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Mechanical systems subject to constraints play a essential role in the field of control engineering, profoundly influencing the design and performance of control strategies. Consequently, there is a compelling need to explore diverse control methods to effectively tackle the complex task of stabilizing nonlinear systems while ensuring the constraints are not violated. In this context, this paper proposes a design procedure for position-constrained controllers in robot manipulators. The solution relies on the construction of a diffeomorphism (a differentiable coordinate transformation) that maps all the trajectories of the constrained dynamics into an unconstrained one. The controller design is carried out in the unconstrained dynamics without dealing directly with the constraints. The proposed family of controllers employ an explicit control law which circumvents the need for additional time-consuming computation for feasibility and/or optimization. Moreover, the proposed controller is parametrized by a class of diffeomorphisms which can be selected by the designer. Exponential stability in constrained and unconstrained position states is achieved, in the certain case. For the uncertain case, the controller is augmented through sliding modes guaranteeing finite-time convergence towards the manifold and keeping the exponential convergence within the manifold dynamics. The approach is validated through experiments in an actual 2 DOF lightweight robot manipulator.

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PURPOSE: Diffeomorphic image registration is essential in many medical imaging applications. Several registration algorithms of such type have been proposed, but primarily for intra-contrast alignment. Currently, efficient inter-modal/contrast diffeomorphic registration, which is vital in numerous applications, remains a challenging task. METHODS: We proposed a novel inter-modal/contrast registration algorithm that leverages Robust PaTch-based cOrrelation Ratio metric to allow inter-modal/contrast image alignment and bandlimited geodesic shooting demonstrated in Fourier-Approximated Lie Algebras (FLASH) algorithm for fast diffeomorphic registration. RESULTS: The proposed algorithm, named DiffeoRaptor, was validated with three public databases for the tasks of brain and abdominal image registration while comparing the results against three state-of-the-art techniques, including FLASH, NiftyReg, and Symmetric image Normalization (SyN). CONCLUSIONS: Our results demonstrated that DiffeoRaptor offered comparable or better registration performance in terms of registration accuracy. Moreover, DiffeoRaptor produces smoother deformations than SyN in inter-modal and contrast registration. The code for DiffeoRaptor is publicly available at https://github.com/nimamasoumi/DiffeoRaptor .

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The Variational Principle (VP) is designed to generate non-folding grids (diffeomorphisms) with prescribed Jacobian determinant (JD) and curl. The solution pool of the original VP is based on an additive formulation and, consequently, is not invariant in the diffeomorphic Lie algebra. The original VP works well when the prescribed pair of JD and curl is calculated from a diffeomorphism, but not necessarily when the prescribed JD and curl are unknown to come from a diffeomorphism. In spite of that, the original VP works effectively in 2D grid generations. To resolve this issue, in this paper, we describe a new version of VP (revised VP), which is based on the composition of transformations and, therefore, is invariant in the Lie algebra. The revised VP seems to have overcome the inaccuracy of the original VP in 3D grid generations. In the following sections, the mathematical derivations are presented. It is shown that the revised VP can calculate the inverse transformation of a known diffeomorphism. Its inverse consistency and transitivity of transformations are also demonstrated numerically. Finally, a new definition of averaging diffeomorphisms based on the revised VP is proposed.

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The paper proposes a formation tracking control method for the uncertain artificial swarm systems under the inequality constraints. Not only can the agents perform swarm behaviors (e.g., convergence, formation and avoidance of collision), but they can also track the fixed targets in a constrained area (which is formulated as the inequality constraints, such as unilateral constraint and bilateral constraint.). The swarm behaviors are creatively considered as the servo constraints or the control objectives for the swarm agents. Based on the Udwadia-Kalaba (U-K) equation, those prescribed behaviors are realized by a model-based control design (that is the servo constraints force model-based feedforward control). To deal with the inequality constraints in the formation tracking process, a differential homeomorphism transformation is used to relieve the environmental constraints for the swarm agents. Moreover, the uncertainty of the swarm agents (i.e., the parameter uncertainty in modeling and the external disturbances) is considered, which is time-varying and unknown (but bounded). An uncertainty estimation method with dead-zone and leakage term is designed to calculate the possible upper bound of the uncertainty. In virtue of the estimated upper bound of the uncertainty, a robust control is designed for the uncertain swarm agents to obey the prescribed swarm behaviors in the formation tracking task. The system performances of the artificial swarm systems under the proposed control are theoretically guaranteed by a range of rigorous theorems and numerically verified by the simulations of three agents.

