Diffeomorphic Image Research Articles

SPECIFIED GLOBAL POINCARE–BENDIXSON ANNULUS WITH THE LIMIT CYCLE OF THE RAYLEIGH SYSTEM

In the work of A. Grin and K. Schneider [1] two algebraic transversal ovals, which form the Poincare–Bendixson annulus 𝐴(𝜆), are analytically constructed. This annulus contains the unique limit cycle of the Rayleigh equation ¨ 𝑥+𝜆( ˙ 𝑥2/3−1) ˙ 𝑥+𝑥 = 0 for all values of the parameter 𝜆 > 0. In the constructed annulus 𝐴(𝜆) inner boundary consists of zero-level set of the Dulac–Cherkas function and outer boundary represents the diffeomorphic image corresponding boundary of such an annulus for the van der Pol system. In this paper new methods of construction two Dulac–Cherkas functions are worked out. With help of these functions a better inner boundary of the Poincare–Bendixson annulus 𝐴(𝜆) depending on the parameter is found. Also, for the Rayleigh system, a procedure for direct finding a polynomial whose zero-level set contains a transversal oval used as the outer boundary 𝐴(𝜆) is proposed. Then we determine the interval for 𝜆 where the better outer boundary of the annulus is a closed contour, which composed from two arcs of the constructed oval and two arcs of non-closed curves belonging to zerolevel set of one from Dulac–Cherkas functions. Thus, finally the specified global Poincare–Bendixson annulus for the limit cycle of the Rayleigh system is presented.

Multiscale Approach for Variational Problem Joint Diffeomorphic Image Registration and Intensity Correction: Theory and Application

Multiscale Approach for Variational Problem Joint Diffeomorphic Image Registration and Intensity Correction: Theory and Application

Explicit–implicit symmetric diffeomorphic deformable image registration with convolutional neural network

AbstractMedical image registration is essential and a key step in many advanced medical image tasks. In recent years, medical image registration has been applied to many clinical diagnoses, but large deformation registration is still a challenge. Deep learning‐based methods typically have higher accuracy but do not involve spatial transformation, which ignores some desirable properties, including topology preservation and the invertibility of transformation, for medical imaging studies. On the other hand, diffeomorphic registration methods achieve a differentiable spatial transformation, which guarantees topology preservation and invertibility of transformation, but registration accuracy is low. Therefore, a diffeomorphic deformation registration with CNN is proposed, based on a symmetric architecture, simultaneously estimating forward and inverse deformation fields. CNN with Efficient Channel Attention is used to better capture the spatial relationship. Deformation fields are optimized explicitly and implicitly to enhance the invertibility of transformations. An extensive experimental evaluation is performed using two 3D datasets. The proposed method is compared with different state‐of‐the‐art methods. The experimental results show excellent registration accuracy while better guaranteeing the diffeomorphic transformation.

Three-Stage Approach for 2D/3D Diffeomorphic Multimodality Image Registration with Textural Control

Three-Stage Approach for 2D/3D Diffeomorphic Multimodality Image Registration with Textural Control

Modelling systematic anatomical uncertainties of head and neck cancer patients during fractionated radiotherapy treatment

Objective. Head and neck cancer patients experience systematic as well as random day to day anatomical changes during fractionated radiotherapy treatment. Modelling the expected systematic anatomical changes could aid in creating treatment plans which are more robust against such changes. Approach. Inter- patient correspondence aligned all patients to a model space. Intra- patient correspondence between each planning CT scan and on treatment cone beam CT scans was obtained using diffeomorphic deformable image registration. The stationary velocity fields were then used to develop B-Spline based patient specific (SM) and population average (AM) models. The models were evaluated geometrically and dosimetrically. A leave-one-out method was used to compare the training and testing accuracy of the models. Main results. Both SMs and AMs were able to capture systematic changes. The average surface distance between the registration propagated contours and the contours generated by the SM was less than 2 mm, showing that the SM are able to capture the anatomical changes which a patient experiences during the course of radiotherapy. The testing accuracy was lower than the training accuracy of the SM, suggesting that the model overfits to the limited data available and therefore, also captures some of the random day to day changes. For most patients the AMs were a better estimate of the anatomical changes than assuming there were no changes, but the AMs could not capture the variability in the anatomical changes seen in all patients. No difference was seen in the training and testing accuracy of the AMs. These observations were highlighted in both the geometric and dosimetric evaluations and comparisons. Significance. In this work, a SM and AM are presented which are able to capture the systematic anatomical changes of some head and neck cancer patients over the course of radiotherapy treatment. The AM is able to capture the overall trend of the population, but there is large patient variability which highlights the need for more complex, capable population models.

