Affine Transformation Research Articles

New cluster algebras from old: integrability beyond Zamolodchikov periodicity

Abstract We consider discrete dynamical systems obtained as deformations of mutations in cluster algebras associated with finite-dimensional simple Lie algebras. The original (undeformed) dynamical systems provide the simplest examples of Zamolodchikov periodicity: they are affine birational maps for which every orbit is periodic with the same period. Following on from preliminary work by one of us with Kouloukas, here we present integrable maps obtained from deformations of cluster mutations related to the simple root systems 
$A_3$, $B_2$, $B_3$ and $D_4$. We further show how new cluster algebras arise, by considering Laurentification, that is, a lifting to a higher-dimensional map expressed in a set of new variables (tau functions), for which the dynamics exhibits the Laurent property. For the integrable map obtained by deformation of type $A_3$, which already appeared in our previous work, we show that there is a commuting map of Quispel-Roberts-Thompson (QRT) type which is built from a composition of mutations and a permutation applied to 
the same cluster algebra of rank 6, with an additional 2 frozen variables. Furthermore, both the deformed $A_3$ map and the QRT map correspond to translation by a generator in the Mordell-Weil group of a rational elliptic surface of rank two, and the underlying cluster algebra comes from a quiver that is mutation equivalent to the $q$-Painlev\'e III quiver found by Okubo. 
The deformed integrable maps of types $B_2$, $B_3$ and $D_4$ are also related to elliptic surfaces. 

From a dynamical systems viewpoint, the message of the paper is that special families of birational maps with completely periodic dynamics under iteration admit natural deformations that are aperiodic yet completely integrable.

Invariant set estimation for piecewise affine dynamical systems using piecewise affine barrier function

This paper introduces an algorithm for estimating the invariant set of closed-loop controlled dynamical systems identified using single-hidden layer Rectified linear units (ReLU) neural networks or piecewise affine (PWA) functions, particularly addressing the challenge of providing safety guarantees for single-hidden layer ReLU networks commonly used in safety–critical applications. The invariant set of PWA dynamical system is estimated using single-hidden layer ReLU networks or its equivalent PWA function. This method entails formulating the barrier function as a PWA function and converting the search process into a linear optimization problem using vertices. We incorporate a domain refinement strategy to increase flexibility in case the optimization does not find a valid barrier function. Moreover, the objective of the optimization is to find a less conservative invariant set based on the current partition. Our experimental results demonstrate the effectiveness and efficiency of our approach, demonstrating its potential for ensuring the safety of PWA dynamical systems.

Group-theoretic analysis of symmetry-preserving deployable structures and metamaterials.

Many deployable structures in nature, as well as human-made mechanisms, preserve symmetry as their configurations evolve. Examples in nature include blooming flowers, dilation of the iris within the human eye, viral capsid maturation and molecular and bacterial motors. Engineered examples include opening umbrellas, elongating scissor jacks, variable apertures in cameras, expanding Hoberman spheres and some kinds of morphing origami structures. In these cases, the structures either preserve a discrete symmetry group or are described as an evolution from one discrete symmetry group to another of the same type as the structure deploys. Likewise, elastic metamaterials built from lattice structures can also preserve symmetry type while passively deforming and changing lattice parameters. A mathematical formulation of such transitions/deployments is articulated here. It is shown that if [Formula: see text] is Euclidean space, [Formula: see text] is a continuous group of motions of Euclidean space and [Formula: see text] is the type of the discrete subgroup of [Formula: see text] describing the symmetries of the deploying structure, then the symmetry of the evolving structure can be described by time-dependent subgroups of [Formula: see text] of the form [Formula: see text], where [Formula: see text] is a time-dependent affine transformation. Then, instead of considering the whole structure in [Formula: see text], a 'sector' of it that lives in the orbit space [Formula: see text] can be considered at each instant in time, and instead of considering all motions in [Formula: see text], only representatives from right cosets in the space [Formula: see text] need to be considered. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1)'.