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Medical image segmentation annotated by experts provides the labeled data sets for many scientific researches. However, due to the unevenly experienced backgrounds of the experts and limited numbers of patients with certain diseases or illnesses, not only do such labeled data sets have smaller samples but their quality and normality also can range in wide variabilities and be ambiguous. In practice, these segmentations are usually assigned to be the ground truths for the scientific studies, so it may undermine the trustworthiness of the resulting findings. Therefore, it is meaningful to consider how to give a more unified opinion of the annotations among different experts. In this paper, a novel approach to form normal distributions of segmentation is proposed based on multiple doctors' annotations for the same patient. The proposed approach is developed through the following steps: (1) utilize a framework7 of averaging images to construct an averaged annotation based on different given annotations; (2) determine the image registration deformations from the averaged annotation to the given annotations; (3) build a joint multivariate Gaussian distribution over the logorithm of Jacobian determinants and curls of the registration deformations; lastly, (4) simulate a normal distribution of segmentation by the joint Gaussian distribution of registration deformation. This work translates the problem of forming a normal distribution of the image segmentation into a problem of forming joint Gaussian distribution of image registration deformations, which the latter can be reasoned by Jacobian determinant (models local size of pixel cells) and curl (models local rotation of pixel cells) information. In the following sections, a detailed walk-through of the proposed approach is provided along with its analytical mathematics and numerical examples for its effectiveness. A synthetic example of 3 manually defined label image is made to show how to construct a mean label image, and an example of a real cancer image annotated by 3 doctors demonstrates the formation of the normal distribution and the effectiveness of the propose method.

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BACKGROUND AND OBJECTIVE: Training a deep convolutional neural network (CNN) for automatic image classification requires a large database with images of labeled samples. However, in some applications such as biology and medicine only a few experts can correctly categorize each sample. Experts are able to identify small changes in shape and texture which go unnoticed by untrained people, as well as distinguish between objects in the same class that present drastically different shapes and textures. This means that currently available databases are too small and not suitable to train deep learning models from scratch. To deal with this problem, data augmentation techniques are commonly used to increase the dataset size. However, typical data augmentation methods introduce artifacts or apply distortions to the original image, which instead of creating new realistic samples, obtain basic spatial variations of the original ones. METHODS: We propose a novel data augmentation procedure which generates new realistic samples, by combining two samples that belong to the same class. Although the idea behind the method described in this paper is to mimic the variations that diatoms experience in different stages of their life cycle, it has also been demonstrated in glomeruli and pollen identification problems. This new data augmentation procedure is based on morphing and image registration methods that perform diffeomorphic transformations. RESULTS: The proposed technique achieves an increase in accuracy over existing techniques of 0.47%, 1.47%, and 0.23% for diatom, glomeruli and pollen problems respectively. CONCLUSIONS: For the Diatom dataset, the method is able to simulate the shape changes in different diatom life cycle stages, and thus, images generated resemble newly acquired samples with intermediate shapes. In fact, the other methods compared obtained worse results than those which were not using data augmentation. For the Glomeruli dataset, the method is able to add new samples with different shapes and degrees of sclerosis (through different textures). This is the case where our proposed DA method is more beneficial, when objects highly differ in both shape and texture. Finally, for the Pollen dataset, since there are only small variations between samples in a few classes and this dataset has other features such as noise which are likely to benefit other existing DA techniques, the method still shows an improvement of the results.

