SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 861-887, June 2024. Abstract.In this paper, we propose a new approach to deformable image registration that captures sliding motions. The large deformation diffeomorphic metric mapping (LDDMM) registration method faces challenges in representing sliding motion since it per construction generates smooth warps. To address this issue, we extend LDDMM by incorporating both zeroth- and first-order momenta with a nondifferentiable kernel. This allows us to represent both discontinuous deformation at switching boundaries and diffeomorphic deformation in homogeneous regions. We provide a mathematical analysis of the proposed deformation model from the viewpoint of discontinuous systems. To evaluate our approach, we conduct experiments on both artificial images and the publicly available DIR-Lab 4DCT dataset. Results show the effectiveness of our approach in capturing plausible sliding motion.
{"title":"Sliding at First-Order: Higher-Order Momentum Distributions for Discontinuous Image Registration","authors":"Lili Bao, Jiahao Lu, Shihui Ying, Stefan Sommer","doi":"10.1137/23m1558665","DOIUrl":"https://doi.org/10.1137/23m1558665","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 861-887, June 2024. <br/> Abstract.In this paper, we propose a new approach to deformable image registration that captures sliding motions. The large deformation diffeomorphic metric mapping (LDDMM) registration method faces challenges in representing sliding motion since it per construction generates smooth warps. To address this issue, we extend LDDMM by incorporating both zeroth- and first-order momenta with a nondifferentiable kernel. This allows us to represent both discontinuous deformation at switching boundaries and diffeomorphic deformation in homogeneous regions. We provide a mathematical analysis of the proposed deformation model from the viewpoint of discontinuous systems. To evaluate our approach, we conduct experiments on both artificial images and the publicly available DIR-Lab 4DCT dataset. Results show the effectiveness of our approach in capturing plausible sliding motion.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"25 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140567735","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Hong Ye Tan, Subhadip Mukherjee, Junqi Tang, Carola-Bibiane Schönlieb
SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 785-819, June 2024. Abstract.Plug-and-Play (PnP) methods are a class of efficient iterative methods that aim to combine data fidelity terms and deep denoisers using classical optimization algorithms, such as ISTA or ADMM, with applications in inverse problems and imaging. Provable PnP methods are a subclass of PnP methods with convergence guarantees, such as fixed point convergence or convergence to critical points of some energy function. Many existing provable PnP methods impose heavy restrictions on the denoiser or fidelity function, such as nonexpansiveness or strict convexity, respectively. In this work, we propose a novel algorithmic approach incorporating quasi-Newton steps into a provable PnP framework based on proximal denoisers, resulting in greatly accelerated convergence while retaining light assumptions on the denoiser. By characterizing the denoiser as the proximal operator of a weakly convex function, we show that the fixed points of the proposed quasi-Newton PnP algorithm are critical points of a weakly convex function. Numerical experiments on image deblurring and super-resolution demonstrate 2–8x faster convergence as compared to other provable PnP methods with similar reconstruction quality.
{"title":"Provably Convergent Plug-and-Play Quasi-Newton Methods","authors":"Hong Ye Tan, Subhadip Mukherjee, Junqi Tang, Carola-Bibiane Schönlieb","doi":"10.1137/23m157185x","DOIUrl":"https://doi.org/10.1137/23m157185x","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 785-819, June 2024. <br/> Abstract.Plug-and-Play (PnP) methods are a class of efficient iterative methods that aim to combine data fidelity terms and deep denoisers using classical optimization algorithms, such as ISTA or ADMM, with applications in inverse problems and imaging. Provable PnP methods are a subclass of PnP methods with convergence guarantees, such as fixed point convergence or convergence to critical points of some energy function. Many existing provable PnP methods impose heavy restrictions on the denoiser or fidelity function, such as nonexpansiveness or strict convexity, respectively. In this work, we propose a novel algorithmic approach incorporating quasi-Newton steps into a provable PnP framework based on proximal denoisers, resulting in greatly accelerated convergence while retaining light assumptions on the denoiser. By characterizing the denoiser as the proximal operator of a weakly convex function, we show that the fixed points of the proposed quasi-Newton PnP algorithm are critical points of a weakly convex function. Numerical experiments on image deblurring and super-resolution demonstrate 2–8x faster convergence as compared to other provable PnP methods with similar reconstruction quality.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"34 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140567734","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 756-783, March 2024. Abstract. In this paper, we consider a low-rank tensor recovery problem. Based on the tensor singular value decomposition (t-SVD), we propose the ratio of the tensor nuclear norm and the tensor Frobenius norm (TNF) as a novel nonconvex surrogate of tensor’s tubal rank. The rationale of the proposed model for enforcing a low-rank structure is analyzed as its theoretical properties. Specifically, we introduce a null space property (NSP) type condition, under which a low-rank tensor is a local minimum for the proposed TNF recovery model. Numerically, we consider a low-rank tensor completion problem as a specific application of tensor recovery and employ the alternating direction method of multipliers (ADMM) to secure a model solution with guaranteed subsequential convergence under mild conditions. Extensive experiments demonstrate the superiority of our proposed model over state-of-the-art methods.