High-quality expert annotations enhance artificial intelligence model accuracy for osteosarcoma X-ray diagnosis.

Primary malignant bone tumors, such as osteosarcoma, significantly affect the pediatric and young adult populations, necessitating early diagnosis for effective treatment. This study developed a high-performance artificial intelligence (AI) model to detect osteosarcoma from X-ray images using highly accurate annotated data to improve diagnostic accuracy at initial consultations. Traditional models trained on unannotated data have shown limited success, with sensitivities of approximately 60%-70%. In contrast, our model used a data-centric approach with annotations from an experienced oncologist, achieving a sensitivity of 95.52%, specificity of 96.21%, and an area under the curve of 0.989. The model was trained using 468 X-ray images from 31 osteosarcoma cases and 378 normal knee images with a strategy to maximize diversity in the training and validation sets. It was evaluated using an independent dataset of 268 osteosarcoma and 554 normal knee images to ensure generalizability. By applying the U-net architecture and advanced image processing techniques such as renormalization and affine transformations, our AI model outperforms existing models, reducing missed diagnoses and enhancing patient outcomes by facilitating earlier treatment. This study highlights the importance of high-quality training data and advocates a shift towards data-centric AI development in medical imaging. These insights can be extended to other rare cancers and diseases, underscoring the potential of AI in transforming diagnostic processes in oncology. The integration of this AI model into clinical workflows could support physicians in early osteosarcoma detection, thereby improving diagnostic accuracy and patient care.

FF-RRT*: A Sampling-Based Planner for Multirobot Global Formation Path Planning

Abstract This article presents a collision-free global path planning approach for multirobot systems called flexible formation—rapidly exploring randomized trees* (FF-RRT*). The algorithm is based on RRT* and provides a flexible and efficient representation of the formation geometry independent of the number of robots. It is designed to generate optimal paths for multirobot systems in a five-dimensional configuration space comprising the formation centroid, orientation, and scaling. An affine transformation is used to convert the path in the configuration space to the workspace of the robots. The proposed method employs a new distance function that eliminates the need for tuning weights and a sampling scheme for scaling factors to avoid inter-robot collisions. FF-RRT* is experimentally demonstrated using nonholonomic wheeled mobile robots and is effective in planning the collective motion of the robots. Robots could collectively reach goals by squeezing through narrow gaps and splitting and merging around large obstacles.

Enhanced Electron Emission Performance and Air-Surface Stability in ScO-Terminated Diamond for Thermionic Energy Converters.

Diamond with negative electron affinity (NEA) and low work function surfaces are suggested as a suitable material for electron-generation applications in vacuum, in particular, as the emitter electrode in thermionic energy converters. Such NEA surfaces can be fabricated by evaporating and then annealing submonolayers of a suitable metal in vacuo onto bare or oxidized diamond. Among the metals studied, scandium termination of bare diamond (100) and (111) surfaces is recently reported to give the largest NEA values reported to date for a metal-diamond system, as well as being thermally stable to 900°C. It is now shown that preoxidized (100) diamond functionalized with 0.25monolayers of Sc also produces a large NEA value of -1.02eV with low work functions (<3.63eV). Moreover, this surface is thermally stable to 700°C and can withstand exposure to air for extended periods. Here, the structural and electronic properties of this Sc─O-functionalized diamond surface are characterized in detail using a variety of surface-science techniques. The results suggest that this material may be the ideal candidate for the fabrication of commercial thermionic energy conversion devices, e.g., for solar-power generation, as well as for various other electronic devices that rely upon electron emission.

Sample tilting for speckle suppression through angular compounding.