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PURPOSE: The traditional learning-based non-rigid registration methods for medical images are trained by an invariant smoothness regularization parameter, which cannot satisfy the registration accuracy and diffeomorphic property simultaneously. The diffeomorphic property reflects the credibility of the registration results. METHOD: To improve the diffeomorphic property in 3D medical image registration, we propose a diffeomorphic cascaded network based on the compressed loss (CL), named LDVoxelMorph. The proposed network has several constituent U-Nets and is trained with deep supervision, which uses a different spatial smoothness regularization parameter in each constituent U-Nets for training. This cascade-variant smoothness regularization parameter can maintain the diffeomorphic property in top cascades with large displacement and achieve precise registration in bottom cascades. Besides, we develop the CL as a penalty for the velocity field, which can accurately limit the velocity field that causes the deformation field overlap after the velocity field integration. RESULTS: In our registration experiments, the dice scores of our method were 0.892 ± 0.040 on liver CT datasets SLIVER37 , 0.848 ± 0.044 on liver CT datasets LiTS38 , 0.689 ± 0.014 on brain MRI datasets LPBA38 , and the number of overlapping voxels of deformation field were 325, 159, and 0, respectively. Ablation study shows that the CL improves the diffeomorphic property more effectively than increases. CONCLUSION: Experiment results show that our method can achieve higher registration accuracy assessed by dice scores and overlapping voxels while maintaining the diffeomorphic property for large deformation.

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Morphological variation of the human pelvis, and particularly the hip bone, mainly results from both female-specific selective pressure related to the give birth of large-headed newborns, and constraints in both sexes for efficient bipedal locomotion, abdominal stability, and adaptation to climate. Hip bone morphology has thus been extensively investigated using several approaches, although the nuances of inter-individual and sex-related variation are still underappreciated, and the effect of sex on ontogenetic patterns is debated. Here, we employ a landmark-free, deformation-based morphometric approach to explore variation in modern human hip bone shape and size from middle adolescence to adulthood. Virtual surface models of the hip bone were obtained from 147 modern human individuals (70 females and 77 males) including adolescents, and young and mature adults. The 3D meshes were registered by rotation, translation, and uniform scaling prior to analysis in Deformetrica. The orientation and amplitude of deviations of individual specimens relative to a global mean were assessed using Principal Component Analysis, while colour maps and vectors were employed for visualisation purposes. Deformation-based morphometrics is a time-efficient and objective method free of observer-dependent biases that allows accurate shape characterisation of general and more subtle morphological variation. Here, we captured nuanced hip bone morphology revealing ontogenetic trends and sex-based variation in arcuate line curvature, greater sciatic notch shape, pubic body and rami length, acetabular expansion, and height-to-width proportions of the ilium. The observed ontogenetic trends showed a higher degree of bone modelling of the lesser pelvis of adolescent females, while male variation was mainly confined to the greater pelvis.

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In MRI studies, spatial normalization is required to infer results at the group level. In the presence of a brain lesion, such as in stroke patients, the normalization process can be affected by tissue loss, spatial deformations, signal intensity changes, and other stroke sequelae that introduce confounds into the group analysis results. Previously, most neuroimaging studies with lesioned brains have used normalization methods optimized for intact brains, raising potential concerns about the accuracy of the resulting transformations and, in turn, their reported group level results. In this study, we demonstrate the benefits of creating an intermediate, cohort-specific template in conjunction with diffeomorphism-based methods to normalize structural MRI images in stroke patients. We show that including this cohort-specific template improves accuracy compared to standard methods for normalizing lesioned brains. Critically, this method reduces overall differences in normalization accuracy between stroke patients and healthy controls, and may improve the localization and connectivity of BOLD signal in functional neuroimaging data.

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We review a modern differential geometric description of fluid isentropic motion and features of it including diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. We analyze the adiabatic liquid dynamics, within which, following the general approach, the nature of the related Poissonian structure on the fluid motion phase space as a semidirect Banach groups product, and a natural reduction of the canonical symplectic structure on its cotangent space to the classical Lie-Poisson bracket on the adjoint space to the corresponding semidirect Lie algebras product are explained in detail. We also present a modification of the Hamiltonian analysis in case of a flow governed by isothermal liquid dynamics. We study the differential-geometric structure of isentropic magneto-hydrodynamic superfluid phase space and its related motion within the Hamiltonian analysis and related invariant theory. In particular, we construct an infinite hierarchy of different kinds of integral magneto-hydrodynamic invariants, generalizing those previously constructed in the literature, and analyzing their differential-geometric origins. A charged liquid dynamics on the phase space invariant with respect to an abelian gauge group transformation is also investigated, and some generalizations of the canonical Lie-Poisson type bracket is presented.