EPDIFF-JF-NET: Adjoint Jacobi Fields for Diffeomorphic Registration Networks

This paper presents a deep learning unsupervisedapproach for diffeomorphic image registrationcalled EPDiff-JF-Net. We propose a novel paralleltransport layer to compute the gradients necessaryfor training with adjoint Jacobi fields. We test ourmethod on two independent brain MRI datasets andobtain state-of-the-art results.

Medical image registration via neural fields

Image registration is an essential step in many medical image analysis tasks. Traditional methods for image registration are primarily optimization-driven, finding the optimal deformations that maximize the similarity between two images. Recent learning-based methods, trained to directly predict transformations between two images, run much faster, but suffer from performance deficiencies due to domain shift. Here we present a new neural network based image registration framework, called NIR (Neural Image Registration), which is based on optimization but utilizes deep neural networks to model deformations between image pairs. NIR represents the transformation between two images with a continuous function implemented via neural fields, receiving a 3D coordinate as input and outputting the corresponding deformation vector. NIR provides two ways of generating deformation field: directly output a displacement vector field for general deformable registration, or output a velocity vector field and integrate the velocity field to derive the deformation field for diffeomorphic image registration. The optimal registration is discovered by updating the parameters of the neural field via stochastic mini-batch gradient descent. We describe several design choices that facilitate model optimization, including coordinate encoding, sinusoidal activation, coordinate sampling, and intensity sampling. NIR is evaluated on two 3D MR brain scan datasets, demonstrating highly competitive performance in terms of both registration accuracy and regularity. Compared to traditional optimization-based methods, our approach achieves better results in shorter computation times. In addition, our methods exhibit performance on a cross-dataset registration task, compared to the pre-trained learning-based methods.

Diffeomorphic transformer-based abdomen MRI-CT deformable image registration.

Stereotactic body radiotherapy (SBRT) is a well-established treatment modality for liver metastases in patients unsuitable for surgery. Both CT and MRI are useful during treatment planning for accurate target delineation and to reduce potential organs-at-risk (OAR) toxicity from radiation. MRI-CT deformable image registration (DIR) is required to propagate the contours defined on high-contrast MRI to CT images. An accurate DIR method could lead to more precisely defined treatment volumes and superior OAR sparing on the treatment plan. Therefore, it is beneficial to develop an accurate MRI-CT DIR for liver SBRT. To create a new deep learning model that can estimate the deformation vector field (DVF) for directly registering abdominal MRI-CT images. The proposed method assumed a diffeomorphic deformation. By using topology-preserved deformation features extracted from the probabilistic diffeomorphic registration model, abdominal motion can be accurately obtained and utilized for DVF estimation. The model integrated Swin transformers, which have demonstrated superior performance in motion tracking, into the convolutional neural network (CNN) for deformation feature extraction. The model was optimized using a cross-modality image similarity loss and a surface matching loss. To compute the image loss, a modality-independent neighborhood descriptor (MIND) was used between the deformed MRI and CT images. The surface matching loss was determined by measuring the distance between the warped coordinates of the surfaces of contoured structures on the MRI and CT images. To evaluate the performance of the model, a retrospective study was carried out on a group of 50 liver cases that underwent rigid registration of MRI and CT scans. The deformed MRI image was assessed against the CT image using the target registration error (TRE), Dice similarity coefficient (DSC), and mean surface distance (MSD) between the deformed contours of the MRI image and manual contours of the CT image. When compared to only rigid registration, DIR with the proposed method resulted in an increase of the mean DSC values of the liver and portal vein from 0.850±0.102 and 0.628±0.129 to 0.903±0.044 and 0.763±0.073, a decrease of the mean MSD of the liver from 7.216±4.513mm to 3.232±1.483mm, and a decrease of the TRE from 26.238±2.769mm to 8.492±1.058mm. The proposed DIR method based on a diffeomorphic transformer provides an effective and efficient way to generate an accurate DVF from an MRI-CT image pair of the abdomen. It could be utilized in the current treatment planning workflow for liver SBRT.

