Affine Transformation Research Articles

Underapproximate Reachability Analysis for a Class of Linear Systems With Inputs

Under-approximations of reachable sets and tubes have been receiving growing research attention due to their important roles in control synthesis and verification. Available under-approximation methods applicable to continuous-time linear systems typically assume the ability to compute transition matrices and their integrals exactly, which is not feasible in general, and/or suffer from high computational costs. In this note, we attempt to overcome these drawbacks for a class of linear time-invariant (LTI) systems, where we propose a novel method to under-approximate finite-time forward reachable sets and tubes, utilizing approximations of the matrix exponential and its integral. In particular, we consider the class of continuous-time LTI systems with an identity input matrix and initial and input values belonging to full dimensional sets that are affine transformations of closed unit balls. The proposed method yields computationally efficient under-approximations of reachable sets and tubes, when implemented using zonotopes, with first-order convergence guarantees in the sense of the Hausdorff distance. To illustrate its performance, we implement our approach in three numerical examples, where linear systems of dimensions ranging between 2 and 200 are considered.

Tracial oscillation zero and stable rank one

Abstract Let A be a separable (not necessarily unital) simple $C^*$ -algebra with strict comparison. We show that if A has tracial approximate oscillation zero, then A has stable rank one and the canonical map $\Gamma $ from the Cuntz semigroup of A to the corresponding lower-semicontinuous affine function space is surjective. The converse also holds. As a by-product, we find that a separable simple $C^*$ -algebra which has almost stable rank one must have stable rank one, provided it has strict comparison and the canonical map $\Gamma $ is surjective.

Development of a New Affinity Gold Polymer Membrane with Immobilized Protein A.

New and highly selective stationary phases for affinity membrane chromatography have the potential to significantly enhance the efficiency and specificity of therapeutic protein purification by reduced mass transfer limitations. This work developed and compared different immobilization strategies for recombinant Protein A ligands to a gold-sputtered polymer membrane for antibody separation in terms of functionalization and immobilization success, protein load, and stability. Successful, functionalization was validated via X-ray photoelectron spectroscopy (XPS). Here, a recombinant Protein A ligand was coupled by N-hydroxysuccinimide (NHS)/N-(3-dimethylaminopropyl)-N'-ethylcarbodiimide (EDC) chemistry to carboxy-functionalized, gold-sputtered membranes. We achieved a binding capacity of up to 104 ± 17 mg of the protein ligand per gram of the gold-sputtered membrane. The developed membranes were able to successfully capture and release the monoclonal antibody (mAb) Trastuzumab, as well as antibodies from fresh frozen human blood plasma in both static and dynamic setups. Therefore, they demonstrated successful functionalization and immobilization strategies. The antibody load was tested using bicinchoninic acid (BCA), ultraviolet-visible spectroscopy (UV-vis) measurements, and sodium dodecyl sulfate polyacrylamide gel electrophoresis (SDS-PAGE). The outcome is a fully functional affinity membrane that can be implemented in a variety of different antibody purification processes, eliminating the need for creating individualized strategies for modifying the surface to suit different substrates or conditions.

Trajectory tracking with integral action of switched periodic affine systems

This paper deals with integral control of a class of switched affine systems characterised by time-varying differential equations that, along with the discontinuities caused by the switching, also depend periodically on an exogenous parameter, which is an affine function of time. The goal is to design a state-dependent switching function with integral action that ensures global asymptotic tracking of a trajectory of interest. The integral action is responsible for providing robustness to the system against uncertainties and load variations, whenever the discrepancy between the model and the real system is sufficiently bounded. Due to the integrator, all the convex combinations of the dynamic matrices are non-Hurwitz. In this case, the main difficulty is to guarantee that the time derivative of the Lyapunov function is strictly negative definite, which is done by means of LMIs and conditions on the affine terms. To the best of our knowledge, this is the first Lyapunov-based switching rule with integral action able to ensure global asymptotic tracking of a pre-specified profile. When applied to AC circuits the advantage is to design the switching rule based on the original system, without resorting to averaged models. The theory is illustrated by the control of a three-phase AC-DC converter.