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Nonlinear registration is critical to many aspects of Neuroimaging research. It facilitates averaging and comparisons across multiple subjects, as well as reporting of data in a common anatomical frame of reference. It is, however, a fundamentally ill-posed problem, with many possible solutions which minimise a given dissimilarity metric equally well. We present a regularisation method capable of selectively driving solutions towards those which would be considered anatomically plausible by penalising unlikely lineal, areal and volumetric deformations. This penalty is symmetric in the sense that geometric expansions and contractions are penalised equally, which encourages inverse-consistency. We demonstrate that this method is able to significantly reduce local volume changes and shape distortions compared to state-of-the-art elastic (FNIRT) and plastic (ANTs) registration frameworks. Crucially, this is achieved whilst simultaneously matching or exceeding the registration quality of these methods, as measured by overlap scores of labelled cortical regions. Extensive leveraging of GPU parallelisation has allowed us to solve this highly computationally intensive optimisation problem while maintaining reasonable run times of under half an hour.

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We construct certain operations on stable moduli spaces and use them to compare cohomology of moduli spaces of closed manifolds with tangential structure. We obtain isomorphisms in a stable range provided the p-adic valuation of the Euler characteristics agree, for all primes p not invertible in the coefficients for cohomology.

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In this work, we adopt a new approach to the construction of a global theory of algebras of generalized functions on manifolds based on the concept of smoothing operators. This produces a generalization of previous theories in a form which is suitable for applications to differential geometry. The generalized Lie derivative is introduced and shown to extend the Lie derivative of Schwartz distributions. A new feature of this theory is the ability to define a covariant derivative of generalized scalar fields which extends the covariant derivative of distributions at the level of association. We end by sketching some applications of the theory. This work also lays the foundations for a nonlinear theory of distributional geometry that is developed in a subsequent paper that is based on Colombeau algebras of tensor distributions on manifolds.

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This paper builds on the theory of nonlinear generalized functions begun in Nigsch & Vickers (Nigsch, Vickers 2021 Proc. R. Soc. A 20200640 (doi:10.1098/rspa.2020.0640)) and extends this to a diffeomorphism-invariant nonlinear theory of generalized tensor fields with the sheaf property. The generalized Lie derivative is introduced and shown to commute with the embedding of distributional tensor fields and the generalized covariant derivative commutes with the embedding at the level of association. The concept of a generalized metric is introduced and used to develop a non-smooth theory of differential geometry. It is shown that the embedding of a continuous metric results in a generalized metric with well-defined connection and curvature and that for C 2 metrics the embedding preserves the curvature at the level of association. Finally, we consider an example of a conical metric outside the Geroch-Traschen class and show that the curvature is associated to a delta function.

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We present a construction of algebras of generalized functions of Colombeau-type which, instead of asymptotic estimates with respect to a regularization parameter, employs only topological estimates on certain spaces of kernels for its definition.

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Robust spatial alignment of post mortem data and in vivo MRI acquisitions from different ages, especially from the early developmental stages, into standard spaces is still a bottleneck hampering easy comparison with the mainstream neuroimaging results. In this paper, we test a landmark-based spatial normalization strategy as a framework for the seamless integration of any macroscopic dataset in the context of the Human Brain Project (HBP). This strategy stems from an approach called DISCO embedding sulcal constraints in a registration framework used to initialize DARTEL, the widely used spatial normalization approach proposed in the SPM software. We show that this strategy is efficient with a heterogeneous dataset including challenging data as preterm newborns, infants, post mortem histological data and a synthetic atlas computed from averaging the ICBM database, as well as more commonly studied data acquired in vivo in adults. We then describe some perspectives for a research program aiming at improving folding pattern matching for atlas inference in the context of the future HBP's portal.