Speckle degrades the quality of optical coherence tomography (OCT) images and impedes their visual interpretation. Current hardware methods for speckle suppression necessitate difficult hardware modifications. As a result, algorithmic approaches for speckle suppression generally lack validation or training with physically meaningful ground truth. Here, we demonstrate angular compounding through tilting of the sample with a motorized rotation stage. Tomograms acquired at different tilt angles are related to each other through a physics-informed affine map, which can be retrieved directly from the measurement data. Using a mechanical sample tilting stage obviates the need to alter the OCT hardware and enables effective angular compounding with existing OCT instruments.

A hierarchical multi-parametric programming approach for dynamic risk-based model predictive quality control

In this work, we present a hierarchical batch quality control strategy with real-time process safety management. It features a multi-time-scale decision-making framework augmenting: (i) Risk-aware model predictive controller for short-term set point tracking and dynamic risk control under disturbances; (ii) Control-aware optimizer for long-term quality and safety optimization over the entire batch operation; (iii) Intermediate surrogate model to bridge the timescale gap by readjusting the optimizer operating decisions for the controller. All of the above problems are solved via multi-parametric mixed-integer quadratic programming with a key advantage to generate offline explicit control/optimization laws as affine functions of process and risk variables. This allows for the design of a fit-for-purpose risk management plan prior to real-time implementation, while reducing the need for repetitive online dynamic optimization. A unified process model is used to underpin the consistency of hierarchical operational optimization. The proposed approach offers a flexible strategy to integrate distinct decision-making time scales which can be selected separately tailored to the process-specific need of control, fault prognosis, and end-batch quality control. A T2 batch reactor case study is presented to showcase this approach to systematically address the interactions and trade-offs of multiple decision layers toward improving process efficiency and safety.

Retraction Note: A color image watermarking scheme based on affine transformation and S4 permutation

Retraction Note: A color image watermarking scheme based on affine transformation and S4 permutation

The genomic potential of photosynthesis in piconanoplankton is functionally redundant but taxonomically structured at a global scale.

Carbon fixation is a key metabolic function shaping marine life, but the underlying taxonomic and functional diversity involved is only partially understood. Using metagenomic resources targeted at marine piconanoplankton, we provide a reproducible machine learning framework to derive the potential biogeography of genomic functions through the multi-output regression of gene read counts on environmental climatologies. Leveraging the Marine Atlas of Tara Oceans Unigenes, we investigate the genomic potential of primary production in the global ocean. The latter is performed by ribulose-1,5-bisphosphate carboxylase/oxygenase (RUBISCO) and is often associated with carbon concentration mechanisms in piconanoplankton, major marine unicellular photosynthetic organisms. We show that the genomic potential supporting C4 enzymes and RUBISCO exhibits strong functional redundancy and important affinity toward tropical oligotrophic waters. This redundancy is taxonomically structured by the dominance of Mamiellophyceae and Prymnesiophyceae in mid and high latitudes. These findings enhance our understanding of the relationship between functional and taxonomic diversity of microorganisms and environmental drivers of key biogeochemical cycles.

Compact Extended Formulations for Low-Rank Functions with Indicator Variables

We study the mixed-integer epigraph of a special class of convex functions with nonconvex indicator constraints, which are often used to impose logical constraints on the support of the solutions. The class of functions we consider are defined as compositions of low-dimensional nonlinear functions with affine functions. Extended formulations describing the convex hull of such sets can easily be constructed via disjunctive programming although a direct application of this method often yields prohibitively large formulations, whose size is exponential in the number of variables. In this paper, we propose a new disjunctive representation of the sets under study, which leads to compact formulations with size exponential in the dimension of the nonlinear function but polynomial in the number of variables. Moreover, we show how to project out the additional variables for the case of dimension one, recovering or generalizing known results for the convex hulls of such sets (in the original space of variables). Our computational results indicate that the proposed approach can significantly improve the performance of solvers in structured problems. Funding: This work was supported by the National Science Foundation Division of Computing and Communication Foundations [Grant 2006762].

Integrating gene expression and imaging data across Visium capture areas with visiumStitched.