Human brain mapping of homotopic functional affinity

Human brain mapping of homotopic functional affinity

Gini-stable Lorenz curves and their relation to the generalised Pareto distribution

We introduce an iterative discrete information production process where we can extend ordered normalised vectors by new elements based on a simple affine transformation, while preserving the predefined level of inequality, G, as measured by the Gini index.Then, we derive the family of empirical Lorenz curves of the corresponding vectors and prove that it is stochastically ordered with respect to both the sample size and G which plays the role of the uncertainty parameter. We prove that asymptotically, we obtain all, and only, Lorenz curves generated by a new, intuitive parametrisation of the finite-mean Pickands' Generalised Pareto Distribution (GPD) that unifies three other families, namely: the Pareto Type II, exponential, and scaled beta distributions. The family is not only totally ordered with respect to the parameter G, but also, thanks to our derivations, has a nice underlying interpretation. Our result may thus shed a new light on the genesis of this family of distributions.Our model fits bibliometric, informetric, socioeconomic, and environmental data reasonably well. It is quite user-friendly for it only depends on the sample size and its Gini index.

A continuous-time neurodynamic approach in matrix form for rank minimization

This article proposes a continuous-time neurodynamic approach for solving the rank minimization under affine constraints. As opposed to the traditional neurodynamic approach, the proposed neurodynamic approach extends the form of the variables from the vector form to the matrix form. First, a continuous-time neurodynamic approach with variables in matrix form is developed by combining the optimal rank r projection and the gradient. Then, the optimality of the proposed neurodynamic approach is rigorously analyzed by demonstrating that the objective function satisfies the functional property which is called as (2r,4r)-restricted strong convexity and smoothness ((2r,4r)-RSCS). Furthermore, the convergence and stability analysis of the proposed neurodynamic approach is rigorously conducted by establishing appropriate Lyapunov functions and considering the relevant restricted isometry property (RIP) condition associated with the affine transformation. Finally, through experiments involving low-rank matrix recovery under affine transformations and the completion of low-rank real image, the effectiveness of this approach has been demonstrated, along with its superiority compared to the vector-based approach.

An analytic approach for the moment-of-fluid interface reconstruction method on tetrahedral meshes

This article presents an original fast and robust approach for solving the moment-of-fluid interface reconstruction problem on tetrahedral meshes. We establish an explicit parametrization of the locus of the centroids under volume constrains using the spherical coordinates on a reference tetrahedron. This parametrization allows to write the moment-of-fluid objective function and its gradient by means of analytic formulas. Then, we show that the minimization problem on arbitrary tetrahedra can be effectively solved on the reference tetrahedron by using an affine transformation. Based on this transformation, we propose a improved technique for obtaining an initial condition for the minimization algorithm. We then describe the Gauss-Newton and the BFGS minimization algorithms, both combined with a trust-region method, to solve this reconstruction problem. Finally, we perform a verification and validation process on a series of increasingly complex test cases. These tests show that the proposed analytic method, paired with the Gauss-Newton algorithm, outperforms the other methods that were tested.

On the Construction of 3D Fibonacci Spirals

The paper aims to extend the classical two-dimensional (2D) Fibonacci spiral into three-dimensional (3D) space by using geometric constructions starting from cubic Fibonacci identities and relying on affine maps and parametrizations of the curves. We have already performed a comprehensive survey of cubic Fibonacci identities, which, to our surprise, uncovered only a handful of homogenous cubic identities. Obviously, the goal here is to show how one could use a particular homogenous cubic Fibonacci identity for generating 3D geometric designs similar in spirit to the way the classical Fibonacci spiral is built in 2D starting from a quadratic Fibonacci identity. This made us realize that for any cubic identity there are many different ways of packing cuboids, while only an insignificant fraction of those possible tilings might allow a smooth spiral-like curve to be drawn through them. After reviewing the state of the art, we present accurate details on ways to construct such 3D spirals using affine maps. We go on to prove the continuity and smoothness of such 3D spirals by giving a parametrization of the intersection of the surfaces that define the curves. Throughout the paper, we visualize the resulting 3D spirals by generating geometrically correct stereoscopic views. Finally, it is to be mentioned that the recursive 3D packing of cuboids tends to lead to fractal structures, which will need further investigations.

