The idea is to estimate the best- case lower bound to the dissimilarity measure between the query and the image. By making use of multiresolution representation, this lower bound becomes tighter as the scale becomes finer. Because image contents are usually pre- extracted and stored, a key issue is how to determine the number of levels used in the representation. We address this issue analytically by estimating the CPU and I O costs, and experimentally by comparing the performance and accuracy of the outcomes of various filtering schemes. Our findings suggest that a 3-level hierarchy is preferred. We also study three strategies for searching multiple resolutions. Our studies indicate that the hybrid strategy with horizontal filtering on the coarse level and vertical filtering on remaining levels is the best choice when using Padding and Reduction Algorithms in the preferred 3-level multiresolution representation. The best 10 desired images can be retrieved efficiently and effectively from a collection of a thousand images in about 3.5 seconds.© (1997) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.