Ridge and valley enhancing filters are widely used in applications such as vessel detection in medical image computing. When images are degraded by noise or include vessels at different scales, such filters are an essential step for meaningful and stable vessel localization. In this work, we propose a novel multi-scale anisotropic fourth-order diffusion equation that allows us to smooth along vessels, while sharpening them in the orthogonal direction. The proposed filter uses a fourth-order diffusion tensor whose eigentensors and eigenvalues are determined from the local Hessian matrix, at a scale that is automatically selected for each pixel. We discuss efficient implementation using a fast explicit diffusion scheme and demonstrate results on synthetic images and vessels in fundus images. Compared to previous isotropic and anisotropic fourth-order filters, as well as established second-order vessel enhancing filters, our newly proposed one better restores the centerlines in all cases.
In a one-dimensional setting, discrete variants of higher order data regularization can be traced back to a 1922 article by Whittaker @cite_1 . A first approach for higher order regularization in image processing involving the absolute value of all second order partial derivatives has been proposed by Scherzer @cite_32 . The resulting method has the drawback of not being rotationally invariant. An extension of classical regularization by choosing @math as argument of the penalizer in the smoothness term has been proposed by You and Kaveh @cite_12 . Their method introduces speckle artifacts around edges that require some post-processing. Both problems can be solved by using the squared Frobenius norm of the Hessian matrix @math as argument of the penalizer. This has been proposed by @cite_7 .
@cite_7 @cite_1 @cite_32 @cite_12
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TV regularization is an L1 penalization of the flow gradient magnitudes, and due to the tendency of the L1 norm to favor sparse solutions (i.e. lots of ‘zeros’), the fill-in effect caused by the regularizer leads to piecewise constant solutions in weakly textured areas. This effect, known as ‘staircasing’ in a 1D setting, can be reduced significantly by using a quadratic penalization for small gradient magnitudes while sticking to linear penalization for larger magnitudes to maintain the discontinuity preserving properties known from TV. A comparison of isotropic TV and isotropic Huber regularity is shown in Fig. 1 by means of rendering the disparities u1 of the Dimetrodon dataset. The color coded flow (cf. Fig. 1(a)) is superimposed as texture. Based on the two observations that motion discontinuities often occur along object boundaries and that in turn object boundaries often coincide Statistical image reconstruction using penalized weighted least-squares (PWLS) criteria can improve image-quality in X-ray computed tomography (CT). However, the huge dynamic range of the statistical weights leads to a highly shift-variant inverse problem making it difficult to precondition and accelerate existing iterative algorithms that attack the statistical model directly. We propose to alleviate the problem by using a variable-splitting scheme that separates the shift-variant and ("nearly") invariant components of the statistical data model and also decouples the regularization term. This leads to an equivalent constrained problem that we tackle using the classical method-of-multipliers framework with alternating minimization. The specific form of our splitting yields an alternating direction method of multipliers (ADMM) algorithm with an inner-step involving a "nearly" shift-invariant linear system that is suitable for FFT-based preconditioning using cone-type filters. The proposed method can efficiently handle a variety of convex regularization criteria including smooth edge-preserving regularizers and non- smooth sparsity-promoting ones based on the l1-norm and total variation. Numerical experiments with synthetic and real in vivo human data illustrate that cone-filter preconditioners accelerate the proposed ADMM resulting in fast convergence of ADMM compared to conventional (nonlinear conjugate gradient, ordered subsets) and state-of-the-art (MFISTA, split-Bregman) algorithms that are applicable for CT. In this paper we introduce a novel higher-order regularization term. The proposed regularizer is a non-local extension of the popular second-order Total Generalized variation, which favors piecewise affine solutions and allows to incorporate soft-segmentation cues into the regularization term. These properties make this regularizer especially appealing for optical flow estimation, where it offers accurately localized motion boundaries and allows to resolve ambiguities in the matching term. We additionally propose a novel matching term which is robust to illumination and scale changes, two major sources of errors in optical flow estimation algorithms. We extensively evaluate the proposed regularizer and data term on two challenging benchmarks, where we are able to obtain state of the art results. Our method is currently ranked first among classical two-frame optical flow methods on the KITTI optical flow benchmark. Regularization is a well studied problem in the context of neural networks. It is usually used to improve the generalization performance when the number of input samples is relatively small or heavily contaminated with noise. The regularization of a parametric model can be achieved in different manners some of which are early stopping (Morgan and Bourlard, 1990), weight decay, output smoothing that are used to avoid overfitting during the training of the considered model. From a Bayesian point of view, many regularization techniques correspond to imposing certain prior distributions on model parameters (Krogh and Hertz, 1991). Using Bishop's approximation (Bishop, 1995) of the objective function when a restricted type of noise is added to the input of a parametric function, we derive the higher order terms of the Taylor expansion and analyze the coefficients of the regularization terms induced by the noisy input. In particular we study the effect of penalizing the Hessian of the mapping function with respect to the input in terms of generalization performance. We also show how we can control independently this coefficient by explicitly penalizing the Jacobian of the mapping function on corrupted inputs.