Journal of Mathematical Imaging and Vision最新文献

Current approaches to generic segmentation start by creating a hierarchy of nested image partitions and then specifying a segmentation from it. Our first contribution is to describe several ways, most of them new, for specifying segmentations using the hierarchy elements. Then, we consider the best hierarchy-induced segmentation specified by a limited number of hierarchy elements. We focus on a common quality measure for binary segmentations, the Jaccard index (also known as IoU). Optimizing the Jaccard index is highly nontrivial, and yet we propose an efficient approach for doing exactly that. This way we get algorithm-independent upper bounds on the quality of any segmentation created from the hierarchy. We found that the obtainable segmentation quality varies significantly depending on the way that the segments are specified by the hierarchy elements, and that representing a segmentation with only a few hierarchy elements is often possible.

Abstract

In this work, we propose a new primal–dual algorithm with adaptive step sizes. The stochastic primal–dual hybrid gradient (SPDHG) algorithm with constant step sizes has become widely applied in large-scale convex optimization across many scientific fields due to its scalability. While the product of the primal and dual step sizes is subject to an upper-bound in order to ensure convergence, the selection of the ratio of the step sizes is critical in applications. Up-to-now there is no systematic and successful way of selecting the primal and dual step sizes for SPDHG. In this work, we propose a general class of adaptive SPDHG (A-SPDHG) algorithms and prove their convergence under weak assumptions. We also propose concrete parameters-updating strategies which satisfy the assumptions of our theory and thereby lead to convergent algorithms. Numerical examples on computed tomography demonstrate the effectiveness of the proposed schemes.

In this work, we tackle the problem of estimating the density ( f_X ) of a random variable ( X ) by successive smoothing, such that the smoothed random variable ( Y ) fulfills the diffusion partial differential equation ( (partial _t - Delta _1)f_Y(,cdot ,, t) = 0 ) with initial condition ( f_Y(,cdot ,, 0) = f_X ). We propose a product-of-experts-type model utilizing Gaussian mixture experts and study configurations that admit an analytic expression for ( f_Y (,cdot ,, t) ). In particular, with a focus on image processing, we derive conditions for models acting on filter, wavelet, and shearlet responses. Our construction naturally allows the model to be trained simultaneously over the entire diffusion horizon using empirical Bayes. We show numerical results for image denoising where our models are competitive while being tractable, interpretable, and having only a small number of learnable parameters. As a by-product, our models can be used for reliable noise level estimation, allowing blind denoising of images corrupted by heteroscedastic noise.

We consider the problem of denoising with the help of prior information taken from a database of clean signals or images. Denoising with variational methods is very efficient if a regularizer well-adapted to the nature of the data is available. Thanks to the maximum a posteriori Bayesian framework, such regularizer can be systematically linked with the distribution of the data. With deep neural networks (DNN), complex distributions can be recovered from a large training database. To reduce the computational burden of this task, we adapt the compressive learning framework to the learning of regularizers parametrized by DNN. We propose two variants of stochastic gradient descent (SGD) for the recovery of deep regularization parameters from a heavily compressed database. These algorithms outperform the initially proposed method that was limited to low-dimensional signals, each iteration using information from the whole database. They also benefit from classical SGD convergence guarantees. Thanks to these improvements we show that this method can be applied for patch-based image denoising.

Topology preservation is a crucial issue in parallel reductions that transform binary pictures by changing only a set of black points to white at a time. In this paper, we present sufficient conditions for topology-preserving parallel reductions on the three types of pictures of the unconventional 3D face-centered cubic (FCC) grid. Some conditions provide methods of verifying that a given parallel reduction always preserves the topology, and the remaining ones directly provide deletion rules of topology-preserving parallel reductions, and make us possible to generate topologically correct thinning algorithms. We give local characterizations of P-simple points, whose simultaneous deletion preserves the topology, and the relationships among the existing universal sufficient conditions for arbitrary types of binary pictures and our new FCC-specific results are also established.

Scale invariance of an algorithm refers to its ability to treat objects equally independently of their size. For neural networks, scale invariance is typically achieved by data augmentation. However, when presented with a scale far outside the range covered by the training set, neural networks may fail to generalize. Here, we introduce the Riesz network, a novel scale- invariant neural network. Instead of standard 2d or 3d convolutions for combining spatial information, the Riesz network is based on the Riesz transform which is a scale-equivariant operation. As a consequence, this network naturally generalizes to unseen or even arbitrary scales in a single forward pass. As an application example, we consider detecting and segmenting cracks in tomographic images of concrete. In this context, ‘scale’ refers to the crack thickness which may vary strongly even within the same sample. To prove its scale invariance, the Riesz network is trained on one fixed crack width. We then validate its performance in segmenting simulated and real tomographic images featuring a wide range of crack widths. An additional experiment is carried out on the MNIST Large Scale data set.