{"title":"A Scale-Invariant Relaxation in Low-Rank Tensor Recovery with an Application to Tensor Completion","authors":"Huiwen Zheng, Yifei Lou, Guoliang Tian, Chao Wang","doi":"10.1137/23m1560847","DOIUrl":"https://doi.org/10.1137/23m1560847","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 756-783, March 2024. <br/> Abstract. In this paper, we consider a low-rank tensor recovery problem. Based on the tensor singular value decomposition (t-SVD), we propose the ratio of the tensor nuclear norm and the tensor Frobenius norm (TNF) as a novel nonconvex surrogate of tensor’s tubal rank. The rationale of the proposed model for enforcing a low-rank structure is analyzed as its theoretical properties. Specifically, we introduce a null space property (NSP) type condition, under which a low-rank tensor is a local minimum for the proposed TNF recovery model. Numerically, we consider a low-rank tensor completion problem as a specific application of tensor recovery and employ the alternating direction method of multipliers (ADMM) to secure a model solution with guaranteed subsequential convergence under mild conditions. Extensive experiments demonstrate the superiority of our proposed model over state-of-the-art methods.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"54 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140325095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 706-755, March 2024. Abstract.This paper proposes a novel method for computing bijective density-equalizing quasiconformal flattening maps for multiply connected open surfaces. In conventional density-equalizing maps, shape deformations are solely driven by prescribed constraints on the density distribution, defined as the population per unit area, while the bijectivity and local geometric distortions of the mappings are uncontrolled. Also, prior methods have primarily focused on simply connected open surfaces but not surfaces with more complicated topologies. Our proposed method overcomes these issues by formulating the density diffusion process as a quasiconformal flow, which allows us to effectively control the local geometric distortion and guarantee the bijectivity of the mapping by solving an energy minimization problem involving the Beltrami coefficient of the mapping. To achieve an optimal parameterization of multiply connected surfaces, we develop an iterative scheme that optimizes both the shape of the target planar circular domain and the density-equalizing quasiconformal map onto it. In addition, landmark constraints can be incorporated into our proposed method for consistent feature alignment. The method can also be naturally applied to simply connected open surfaces. By changing the prescribed population, a large variety of surface flattening maps with different desired properties can be achieved. The method is tested on both synthetic and real examples, demonstrating its efficacy in various applications in computer graphics and medical imaging.
{"title":"Bijective Density-Equalizing Quasiconformal Map for Multiply Connected Open Surfaces","authors":"Zhiyuan Lyu, Gary P. T. Choi, Lok Ming Lui","doi":"10.1137/23m1594376","DOIUrl":"https://doi.org/10.1137/23m1594376","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 706-755, March 2024. <br/> Abstract.This paper proposes a novel method for computing bijective density-equalizing quasiconformal flattening maps for multiply connected open surfaces. In conventional density-equalizing maps, shape deformations are solely driven by prescribed constraints on the density distribution, defined as the population per unit area, while the bijectivity and local geometric distortions of the mappings are uncontrolled. Also, prior methods have primarily focused on simply connected open surfaces but not surfaces with more complicated topologies. Our proposed method overcomes these issues by formulating the density diffusion process as a quasiconformal flow, which allows us to effectively control the local geometric distortion and guarantee the bijectivity of the mapping by solving an energy minimization problem involving the Beltrami coefficient of the mapping. To achieve an optimal parameterization of multiply connected surfaces, we develop an iterative scheme that optimizes both the shape of the target planar circular domain and the density-equalizing quasiconformal map onto it. In addition, landmark constraints can be incorporated into our proposed method for consistent feature alignment. The method can also be naturally applied to simply connected open surfaces. By changing the prescribed population, a large variety of surface flattening maps with different desired properties can be achieved. The method is tested on both synthetic and real examples, demonstrating its efficacy in various applications in computer graphics and medical imaging.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"138 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140314602","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 672-705, March 2024. Abstract. We develop a boundary integral equation–based numerical method to solve for the electrostatic potential in two dimensions, inside a medium with piecewise constant conductivity, where the boundary condition is given by the complete electrode model (CEM). The CEM is seen as the most accurate model of the physical setting where electrodes are placed on the surface of an electrically conductive body, currents are injected through the electrodes, and the resulting voltages are measured again on these same electrodes. The integral equation formulation is based on expressing the electrostatic potential as the solution to a finite number of Laplace equations which are coupled through boundary matching conditions. This allows us to re-express the solution in terms of single-layer potentials; the problem is thus recast as a system of integral equations on a finite number of smooth curves. We discuss an adaptive method for the solution of the resulting system of mildly singular integral equations. This forward solver is both fast and accurate. We then present a numerical inverse solver for electrical impedance tomography which uses our forward solver at its core. To demonstrate the applicability of our results we test our numerical methods on an open electrical impedance tomography data set provided by the Finnish Inverse Problems Society.