Visium is a widely-used spatially-resolved transcriptomics assay available from 10x Genomics. Standard Visium capture areas (6.5mm by 6.5mm) limit the survey of larger tissue structures, but combining overlapping images and associated gene expression data allow for more complex study designs. Current software can handle nested or partial image overlaps, but is designed for merging up to two capture areas, and cannot account for some technical scenarios related to capture area alignment. We generated Visium data from a postmortem human tissue sample such that two capture areas were partially overlapping and a third one was adjacent. We developed the R/Bioconductor package visiumStitched, which facilitates stitching the images together with Fiji (ImageJ), and constructing SpatialExperiment R objects with the stitched images and gene expression data. visiumStitched constructs an artificial hexagonal array grid which allows seamless downstream analyses such as spatially-aware clustering without discarding data from overlapping spots. Data stitched with visiumStitched can then be interactively visualized with spatialLIBD. visiumStitched provides a simple, but flexible framework to handle various multi-capture area study design scenarios. Specifically, it resolves a data processing step without disrupting analysis workflows and without discarding data from overlapping spots. visiumStiched relies on affine transformations by Fiji, which have limitations and are less accurate when aligning against an atlas or other situations. visiumStiched provides an easy-to-use solution which expands possibilities for designing multi-capture area study designs.

UnWarpME: Unsupervised warping map estimation in real-world scenarios

This paper introduces a novel approach to warp an RGB image to a new target pose, where it learns to inpaint the invisible parts of the image by sampling pixels from the visible parts. This technique is particularly relevant in applications such as novel pose image generation, where the quality of the generated samples heavily relies on the warped images. Conventional methods typically utilize an affine warping map estimator and learn to estimate the warping map through a downstream image generation task. However, this approach results in an affine function being returned regardless of the complexity of the deformation between the source and target poses. In contrast, our proposed method involves estimating the warping map using a convolutional function, which learns through alternating between two downstream tasks. Initially, the estimation treats the warping as part of an image generation task, similar to existing methods but without the constraint of an affine transformation. This encourages the estimation of complex transformations beyond affine deformations. Subsequently, disregarding the image generation task, we supervise the convolutional warping map to learn any potential affine transformation between the guiding poses of the sample. Our experimental results demonstrate the high effectiveness of our method in preserving image textures while transforming it into highly complex poses.

Theorem of Alternative and Scalarization of Optimization Problems in Topological Vector Spaces

In this paper, we introduce the definitions of affinelike, preaffinelike, generalized affinelike, and generalized preaffinelike functions by use of "pointed convex cone", and prove that those definitions of generalized affine functions are all different. We discuss optimization problems in topological vector spaces and obtain a theorem of alternative and a scalarization theorem. Our inequalities are given by partial order relations. Our generalized affineness may be used for many discussions in mathematics or applied mathematics wherever the affineness is a condition.