Abstract

The primal–dual hybrid gradient (PDHG) algorithm has been applied for solving linearly constrained convex problems. However, it was shown that without some additional assumptions, convergence may fail. In this work, we propose a new competitive prediction–correction primal–dual hybrid gradient algorithm to solve this kind of problem. Under some conditions, we prove the global convergence for the proposed algorithm with the rate of O(1/T) in a nonergodic sense, and also in the ergodic sense, in terms of the objective function value gap and the constraint violation. Comparative performance analysis of our method with other related methods on some matrix completion and wavelet-based image inpainting test problems shows the outperformance of our approach, in terms of iteration number and CPU time.

Incorporating prior knowledge into a segmentation task, whether it be under the form of geometrical constraints (area/volume penalisation, convexity enforcement, etc.) or of topological constraints (to preserve the contextual relations between objects, to monitor the number of connected components), proves to increase accuracy in medical image segmentation. In particular, it allows to compensate for the issue of weak boundary definition, of imbalanced classes, and to be more in line with anatomical consistency even though the data do not explicitly exhibit those features. This observation underpins the introduced contribution that aims, in a hybrid setting, to leverage the best of both worlds that variational methods and supervised deep learning approaches embody: (a) versatility and adaptability in the mathematical formulation of the problem to encode geometrical/topological constraints, (b) interpretability of the results for the former formalism, while (c) more efficient and effective processing models, (d) ability to become more proficient at learning intricate features and executing more computationally intensive tasks, for the latter one. To be more precise, a unified variational framework involving topological prescriptions in the training of convolutional neural networks through the design of a suitable penalty in the loss function is provided. These topological constraints are implicitly enforced by viewing the segmentation procedure as a registration task between the processed image and its associated ground truth under incompressibility conditions, thus making them homeomorphic. A very preliminary version (Lambert et al., in Calatroni, Donatelli, Morigi, Prato, Santacesaria (eds) Scale space and variational methods in computer vision, Springer, Berlin, 2023, pp. 363–375) of this work has been published in the proceedings of the Ninth International Conference on Scale Space and Variational Methods in Computer Vision, 2023. It contained neither all the theoretical results, nor the detailed related proofs, nor did it include the numerical analysis of the designed algorithm. Besides these more involved developments in the present version, a more complete, systematic and thorough analysis of the numerical experiments is also conducted, addressing several issues: (i) limited amount of labelled data in the training phase, (ii) low contrast or imbalanced classes exhibited by the data, and (iii) explainability of the results.

We introduce a data-driven version of the plus Cartan connection on the homogeneous space ({mathbb {M}}_2) of 2D positions and orientations. We formulate a theorem that describes all shortest and straight curves (parallel velocity and parallel momentum, respectively) with respect to this new data-driven connection and corresponding Riemannian manifold. Then we use these shortest curves for geodesic tracking of complex vasculature in multi-orientation image representations defined on ({mathbb {M}}_{2}). The data-driven Cartan connection characterizes the Hamiltonian flow of all geodesics. It also allows for improved adaptation to curvature and misalignment of the (lifted) vessel structure that we track via globally optimal geodesics. We compute these geodesics numerically via steepest descent on distance maps on ({mathbb {M}}_2) that we compute by a new modified anisotropic fast-marching method.Our experiments range from tracking single blood vessels with fixed endpoints to tracking complete vascular trees in retinal images. Single vessel tracking is performed in a single run in the multi-orientation image representation, where we project the resulting geodesics back onto the underlying image. The complete vascular tree tracking requires only two runs and avoids prior segmentation, placement of extra anchor points, and dynamic switching between geodesic models. Altogether we provide a geodesic tracking method using a single, flexible, transparent, data-driven geodesic model providing globally optimal curves which correctly follow highly complex vascular structures in retinal images. All experiments in this article can be reproduced via documented Mathematica notebooks available at van den Berg (Data-driven left-invariant tracking in Mathematica, 2022).

In this paper, we aim to clarify the statistical and geometric properties of linear resolution conversion for registration between different resolutions observed using the same modality. The pyramid transform is achieved by smoothing and downsampling. The dual operation of the pyramid transform is achieved by linear smoothing after upsampling. The rational-order pyramid transform is decomposed into upsampling for smoothing and the conventional integer-order pyramid transform. By controlling the ratio between upsampling for smoothing and downsampling in the pyramid transform, the rational-order pyramid transform is computed. The tensor expression of the multiway pyramid transform implies that the transform yields orthogonal base systems for any ratio of the rational pyramid transform. The numerical evaluation of the transform shows that the rational-order pyramid transform preserves the normalised distribution of greyscale in images.