{"title":"A Boundary Integral Equation Method for the Complete Electrode Model in Electrical Impedance Tomography with Tests on Experimental Data","authors":"Teemu Tyni, Adam R. Stinchcombe, Spyros Alexakis","doi":"10.1137/23m1585696","DOIUrl":"https://doi.org/10.1137/23m1585696","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 672-705, March 2024. <br/> Abstract. We develop a boundary integral equation–based numerical method to solve for the electrostatic potential in two dimensions, inside a medium with piecewise constant conductivity, where the boundary condition is given by the complete electrode model (CEM). The CEM is seen as the most accurate model of the physical setting where electrodes are placed on the surface of an electrically conductive body, currents are injected through the electrodes, and the resulting voltages are measured again on these same electrodes. The integral equation formulation is based on expressing the electrostatic potential as the solution to a finite number of Laplace equations which are coupled through boundary matching conditions. This allows us to re-express the solution in terms of single-layer potentials; the problem is thus recast as a system of integral equations on a finite number of smooth curves. We discuss an adaptive method for the solution of the resulting system of mildly singular integral equations. This forward solver is both fast and accurate. We then present a numerical inverse solver for electrical impedance tomography which uses our forward solver at its core. To demonstrate the applicability of our results we test our numerical methods on an open electrical impedance tomography data set provided by the Finnish Inverse Problems Society.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"363 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140203035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Julien Flamant, Konstantin Usevich, Marianne Clausel, David Brie
SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 632-671, March 2024. Abstract. This work introduces polarimetric Fourier phase retrieval (PPR), a physically inspired model to leverage polarization of light information in Fourier phase retrieval problems. We provide a complete characterization of its uniqueness properties by unraveling equivalencies with two related problems, namely, bivariate phase retrieval and a polynomial autocorrelation factorization problem. In particular, we show that the problem admits a unique solution, which can be formulated as a greatest common divisor (GCD) of measurement polynomials. As a result, we propose algebraic solutions for PPR based on approximate GCD computations using the null-space properties of Sylvester matrices. Alternatively, existing iterative algorithms for phase retrieval, semidefinite positive relaxation and Wirtinger flow, are carefully adapted to solve the PPR problem. Finally, a set of numerical experiments permits a detailed assessment of the numerical behavior and relative performances of each proposed reconstruction strategy. They further demonstrate the fruitful combination of algebraic and iterative approaches toward a scalable, computationally efficient, and robust to noise reconstruction strategy for PPR.