Optimizing ADMM and Over-Relaxed ADMM Parameters for Linear Quadratic Problems

The Alternating Direction Method of Multipliers (ADMM) has gained significant attention across a broad spectrum of machine learning applications. Incorporating the over-relaxation technique shows potential for enhancing the convergence rate of ADMM. However, determining optimal algorithmic parameters, including both the associated penalty and relaxation parameters, often relies on empirical approaches tailored to specific problem domains and contextual scenarios. Incorrect parameter selection can significantly hinder ADMM's convergence rate. To address this challenge, in this paper we first propose a general approach to optimize the value of penalty parameter, followed by a novel closed-form formula to compute the optimal relaxation parameter in the context of linear quadratic problems (LQPs). We then experimentally validate our parameter selection methods through random instantiations and diverse imaging applications, encompassing diffeomorphic image registration, image deblurring, and MRI reconstruction.

A bi-variant variational model for diffeomorphic image registration with relaxed Jacobian determinant constraints

Diffeomorphic registration is a widely used technique for finding a smooth and invertible transformation between two coordinate systems, which are measured using template and reference images. The point-wise volume-preserving constraint det⁡(∇φ(x))=1 is effective in some cases, but may be too restrictive in others, especially when local deformations are relatively large. This can result in poor matching when enforcing large local deformations. In this paper, we propose a new bi-variant diffeomorphic image registration model that introduces a soft constraint on the Jacobian equation det⁡(∇φ(x))=f(x)>0. This allows local deformations to shrink and grow within a flexible range 0<κm<det⁡(∇φ(x))<κM. The Jacobian determinant of transformation is explicitly controlled by optimizing the relaxation function f(x). To prevent deformation folding and improve the smoothness of the transformation, a positive constraint is imposed on the optimization of the relaxation function f(x), and a regularizer is used to ensure the smoothness of f(x). Furthermore, the positivity constraint ensures that f(x) is as close to one as possible, which helps to achieve a volume-preserving transformation on average. We also analyze the existence of the minimizer for the variational model and propose a penalty-splitting algorithm with a multilevel strategy to solve this model. Numerical experiments demonstrate the convergence of the proposed algorithm and show that the positivity constraint can effectively control the range of relative volume without compromising the accuracy of the registration. Moreover, the proposed model generates diffeomorphic maps for large local deformations and outperforms several existing registration models in terms of performance.

SMILE: Siamese Multi-scale Interactive-representation LEarning for Hierarchical Diffeomorphic Deformable image registration

Deformable medical image registration plays an important role in many clinical applications. It aims to find a dense deformation field to establish point-wise correspondences between a pair of fixed and moving images. Recently, unsupervised deep learning-based registration methods have drawn more and more attention because of fast inference at testing stage. Despite remarkable progress, existing deep learning-based methods suffer from several limitations including: (a) they often overlook the explicit modeling of feature correspondences due to limited receptive fields; (b) the performance on image pairs with large spatial displacements is still limited since the dense deformation field is regressed from features learned by local convolutions; and (c) desirable properties, including topology-preservation and the invertibility of transformation, are often ignored. To address above limitations, we propose a novel Convolutional Neural Network (CNN) consisting of a Siamese Multi-scale Interactive-representation LEarning (SMILE) encoder and a Hierarchical Diffeomorphic Deformation (HDD) decoder. Specifically, the SMILE encoder aims for effective feature representation learning and spatial correspondence establishing while the HDD decoder seeks to regress the dense deformation field in a coarse-to-fine manner. We additionally propose a novel Local Invertible Loss (LIL) to encourage topology-preservation and local invertibility of the regressed transformation while keeping high registration accuracy. Extensive experiments conducted on two publicly available brain image datasets demonstrate the superiority of our method over the state-of-the-art (SOTA) approaches. Specifically, on the Neurite-OASIS dataset, our method achieved an average DSC of 0.815 and an average ASSD of 0.633 mm.