The Generic Failure of Lower-Semicontinuity for the Linear Distortion Functional

AbstractWe consider the convexity properties of distortion functionals, particularly the linear distortion, defined for homeomorphisms of domains in Euclidean n-spaces, $$n\ge 3$$ n ≥ 3 . The inner and outer distortion functionals are lower semi-continuous in all dimensions and so for the curve modulus or analytic definitions of quasiconformality it ifollows that if $$ \{ f_{n} \}_{n=1}^{\infty } $$ { f n } n = 1 ∞ is a sequence of K-quasiconformal mappings (here K depends on the particular distortion functional but is the same for every element of the sequence) which converges locally uniformly to a mapping f, then this limit function is also K-quasiconformal. Despite a widespread belief that this was also true for the geometric definition of quasiconformality (defined through the linear distortion $$H({f_{n}})$$ H ( f n ) ), T. Iwaniec gave a specific and surprising example to show that the linear distortion functional is not always lower-semicontinuous on uniformly converging sequences of quasiconformal mappings. Here we show that this failure of lower-semicontinuity is common, perhaps generic in the sense that under mild restrictions on a quasiconformal f, there is a sequence $$ \{f_{n} \}_{n=1}^{\infty } $$ { f n } n = 1 ∞ with $$ {f_{n}}\rightarrow {f}$$ f n → f locally uniformly and with $$\limsup _{n\rightarrow \infty } H( {f_{n}})&lt;H( {f})$$ lim sup n → ∞ H ( f n ) &lt; H ( f ) . Our main result shows this is true for affine mappings. Addressing conjectures of Gehring and Iwaniec we show the jump up in the limit can be arbitrarily large and give conjecturally sharp bounds: for each $$\alpha &lt;\sqrt{2}$$ α &lt; 2 there is $${f_{n}}\rightarrow {f}$$ f n → f locally uniformly with f affine and $$\begin{aligned} \alpha \; \limsup _{n\rightarrow \infty } H( {f_{n}}) &lt; H( {f}) \end{aligned}$$ α lim sup n → ∞ H ( f n ) &lt; H ( f ) We conjecture $$\sqrt{2}$$ 2 to be best possible.

Uncertain Scheduling of the Power System Based on Wasserstein Distributionally Robust Optimization and Improved Differential Evolution Algorithm

The rapid development of renewable energy presents challenges to the security and stability of power systems. Aiming at addressing the power system scheduling problem with load demand and wind power uncertainty, this paper proposes the establishment of different error fuzzy sets based on the Wasserstein probability distance to describe the uncertainties of load and wind power separately. Based on these Wasserstein fuzzy sets, a distributed robust chance-constrained scheduling model was established. In addition, the scheduling model was transformed into a linear programming problem through affine transformation and CVaR approximation. The simplex method and an improved differential evolution algorithm were used to solve the model. Finally, the model and algorithm proposed in this paper were applied to model and solve the economic scheduling problem for the IEEE 6-node system with a wind farm. The results show that the proposed method has better optimization performance than the traditional method.

Assessing spatial distribution of bioindicator elements in various cutaneous tumors using correlative imaging with laser-ablation-based analytical methods

Correlative imaging of cutaneous tumors provides additional information to the standard histopathologic examination. However, the joint progress in the establishment of analytical techniques, such as Laser-Induced Breakdown Spectroscopy (LIBS) and Laser Ablation Inductively Coupled Plasma Mass Spectrometry (LA-ICP-MS) in clinical practice is still limited. Their combination provides complementary information as it is also shown in our study in terms of major biotic (Ca, Mg, and P) and trace (Cu and Zn) elements. To elucidate changes in the elemental composition in tumors, we have compiled a set of malignant tumors (Squamous Cell Carcinoma, Basal Cell Carcinoma, Malignant Melanoma, and Epithelioid Angiosarcoma), one benign tumor (Pigmented Nevus) and one healthy-skin sample. The data processing was based on a methodological pipeline involving binary image registration and affine transformation. Thus, our paper brings a feasibility study of a practical methodological concept that enables us to compare LIBS and LA-ICP-MS results despite the mutual spatial distortion of original elemental images. Moreover, we also show that LIBS could be a sufficient pre-screening method even for a larger number of samples according to the speed and reproducibility of the analyses. Whereas LA-ICP-MS could serve as a ground truth and reference technique for preselected samples.

Crop Water Status Analysis from Complex Agricultural Data Using UMAP-Based Local Biplot