{"title":"Polarimetric Fourier Phase Retrieval","authors":"Julien Flamant, Konstantin Usevich, Marianne Clausel, David Brie","doi":"10.1137/23m1570971","DOIUrl":"https://doi.org/10.1137/23m1570971","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 632-671, March 2024. <br/> Abstract. This work introduces polarimetric Fourier phase retrieval (PPR), a physically inspired model to leverage polarization of light information in Fourier phase retrieval problems. We provide a complete characterization of its uniqueness properties by unraveling equivalencies with two related problems, namely, bivariate phase retrieval and a polynomial autocorrelation factorization problem. In particular, we show that the problem admits a unique solution, which can be formulated as a greatest common divisor (GCD) of measurement polynomials. As a result, we propose algebraic solutions for PPR based on approximate GCD computations using the null-space properties of Sylvester matrices. Alternatively, existing iterative algorithms for phase retrieval, semidefinite positive relaxation and Wirtinger flow, are carefully adapted to solve the PPR problem. Finally, a set of numerical experiments permits a detailed assessment of the numerical behavior and relative performances of each proposed reconstruction strategy. They further demonstrate the fruitful combination of algebraic and iterative approaches toward a scalable, computationally efficient, and robust to noise reconstruction strategy for PPR.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"42 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140108004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 540-594, March 2024. Abstract. For problems in image processing and many other fields, a large class of effective neural networks has encoder-decoder-based architectures. Although these networks have shown impressive performance, mathematical explanations of their architectures are still underdeveloped. In this paper, we study the encoder-decoder-based network architecture from the algorithmic perspective and provide a mathematical explanation. We use the two-phase Potts model for image segmentation as an example for our explanations. We associate the segmentation problem with a control problem in the continuous setting. Then, the continuous control model is time discretized by an operator-splitting scheme, the PottsMGNet, and space discretized by the multigrid method. We show that the resulting discrete PottsMGNet is equivalent to an encoder-decoder-based network. With minor modifications, it is shown that a number of the popular encoder-decoder-based neural networks are just instances of the proposed PottsMGNet. By incorporating the soft-threshold-dynamics into the PottsMGNet as a regularizer, the PottsMGNet has shown to be robust with the network parameters such as network width and depth and has achieved remarkable performance on datasets with very large noise. In nearly all our experiments, the new network always performs better than or as well as on accuracy and dice score compared to existing networks for image segmentation.
{"title":"PottsMGNet: A Mathematical Explanation of Encoder-Decoder Based Neural Networks","authors":"Xue-Cheng Tai, Hao Liu, Raymond Chan","doi":"10.1137/23m1586355","DOIUrl":"https://doi.org/10.1137/23m1586355","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 540-594, March 2024. <br/> Abstract. For problems in image processing and many other fields, a large class of effective neural networks has encoder-decoder-based architectures. Although these networks have shown impressive performance, mathematical explanations of their architectures are still underdeveloped. In this paper, we study the encoder-decoder-based network architecture from the algorithmic perspective and provide a mathematical explanation. We use the two-phase Potts model for image segmentation as an example for our explanations. We associate the segmentation problem with a control problem in the continuous setting. Then, the continuous control model is time discretized by an operator-splitting scheme, the PottsMGNet, and space discretized by the multigrid method. We show that the resulting discrete PottsMGNet is equivalent to an encoder-decoder-based network. With minor modifications, it is shown that a number of the popular encoder-decoder-based neural networks are just instances of the proposed PottsMGNet. By incorporating the soft-threshold-dynamics into the PottsMGNet as a regularizer, the PottsMGNet has shown to be robust with the network parameters such as network width and depth and has achieved remarkable performance on datasets with very large noise. In nearly all our experiments, the new network always performs better than or as well as on accuracy and dice score compared to existing networks for image segmentation.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"123 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140075565","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Gaik Ambartsoumian, Mohammad J. Latifi Jebelli, Rohit K. Mishra
SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 595-631, March 2024. Abstract.The paper discusses numerical implementations of various inversion schemes for generalized V-line transforms on vector fields introduced in [G. Ambartsoumian, M. J. Latifi, and R. K. Mishra, Inverse Problems, 36 (2020), 104002]. It demonstrates the possibility of efficient recovery of an unknown vector field from five different types of data sets, with and without noise. We examine the performance of the proposed algorithms in a variety of setups, and illustrate our results with numerical simulations on different phantoms.