Robust registration of multi-modal remote sensing images based on multi-dimensional oriented self-similarity features

Registration of multi-modal remote sensing images (MRSI) is crucial for unlocking the full potential of heterogeneous remote sensing imagery. However, achieving accurate registration among MRSI is challenging due to the trade-off between geometric invariance and matching accuracy, caused by differences in signal-to-noise ratio and nonlinear radiometric distortion (NRD) arising from varying imaging mechanisms. To tackle the challenge, this paper proposes a lightweight and hybrid feature-guided registration algorithm for MRSI called the hybrid registration algorithm based on multi-dimensional oriented self-similarity features (MOSS). MOSS leverages the advantages of multi-dimensional oriented self-similarity features to progressively enhance registration performance. In the hybrid feature coarse matching stage, oriented self-similarity features are extracted from MRSI, and their directional information is utilized for feature description to estimate the initial affine transformation. The fine matching under multi-dimensional oriented self-similarity features stage takes the outputs of the coarse matching stage to perform a template-like matching process. To evaluate the performance of MOSS, comprehensive experiments are conducted using six different combinations of MRSI, and seven state-of-the-art registration algorithms are selected for comparison. The experimental results demonstrate that MOSS outperforms the compared methods, with the number of correct matches being at least about 1.6 times higher than the comparison methods. Moreover, MOSS exhibits the lowest root mean square error across all experiments, with an average RMSE of 1.86 pixels, achieving an RMSE within 2 pixels. This highlights its effectiveness in achieving precise alignment and robust registration of MRSI.

Attentional pixel-wise deformation for pose-based human image generation

Human pose transfer aims to synthesize referred human images with target pose, bringing the substantial economic potential for E-commerce or virtual reality. In this paper, we propose a novel method, the Attentional Pixel-wise Deformation Network (APD-Net), for synthesizing human images with guided pose and referred images. Specifically, we leverage attention-based spatial transformation modules and affine transformation modules to generate accurate appearance and extract pixel-wise details in local regions to generate intermediate results. Additionally, we introduce a confidence map to refine spatial information during the final image synthesis. Domain alignment loss, cycle loss, perceptual and feature matching loss and contextual loss are applied to constrain the synthesized images while attention loss and fusion loss benefit warp images generation. We verify the efficacy of the model on the Market-1501 and DeepFashion datasets, using quantitative and qualitative measures. Our approach surpasses all previously published state-of-the-art results on most evaluation metrics, e.g., achieving 0.780 SSIM score, 9.55 Sliced Wasserstein Distance score, and a 0.963 Semantic Consistency score on DeepFashion and obtaining 0.303 SSIM score, 16.971 Sliced Wasserstein Distance score and 0.729 Semantic Consistency score on Market-1501 Code and pretrained models are available at: https://github.com/LiaoFJ/APD-Net/.

Enhancing fine-detail image synthesis from text descriptions by text aggregation and connection fusion module

Synthesizing images with fine details from text descriptions is a challenge. The existing single-stage generative adversarial networks (GANs) fuse sentence features into the image generation process through affine transformation, which alleviate the problems of missing details and large computation from stacked networks. However, existing single-stage networks ignore the word features in the text description, resulting in a lack of detail in the generated image. To address this issue, we proposed a text aggregation module (TAM) to fuse sentence features and word features in a text by a simple spatial attention mechanism. Then we built a text connection fusion (TCF) block consisting mainly of gated recurrent unit (GRU) and up-sampled block. It can connect text features used in the up-sampled blocks to improve text utilization. Besides, to further improve the semantic consistency between text and the generated images, we introduce the deep attentional multimodal similarity model (DAMSM) loss, which monitors the similarity between text and improves semantic consistency. Experimental results prove that our method is superior to the state-of-the-art models on the CUB and COCO datasets, regarding both image fidelity and semantic consistency with the text.

On the definition of curvature in Regge calculus

Abstract Regge calculus defines a curvature for piecewise constant metrics on simplicial complexes subject to a partial continuity requirement. We prove that if a part of the complex is embedded in a Euclidean space, by a piecewise affine map, and we perform smoothing by convolution there, then the smoothed metrics have a densitized scalar curvature that converges, in the sense of measures, to that defined by Regge, as the smoothing parameter goes to zero.