R2Net: Efficient and flexible diffeomorphic image registration using Lipschitz continuous residual networks.

Classical diffeomorphic image registration methods, while being accurate, face the challenges of high computational costs. Deep learning based approaches provide a fast alternative to address these issues; however, most existing deep solutions either lose the good property of diffeomorphism or have limited flexibility to capture large deformations, under the assumption that deformations are driven by stationary velocity fields (SVFs). Also, the adopted squaring and scaling technique for integrating SVFs is time- and memory-consuming, hindering deep methods from handling large image volumes. In this paper, we present an unsupervised diffeomorphic image registration framework, which uses deep residual networks (ResNets) as numerical approximations of the underlying continuous diffeomorphic setting governed by ordinary differential equations, which is parameterized by either SVFs or time-varying (non-stationary) velocity fields. This flexible parameterization in our Residual Registration Network (R2Net) not only provides the model's ability to capture large deformation but also reduces the time and memory cost when integrating velocity fields for deformation generation. Also, we introduce a Lipschitz continuity constraint into the ResNet block to help achieve diffeomorphic deformations. To enhance the ability of our model for handling images with large volume sizes, we employ a hierarchical extension with a multi-phase learning strategy to solve the image registration task in a coarse-to-fine fashion. We demonstrate our models on four 3D image registration tasks with a wide range of anatomies, including brain MRIs, cine cardiac MRIs, and lung CT scans. Compared to classical methods SyN and diffeomorphic VoxelMorph, our models achieve comparable or better registration accuracy with much smoother deformations. Our source code is available online at https://github.com/ankitajoshi15/R2Net.

Deep Learning Based Parameterization of Diffeomorphic Image Registration for Cardiac Image Segmentation.

Cardiac segmentation from magnetic resonance imaging (MRI) is one of the essential tasks in analyzing the anatomy and function of the heart for the assessment and diagnosis of cardiac diseases. However, cardiac MRI generates hundreds of images per scan, and manual annotation of them is challenging and time-consuming, and therefore processing these images automatically is of interest. This study proposes a novel end-to-end supervised cardiac MRI segmentation framework based on a diffeomorphic deformable registration that can segment cardiac chambers from 2D and 3D images or volumes. To represent actual cardiac deformation, the method parameterizes the transformation using radial and rotational components computed via deep learning, with a set of paired images and segmentation masks used for training. The formulation guarantees transformations that are invertible and prevents mesh folding, which is essential for preserving the topology of the segmentation results. A physically plausible transformation is achieved by employing diffeomorphism in computing the transformations and activation functions that constrain the range of the radial and rotational components. The method was evaluated over three different data sets and showed significant improvements compared to exacting learning and non-learning based methods in terms of the Dice score and Hausdorff distance metrics.

Rhombus deformation of retinal lateral displacement after epiretinal membrane removal revealed by diffeomorphic image registration.

To establish an analysis method using diffeomorphic image registration and evaluate microvascular displacement through epiretinal membrane (ERM) removal. Medical records of eyes that underwent vitreous surgery for ERM were reviewed. Postoperative optical coherence tomography angiography (OCTA) image s were converted to the corresponding preoperative images according to a configured algorithm using diffeomorphism. Thirty-seven eyes with ERM were examined. Measured changes in the foveal avascular zone (FAZ) area showed a significant negative correlation with central foveal thickness (CFT). The average amplitude of microvascular displacement calculated for each pixel was 69±27 μm in the nasal area, which was relatively smaller than that in other areas. The vector map, which included both the amplitude and vector of microvasculature displacement, showed a unique vector flow pattern called the rhombus deformation sign in 17 eyes. Eyes with this deformation sign showed less surgery-induced changes in the FAZ area and CFT and a milder ERM stage than those without this sign. We calculated and visualized microvascular displacement using diffeomorphism. We found a unique pattern (rhombus deformation) of retinal lateral displacement through ERM removal, which was significantly associated with the severity of ERM.