SIAM 影像科学杂志》第 17 卷第 1 期第 595-631 页，2024 年 3 月。 摘要：本文讨论了[G. Ambartsoumian, M. J. Latifi, and R. K. Mishra, Inverse Problems, 36 (2020), 104002]中介绍的矢量场广义 V 线变换的各种反演方案的数值实现。它展示了从有噪声和无噪声的五种不同类型数据集中高效恢复未知向量场的可能性。我们检验了所提算法在各种设置下的性能，并通过在不同模型上的数值模拟说明了我们的结果。
{"title":"Numerical Implementation of Generalized V-Line Transforms on 2D Vector Fields and their Inversions","authors":"Gaik Ambartsoumian, Mohammad J. Latifi Jebelli, Rohit K. Mishra","doi":"10.1137/23m1573112","DOIUrl":"https://doi.org/10.1137/23m1573112","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 595-631, March 2024. <br/> Abstract.The paper discusses numerical implementations of various inversion schemes for generalized V-line transforms on vector fields introduced in [G. Ambartsoumian, M. J. Latifi, and R. K. Mishra, Inverse Problems, 36 (2020), 104002]. It demonstrates the possibility of efficient recovery of an unknown vector field from five different types of data sets, with and without noise. We examine the performance of the proposed algorithms in a variety of setups, and illustrate our results with numerical simulations on different phantoms.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"7 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140055286","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 501-539, March 2024. Abstract. Many imaging problems can be formulated as mapping problems. A general mapping problem aims to obtain an optimal mapping that minimizes an energy functional subject to the given constraints. Existing methods to solve the mapping problems are often inefficient and can sometimes get trapped in local minima. An extra challenge arises when the optimal mapping is required to be diffeomorphic. In this work, we address the problem by proposing a deep-learning framework based on the Quasiconformal (QC) Teichmüller theories. The main strategy is to learn the Beltrami coefficient (BC) that represents a mapping as the latent feature vector in the deep neural network. The BC measures the local geometric distortion under the mapping, with which the interpretability of the deep neural network can be enhanced. Under this framework, the diffeomorphic property of the mapping can be controlled via a simple activation function within the network. The optimal mapping can also be easily regularized by integrating the BC into the loss function. A crucial advantage of the proposed framework is that once the network is successfully trained, the optimized mapping corresponding to each input data information can be obtained in real time. To examine the efficacy of the proposed framework, we apply the method to the diffeomorphic image registration problem. Experimental results outperform other state-of-the-art registration algorithms in both efficiency and accuracy, which demonstrate the effectiveness of our proposed framework to solve the mapping problem.
{"title":"A Deep Learning Framework for Diffeomorphic Mapping Problems via Quasi-conformal Geometry Applied to Imaging","authors":"Qiguang Chen, Zhiwen Li, Lok Ming Lui","doi":"10.1137/22m1516099","DOIUrl":"https://doi.org/10.1137/22m1516099","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 501-539, March 2024. <br/> Abstract. Many imaging problems can be formulated as mapping problems. A general mapping problem aims to obtain an optimal mapping that minimizes an energy functional subject to the given constraints. Existing methods to solve the mapping problems are often inefficient and can sometimes get trapped in local minima. An extra challenge arises when the optimal mapping is required to be diffeomorphic. In this work, we address the problem by proposing a deep-learning framework based on the Quasiconformal (QC) Teichmüller theories. The main strategy is to learn the Beltrami coefficient (BC) that represents a mapping as the latent feature vector in the deep neural network. The BC measures the local geometric distortion under the mapping, with which the interpretability of the deep neural network can be enhanced. Under this framework, the diffeomorphic property of the mapping can be controlled via a simple activation function within the network. The optimal mapping can also be easily regularized by integrating the BC into the loss function. A crucial advantage of the proposed framework is that once the network is successfully trained, the optimized mapping corresponding to each input data information can be obtained in real time. To examine the efficacy of the proposed framework, we apply the method to the diffeomorphic image registration problem. Experimental results outperform other state-of-the-art registration algorithms in both efficiency and accuracy, which demonstrate the effectiveness of our proposed framework to solve the mapping problem.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"55 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140036956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 476-500, March 2024. Abstract.Via chirp functions from fractional Fourier transforms, we introduce fractional Riesz potentials related to chirp functions, which are further used to give a new image encryption method with double phase coding. In a comparison with the image encryption method based on fractional Fourier transforms, via a series of image encryption and decryption experiments, we demonstrate that the symbols of fractional Riesz potentials related to chirp functions and the order of fractional Fourier transforms essentially provide greater flexibility and information security. We also establish the relations of fractional Riesz potentials related to chirp functions with fractional Fourier transforms, fractional Laplace operators, and fractional Riesz transforms, and we obtain their boundedness on rotation invariant spaces.
{"title":"Fractional Fourier Transforms Meet Riesz Potentials and Image Processing","authors":"Zunwei Fu, Yan Lin, Dachun Yang, Shuhui Yang","doi":"10.1137/23m1555442","DOIUrl":"https://doi.org/10.1137/23m1555442","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 476-500, March 2024. <br/>Abstract.Via chirp functions from fractional Fourier transforms, we introduce fractional Riesz potentials related to chirp functions, which are further used to give a new image encryption method with double phase coding. In a comparison with the image encryption method based on fractional Fourier transforms, via a series of image encryption and decryption experiments, we demonstrate that the symbols of fractional Riesz potentials related to chirp functions and the order of fractional Fourier transforms essentially provide greater flexibility and information security. We also establish the relations of fractional Riesz potentials related to chirp functions with fractional Fourier transforms, fractional Laplace operators, and fractional Riesz transforms, and we obtain their boundedness on rotation invariant spaces.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"36 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140006573","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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