Finite approximation properties of C*-modules III

Abstract We introduce and study a notion of module nuclear dimension for a C * \mathrm{C}^{*} -algebra A which is a C * \mathrm{C}^{*} -module over another C * \mathrm{C}^{*} -algebra 𝔄 {\mathfrak{A}} with compatible actions. We show that the module nuclear dimension of A is zero if A is 𝔄 {\mathfrak{A}} -NF. The converse is shown to hold when 𝔄 {\mathfrak{A}} is a C ⁢ ( X ) {C(X)} -algebra with simple fibers, with X compact and totally disconnected. We also introduce a notion of module decomposition rank, and show that when 𝔄 {\mathfrak{A}} is unital and simple, if the module decomposition rank of A is finite then A is 𝔄 {\mathfrak{A}} -QD. We study the set 𝒯 𝔄 ⁢ ( A ) {\mathcal{T}_{\mathfrak{A}}(A)} of 𝔄 {\mathfrak{A}} -valued module traces on A and relate the Cuntz semigroup of A with lower semicontinuous affine functions on the set 𝒯 𝔄 ⁢ ( A ) {\mathcal{T}_{\mathfrak{A}}(A)} . Along the way, we also prove a module Choi–Effros lifting theorem. We give estimates of the module nuclear dimension for a class of examples.

Template-based piecewise affine regression

Abstract We study the problem of fitting a piecewise affine (PWA) function to input–output data. Our algorithm divides the input domain into finitely many regions whose shapes are specified by a user-provided template and such that the input–output data in each region are fit by an affine function within a user-provided error tolerance. We first prove that this problem is NP-hard. Then, we present a top-down algorithmic approach for solving the problem. The algorithm considers subsets of the data points in a systematic manner, trying to fit an affine function for each subset using linear regression. If regression fails on a subset, the algorithm extracts a minimal set of points from the subset (an unsatisfiable core) that is responsible for the failure. The identified core is then used to split the current subset into smaller ones. By combining this top-down scheme with a set-covering algorithm, we derive an overall approach that provides optimal PWA models for a given error tolerance, where optimality refers to minimizing the number of pieces of the PWA model. We demonstrate our approach on three numerical examples that include PWA approximations of a widely used nonlinear insulin–glucose regulation model and a double inverted pendulum with soft contacts.

Recurrent Affine Transformation for Text-to-Image Synthesis

Text-to-image synthesis aims to generate realistic images conditioned on text descriptions. To fuse text information into synthesized images, conditional affine transformations (CATs), such as conditional batch normalization (CBN) and conditional instance normalization (CIN), are usually used to predict batch statistics of different layers. However, ordinary CAT blocks control the batch statistics independently disregarding the consistency among neighboring layers. To address the above issue, we propose a new fusion approach names recurrent affine transformation (RAT) for synthesizing images conditioned on text information. RAT connects all the CAT blocks with recurrent connections for explicitly fitting the temporal consistency between CAT blocks. To verify the effectiveness of RAT, we propose a novel visualization method to show how generative adversarial network (GAN) fuses conditional information. Our microscopic and macroscopic visualizations not only demonstrate the effectiveness of RAT but also turn out to be a useful perspective to analyze how GAN fuses conditional information. In addition, we propose a more stable spatial attention mechanism for the discriminator, which helps the text description to supervise the generator to synthesize more relevant image contents. Extensive experiments on the CUB, Oxford-102, and COCO datasets demonstrate the proposed model's superiority in comparison to state-of-the-art models. Our code is available on <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/senmaoy/RAT-GAN</uri> .

Multimedia Interactive Technology for Improving Language Perception in English Teaching and Learning

Abstract Based on Direct Show software, this paper acquires English teaching videos and stores them in the corresponding files. As the position of the teaching projection screen may be different in height, it will lead to the phenomenon that the captured image will be tilted, and it is necessary to use the affine transformation and perspective transformation methods to realize the correction of the teaching projection image. In order to promote the general application of multimedia interactive technology in the teaching process, dynamic gesture tracking with Kalzmann filtering is utilized to construct a non-contact human-computer interaction system, which is applied to the process of English teaching, thus designing an English teaching model incorporating multimedia interactive technology and analyzing the effect of the model. It was found that the mean score of the experimental group’s pre-experimental evaluation was 19.225, and the mean score for the post-experimental assessment was 23.087, t=2.274, P (two-sided)=0.018&lt;0.05 at the 0.05 significance level, which is a significant difference. It shows that compared with traditional English teaching methods, the teaching mode that integrates multimedia interactive technology can effectively improve the English language perception ability of college students in China.