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There is a widespread interest in applying pattern recognition methods to anatomical neuroimaging data, but so far, there has been relatively little investigation into how best to derive image features in order to make the most accurate predictions. In this work, a Gaussian Process machine learning approach was used for predicting age, gender and body mass index (BMI) of subjects in the IXI dataset, as well as age, gender and diagnostic status using the ABIDE and COBRE datasets. MRI data were segmented and aligned using SPM12, and a variety of feature representations were derived from this preprocessing. We compared classification and regression accuracy using the different sorts of features, and with various degrees of spatial smoothing. Results suggested that feature sets that did not ignore the implicit background tissue class, tended to result in better overall performance, whereas some of the most commonly used feature sets performed relatively poorly.

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Image registration is a well-known problem in the field of medical imaging. In this paper, we focus on the registration of chest inspiratory and expiratory computed tomography (CT) scans from the same patient. Our method recovers the diffeomorphic elastic displacement vector field (DVF) by jointly regressing the direct and the inverse transformation. Our architecture is based on the RegNet network but we implement a reinforced learning strategy that can accommodate a large training dataset. Our results show that our method performs with a lower estimation error for the same number of epochs than the RegNet approach.

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PURPOSE: OCT offers high in-plane micrometer resolution, enabling studies of neurodegenerative and ocular-disease mechanisms via imaging of the retina at low cost. An important component to such studies is inter-scanner deformable image registration. Image quality of OCT, however, is suboptimal with poor signal-to-noise ratio and through-plane resolution. Geometry of OCT is additionally improperly defined. We developed a diffeomorphic deformable registration method incorporating constraints accommodating the improper geometry and a decentralized-modality-insensitive-neighborhood-descriptors (D-MIND) robust against degradation of OCT image quality and inter-scanner variability. METHOD: The method, called D-MIND Demons, estimates diffeomorphisms using D-MINDs under constraints on the direction of velocity fields in a MIND-Demons framework. Descriptiveness of D-MINDs with/without denoising was ranked against four other shape/texture-based descriptors. Performance of D-MIND Demons and its variants incorporating other descriptors was compared for cross-scanner, intra- and inter-subject deformable registration using clinical retina OCT data. RESULT: D-MINDs outperformed other descriptors with the difference in mutual descriptiveness between high-contrast and homogenous regions > 0.2. Among Demons variants, D-MIND-Demons was computationally efficient, demonstrating robustness against OCT image degradation (noise, speckle, intensity-non-uniformity, and poor through-plane resolution) and consistent registration accuracy [(4±4 µm) and (4±6 µm) in cross-scanner intra- and inter-subject registration] regardless of denoising. CONCLUSIONS: A promising method for cross-scanner, intra- and inter-subject OCT image registration has been developed for ophthalmological and neurological studies of retinal structures. The approach could assist image segmentation, evaluation of longitudinal disease progression, and patient population analysis, which in turn, facilitate diagnosis and patient-specific treatment.

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Atlases provide a framework for spatially mapping information from diverse sources into a common reference space. Specifically, brain atlases allow annotation of gene expression, cell morphology, connectivity, and activity. In larval zebrafish, advances in genetics, imaging, and computational methods now allow the collection of such information brain-wide. However, due to technical considerations, disparate datasets may use different references and may not be aligned to the same coordinate space. Two recent larval zebrafish atlases exemplify this problem: Z-Brain, containing gene expression, neural activity, and neuroanatomical segmentations, was acquired using immunohistochemical stains, while the Zebrafish Brain Browser (ZBB) was constructed from live scans of fluorescent reporters in transgenic larvae. Although different references were used, the atlases included several common transgenic patterns that provide potential "bridges" for transforming each into the other's coordinate space. We tested multiple bridging channels and registration algorithms and found that the symmetric diffeomorphic normalization algorithm improved live brain registration precision while better preserving cell morphology than B-spline-based registrations. Symmetric diffeomorphic normalization also corrected for tissue distortion introduced during fixation. Multi-reference channel optimization provided a transformation that enabled Z-Brain and ZBB to be co-aligned with precision of approximately a single cell diameter and minimal perturbation of cell and tissue morphology. Finally, we developed software to visualize brain regions in 3 dimensions, including a virtual reality neuroanatomy explorer. This study demonstrates the feasibility of integrating whole brain datasets, despite disparate reference templates and acquisition protocols, when sufficient information is present for bridging. Increased accuracy and interoperability of zebrafish digital brain atlases will facilitate neurobiological studies.

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