To optimize growth and management, precision agriculture relies on a deep understanding of agricultural dynamics, particularly crop water status analysis. Leveraging unmanned aerial vehicles, we can efficiently acquire high-resolution spatiotemporal samples by utilizing remote sensors. However, non-linear relationships among data features, localized within specific subgroups, frequently emerge in agricultural data. Interpreting these complex patterns requires sophisticated analysis due to the presence of noise, high variability, and non-stationarity behavior in the collected samples. Here, we introduce Local Biplot, a methodological framework tailored for discerning meaningful data patterns in non-stationary contexts for precision agriculture. Local Biplot relies on the well-known uniform manifold approximation and projection method, such as UMAP, and local affine transformations to codify non-stationary and non-linear data patterns while maintaining interpretability. This lets us find important clusters for transformation and projection within a single global axis pair. Hence, our framework encompasses variable and observational contributions within individual clusters. At the same time, we provide a relevance analysis strategy to help explain why those clusters exist, facilitating the understanding of data dynamics while favoring interpretability. We demonstrated our method’s capabilities through experiments on both synthetic and real-world datasets, covering scenarios involving grass and rice crops. Moreover, we use random forest and linear regression models to predict water status variables from our Local Biplot-based feature ranking and clusters. Our findings revealed enhanced clustering and prediction capability while emphasizing the importance of input features in precision agriculture. As a result, Local Biplot is a useful tool to visualize, analyze, and compare the intricate underlying patterns and internal structures of complex agricultural datasets.

Automatic reorientation to generate short-axis myocardial PET images

BackgroundAccurately redirecting reconstructed Positron emission tomography (PET) images into short-axis (SA) images shows great significance for subsequent clinical diagnosis. We developed a system for automatic redirection and quantitative analysis of myocardial PET images.MethodsA total of 128 patients were enrolled for 18 F-FDG PET/CT myocardial metabolic images (MMIs), including 3 image classifications: without defects, with defects, and excess uptake. The automatic reorientation system includes five modules: regional division, myocardial segmentation, ellipsoid fitting, image rotation and quantitative analysis. First, the left ventricular geometry-based canny edge detection (LVG-CED) was developed and compared with the other 5 common region segmentation algorithms, the optimized partitioning was determined based on partition success rate. Then, 9 myocardial segmentation methods and 4 ellipsoid fitting methods were combined to derive 36 cross combinations for diagnostic performance in terms of Pearson correlation coefficient (PCC), Kendall correlation coefficient (KCC), Spearman correlation coefficient (SCC), and determination coefficient. Finally, the deflection angles were computed by ellipsoid fitting and the SA images were derived by affine transformation. Furthermore, the polar maps were used for quantitative analysis of SA images, and the redirection effects of 3 different image classifications were analyzed using correlation coefficients.ResultsOn the dataset, LVG-CED outperformed other methods in the regional division module with a 100% success rate. In 36 cross combinations, PSO-FCM and LLS-SVD performed the best in terms of correlation coefficient. The linear results indicate that our algorithm (LVG-CED, PSO-FCM, and LLS-SVD) has good consistency with the reference manual method. In quantitative analysis, the similarities between our method and the reference manual method were higher than 96% at 17 segments. Moreover, our method demonstrated excellent performance in all 3 image classifications.ConclusionOur algorithm system could realize accurate automatic reorientation and quantitative analysis of PET MMIs, which is also effective for images suffering from interference.

Impact of face swapping and data augmentation on sign language recognition

AbstractThis study addresses the challenge of improving communication between the deaf and hearing community by exploring different sign language recognition (SLR) techniques. Due to privacy issues and the need for validation by interpreters, creating large-scale sign language (SL) datasets can be difficult. The authors address this by presenting a new Spanish isolated sign language recognition dataset, CALSE-1000, consisting of 5000 videos representing 1000 glosses, with various signers and scenarios. The study also proposes using different computer vision techniques, such as face swapping and affine transformations, to augment the SL dataset and improve the accuracy of the model I3D trained using them. The results show that the inclusion of these augmentations during training leads to an improvement in accuracy in top-1 metrics by up to 11.7 points, top-5 by up to 8.8 points and top-10 by up to 9 points. This has great potential to improve the state of the art in other datasets and other models. Furthermore, the analysis confirms the importance of facial expressions in the model by testing with a facial omission dataset and shows how face swapping can be used to include new anonymous signers without the costly and time-consuming process of recording.