An image quality assessment index based on image features and keypoints for X-ray CT images.

Optimization tasks in diagnostic radiological imaging require objective quantitative metrics that correlate with the subjective perception of observers. However, although one such metric, the structural similarity index (SSIM), is popular, it has limitations across various aspects in its application to medical images. In this study, we introduce a novel image quality evaluation approach based on keypoints and their associated unique image feature values, focusing on developing a framework to address the need for robustness and interpretability that are lacking in conventional methodologies. The proposed index quantifies and visualizes the distance between feature vectors associated with keypoints, which varies depending on changes in the image quality. This metric was validated on images with varying noise levels and resolution characteristics, and its applicability and effectiveness were examined by evaluating images subjected to various affine transformations. In the verification of X-ray computed tomography imaging using a head phantom, the distances between feature descriptors for each keypoint increased as the image quality degraded, exhibiting a strong correlation with the changes in the SSIM. Notably, the proposed index outperformed conventional full-reference metrics in terms of robustness to various transformations which are without changes in the image quality. Overall, the results suggested that image analysis performed using the proposed framework could effectively visualize the corresponding feature points, potentially harnessing lost feature information owing to changes in the image quality. These findings demonstrate the feasibility of applying the novel index to analyze changes in the image quality. This method may overcome limitations inherent in conventional evaluation methodologies and contribute to medical image analysis in the broader domain.

8×8 S-boxes over Klein four-group and Galois field $ {GF}\left({2}^{4}\right) $: AES redesign

&lt;abstract&gt; &lt;p&gt;This research paper is supplemented with a unique formation to design state-of-the-art S-boxes. The invented approach is simple but has the capability of creating confusion in our newly proposed algorithm. Our core planned work refined the method of already designed S-boxes to accomplish more compact ones. Various structures were merged here, namely affine transformation, fractional linear transformation, structure of Klein four-group, and the algebraic structures of the Galois fields, $ GF\left({2}^{4}\right) $ and $ GF\left({2}^{8}\right). $ These structures were utilized to synthesize newly $ 1600 $ robust S-boxes. Besides, we discussed encryption steps of AES with these newly generated S-boxes. We highlighted some specific characteristics, performance of parameter's improvement, and their utilization. Nonlinear properties were mainly set to inspect the behavior of I/O bits and could apply image encryption. Then, the performance of proposed S-boxes and newly structured AES was tested in comparison with other prevailing S-boxes.&lt;/p&gt; &lt;/abstract&gt;

Deep learning-based Alzheimer's disease detection: reproducibility and the effect of modeling choices.

Machine Learning (ML) has emerged as a promising approach in healthcare, outperforming traditional statistical techniques. However, to establish ML as a reliable tool in clinical practice, adherence to best practices in data handling, and modeling design and assessment is crucial. In this work, we summarize and strictly adhere to such practices to ensure reproducible and reliable ML. Specifically, we focus on Alzheimer's Disease (AD) detection, a challenging problem in healthcare. Additionally, we investigate the impact of modeling choices, including different data augmentation techniques and model complexity, on overall performance. We utilize Magnetic Resonance Imaging (MRI) data from the ADNI corpus to address a binary classification problem using 3D Convolutional Neural Networks (CNNs). Data processing and modeling are specifically tailored to address data scarcity and minimize computational overhead. Within this framework, we train 15 predictive models, considering three different data augmentation strategies and five distinct 3D CNN architectures with varying convolutional layers counts. The augmentation strategies involve affine transformations, such as zoom, shift, and rotation, applied either concurrently or separately. The combined effect of data augmentation and model complexity results in up to 10% variation in prediction accuracy. Notably, when affine transformation are applied separately, the model achieves higher accuracy, regardless the chosen architecture. Across all strategies, the model accuracy exhibits a concave behavior as the number of convolutional layers increases, peaking at an intermediate value. The best model reaches excellent performance both on the internal and additional external testing set. Our work underscores the critical importance of adhering to rigorous experimental practices in the field of ML applied to healthcare. The results clearly demonstrate how data augmentation and model depth-often overlooked factors- can dramatically impact final performance if not thoroughly investigated. This highlights both the necessity of exploring neglected modeling aspects and the need to comprehensively report all modeling choices to ensure reproducibility and facilitate meaningful comparisons across studies.