Craniofacial Soft-Tissue Anthropomorphic Database with Magnetic Resonance Imaging and Unbiased Diffeomorphic Registration.

Objective assessment of craniofacial surgery outcomes in a pediatric population is challenging because of the complexity of patient presentations, diversity of procedures performed, and rapid craniofacial growth. There is a paucity of robust methods to quantify anatomical measurements by age and objectively compare craniofacial dysmorphology and postoperative outcomes. Here, the authors present data in developing a racially and ethnically sensitive anthropomorphic database, providing plastic and craniofacial surgeons with "normal" three-dimensional anatomical parameters with which to appraise and optimize aesthetic and reconstructive outcomes. Patients with normal craniofacial anatomy undergoing head magnetic resonance imaging (MRI) scans from 2008 to 2021 were included in this retrospective study. Images were used to construct composite (template) images with diffeomorphic image registration method using the Advanced Normalization Tools package. Composites were thresholded to generate binary three-dimensional segmentations used for anatomical measurements in Materalise Mimics. High-resolution MRI scans from 130 patients generated 12 composites from an average of 10 MRI sequences each: four 3-year-olds, four 4-year-olds, and four 5-year-olds (two male, two female, two Black, and two White). The average head circumference of 3-, 4-, and 5-year-old composites was 50.3, 51.5, and 51.7 cm, respectively, comparable to normative data published by the World Health Organization. Application of diffeomorphic registration-based image template algorithm to MRI is effective in creating composite templates to represent "normal" three-dimensional craniofacial and soft-tissue anatomy. Future research will focus on development of automated computational tools to characterize anatomical normality, generation of indices to grade preoperative severity, and quantification of postoperative results to reduce subjectivity bias.

PD22-12 STATISTICAL SPATIAL MAPPING OF KIDNEY STONES BASED ON A KIDNEY ATLAS DERIVED FROM CT SCANS

You have accessJournal of UrologyCME1 Apr 2023PD22-12 STATISTICAL SPATIAL MAPPING OF KIDNEY STONES BASED ON A KIDNEY ATLAS DERIVED FROM CT SCANS Jiong Wu, Katherine Fischer, Yuemeng Li, Benjamin Schurhamer, Axel Largent, Joey Logan, Abhay Singh, Joanie Garratt, Justin Ziemba, Gregory Tasian, and Yong Fan Jiong WuJiong Wu More articles by this author , Katherine FischerKatherine Fischer More articles by this author , Yuemeng LiYuemeng Li More articles by this author , Benjamin SchurhamerBenjamin Schurhamer More articles by this author , Axel LargentAxel Largent More articles by this author , Joey LoganJoey Logan More articles by this author , Abhay SinghAbhay Singh More articles by this author , Joanie GarrattJoanie Garratt More articles by this author , Justin ZiembaJustin Ziemba More articles by this author , Gregory TasianGregory Tasian More articles by this author , and Yong FanYong Fan More articles by this author View All Author Informationhttps://doi.org/10.1097/JU.0000000000003295.12AboutPDF ToolsAdd to favoritesDownload CitationsTrack CitationsPermissionsReprints ShareFacebookLinked InTwitterEmail Abstract INTRODUCTION AND OBJECTIVE: Kidney stones vary in size and location. Precise characterization of the kidney stones’ size and location may facilitate effective treatment planning, and spatial mapping of the kidney stones of a group of patients offers a quantitative means to understand how kidney stones spatially distribute across subjects within the kidney. This study evaluated the feasibility to create a kidney atlas, assess individual kidney stones’ location and size, and statistically map kidney stones of a group of patients based on their CT scans using image segmentation and registration algorithms. METHODS: Non-contrast CT scans of 122 patients with kidney stones were retrospectively identified. A deep learning (DL) image segmentation model was built to segment kidneys automatically. Kidney stones were automatically identified as outlier voxels with Hounsfield Unit (HU) larger than mean plus 5-sigma of HUs of all voxels within the kidney, and a kidney stone image was generated to have unit intensity value at locations of stone voxels and zero otherwise. Bilateral kidney atlases were created by aligning all kidney images to a manually selected one using affine-transformation and diffeomorphic deformable image registration algorithms subsequently. The kidney stone image of each kidney was spatially transformed to the corresponding left or right kidney atlas with the same image deformation information obtained for creating the atlases. Statistical spatial maps of the kidney stones were generated for the bilateral kidneys separately to quantify frequency of the stones at every voxel of the kidney atlases. RESULTS: The DL image segmentation model segmented the kidneys with an average dice value of 0.96. The automatic stone identification algorithm detected individual kidney stones with 100% sensitivity. Statistical spatial maps of the kidney stones quantified spatial frequency of the kidney stones across subjects, with the highest frequency up to 6%. CONCLUSIONS: The statistical spatial mapping of kidney stones provide a quantitative means to characterize frequency of stones within the kidneys. The method can be used to assess kidney stones’ location for individual subjects with reference to group atlases that encode statistical spatial distribution information at a group level. Source of Funding: NIDDK P20 CHOP/ Penn Center for Machine Learning in Urology (P20DK127488)AUA Care Foundation and SPU Sushil Lacy Research Scholar Award (KMF) © 2023 by American Urological Association Education and Research, Inc.FiguresReferencesRelatedDetails Volume 209Issue Supplement 4April 2023Page: e669 Advertisement Copyright & Permissions© 2023 by American Urological Association Education and Research, Inc.MetricsAuthor Information Jiong Wu More articles by this author Katherine Fischer More articles by this author Yuemeng Li More articles by this author Benjamin Schurhamer More articles by this author Axel Largent More articles by this author Joey Logan More articles by this author Abhay Singh More articles by this author Joanie Garratt More articles by this author Justin Ziemba More articles by this author Gregory Tasian More articles by this author Yong Fan More articles by this author Expand All Advertisement PDF downloadLoading ...

A 2D diffeomorphic image registration model with inequality constraint

A 2D diffeomorphic image registration model with inequality constraint

3D diffeomorphic image registration with Cauchy–Riemann constraint and lower bounded deformation divergence

In order to eliminate mesh folding in 3D image registration problem, we propose a 3D diffeomorphic image registration model with Cauchy–Riemann constraint and lower bounded deformation divergence. This model preserves the local shape and ensures no mesh folding. The existence of solution for the proposed model is proved. Furthermore, an alternating directional projection 3D image registration algorithm is presented to solve the proposed model. Moreover, numerical tests show that the proposed algorithm is competitive compared with the other four algorithms.

Asymptotics of pure dimer coverings on rail yard graphs

Abstract We study the asymptotic limit of random pure dimer coverings on rail yard graphs when the mesh sizes of the graphs go to 0. Each pure dimer covering corresponds to a sequence of interlacing partitions starting with an empty partition and ending in an empty partition. Under the assumption that the probability of each dimer covering is proportional to the product of weights of present edges, we obtain the limit shape (law of large numbers) of the rescaled height functions and the convergence of the unrescaled height fluctuations to a diffeomorphic image of the Gaussian free field (Central Limit Theorem), answering a question in [7]. Applications include the limit shape and height fluctuations for pure steep tilings [9] and pyramid partitions [20; 36; 39; 38]. The technique to obtain these results is to analyze a class of Macdonald processes which involve dual partitions as well.

Global algebraic Poincaré–Bendixson annulus for the Rayleigh equation

We consider the Rayleigh equation x ¨ + λ ( x ˙ 2 / 3 − 1 ) x ˙ + x = 0 depending on the real parameter λ and construct a Poincaré–Bendixson annulus A λ in the phase plane containing the unique limit cycle Γ λ of the Rayleigh equation for all λ &gt; 0 . The novelty of this annulus consists in the fact that its boundaries are algebraic curves depending on λ . The polynomial defining the interior boundary represents a special Dulac–Cherkas function for the Rayleigh equation which immediately implies that the Rayleigh equation has at most one limit cycle. The outer boundary is the diffeomorphic image of the corresponding boundary for the van der Pol equation. Additionally we present some equations which are linearly topologically equivalent to the Rayleigh equation and provide also for these equations global algebraic Poincaré–Bendixson annuli.