Geometric Translation Research Articles

Improving the robustness of adversarial attacks using an affine-invariant gradient estimator

As designers of artificial intelligence try to outwit hackers, both sides continue to hone in on AI’s inherent vulnerabilities. Designed and trained from certain statistical distributions of data, deep neural networks (DNNs) remain vulnerable to deceptive inputs that violate a DNN’s statistical, predictive assumptions. Before being fed into a neural network, however, most existing adversarial examples cannot maintain malicious functionality when applied to an affine transformation. For practical purposes, maintaining that malicious functionality serves as an important measure of the robustness of adversarial attacks. To help DNNs learn to defend themselves more thoroughly against attacks, we propose an affine-invariant adversarial attack, which can consistently produce more robust adversarial examples over affine transformations. For efficiency, we propose to disentangle current affine-transformation strategies from the Euclidean geometry coordinate plane with its geometric translations, rotations and dilations; we reformulate the latter two in polar coordinates. Afterwards, we construct an affine-invariant gradient estimator by convolving the gradient at the original image with derived kernels, which can be integrated with any gradient-based attack methods. Extensive experiments on ImageNet, including some experiments under physical condition, demonstrate that our method can significantly improve the affine invariance of adversarial examples and, as a byproduct, improve the transferability of adversarial examples, compared with alternative state-of-the-art methods.

Using Prior Knowledge to Advance Students’ Mathematical Understanding: Geometric Translation as a Function

Matematik eğitimi araştırmacıları ve öğretim programı geliştiricileri, matematik öğretiminin öğrencilerin ön bilgileri dikkate alınarak tasarlanması gerektiği konusunda hemfikirdir. Bu araştırmanın amacı, öğrencilerin herhangi bir matematiksel konuyla ilgili matematiksel anlamalarını geliştirmek için, ön bilgilerin öğretim dizaynında nasıl kullanılabileceğine dair bir yaklaşım sunmaktır. Konu olarak, okul matematiği ve matematik öğretmen adaylarının eğitimi için önemli konularından birisi olan öteleme seçilmiş ve Pirie-Kieren teorisinde yer alan geriye katlamalar ile öğretim müdahaleleri temel alınarak bir eylem araştırması dizayn edilmiştir. Katılımcılar, bir devlet üniversitesinin Matematik Öğretmenliği programının dördüncü sınıfında öğrenim gören 28 matematik öğretmen adayıdır. Araştırmanın verileri, öteleme dersleri başlamadan önce uygulanan ön teste katılımcıların verdiği yazılı cevaplar, dersler boyunca gerçekleşen matematiksel tartışmalara yönelik kayıtlar, ders gözlem notları ile dersi yürüten eğitimcilerin ders öncesi veya sonrası görüşmelerinde alınan notlardan oluşturmaktadır. Dersler boyunca, öğretmen adaylarının ön bilmeleri üzerinde çalışmalarını sağlayan geriye katlamaları teşvik eden öğretim müdahalelerin, adayların ötelemeyi bir fonksiyon olarak anlamlandırmalarına yardımcı olduğu belirlenmiştir. Araştırmanın, öğrencilerin matematiksel anlamalarını desteklemek amacıyla ön bilmeleri referans alarak öğretim tasarlamak isteyen matematik eğitimcilerine fikir vereceği düşünülmektedir.

Segmentation mask and feature similarity loss guided GAN for object-oriented image-to-image translation

While image-to-image translation has been extensively studied, there are a number of limitations in existing methods designed for transformation between instances of different shapes from different domains. In this paper, a novel approach was proposed (hereafter referred to as ObjectVariedGAN) to handle geometric translation. One may encounter large and significant shape changes during image-to-image translation, especially object transfiguration. Thus, we focus on synthesizing the desired results to maintain the shape of the foreground object without requiring paired training data. Specifically, our proposed approach learns the mapping between source domains and target domains, where the shapes of objects differ significantly. Feature similarity loss is introduced to encourage generative adversarial networks (GANs) to obtain the structure attribute of objects (e.g., object segmentation masks). Additionally, to satisfy the requirement of utilizing unaligned datasets, cycle-consistency loss is combined with context-preserving loss. Our approach feeds the generator with source image(s), incorporated with the instance segmentation mask, and guides the network to generate the desired target domain output. To verify the effectiveness of proposed approach, extensive experiments are conducted on pre-processed examples from the MS-COCO datasets. A comparative summary of the findings demonstrates that ObjectVariedGAN outperforms other competing approaches, in the terms of Inception Score, Frechet Inception Distance, and human cognitive preference.

Improving the Robustness of Adversarial Attacks Using an Affine-Invariant Gradient Estimator

As designers of artificial intelligence try to outwit hackers, both sides continue to hone in on AI's inherent vulnerabilities. Designed and trained from certain statistical distributions of data, AI's deep neural networks (DNNs) remain vulnerable to deceptive inputs that violate a DNN's statistical, predictive assumptions. Before being fed into a neural network, however, most existing adversarial examples cannot maintain malicious functionality when applied to an affine transformation. For practical purposes, maintaining that malicious functionality serves as an important measure of the robustness of adversarial attacks. To help DNNs learn to defend themselves more thoroughly against attacks, we propose an affine-invariant adversarial attack, which can consistently produce more robust adversarial examples over affine transformations. For efficiency, we propose to disentangle current affine-transformation strategies from the Euclidean geometry coordinate plane with its geometric translations, rotations and dilations; we reformulate the latter two in polar coordinates. Afterwards, we construct an affine-invariant gradient estimator by convolving the gradient at the original image with derived kernels, which can be integrated with any gradient-based attack methods. Extensive experiments on ImageNet, including some experiments under physical condition, demonstrate that our method can significantly improve the affine invariance of adversarial examples and, as a byproduct, improve the transferability of adversarial examples, compared with alternative state-of-the-art methods.

Mapping pre-service teachers’ faulty reasoning in geometric translations to the design of Van Hiele phase-based instruction

Background: Pre-service teachers (PSTs) training does not equip students with adequate skills and knowledge of geometry to enable them to teach this section of mathematics competently. Inadequate teacher knowledge of transformation geometry, in particular, requires intervention that targets PSTs’ faulty reasoning displayed in errors they make.Aim: The aim of this study was to explore the use of Bachelor of Education (BEd) students’ faulty reasoning in geometric translations, in designing a Van Hiele phase-based instructional programme that could address such faulty reasoning.Setting: The setting for the study was a newly established rural university in South Africa.Methods: Tests on geometric translations were administered to BEd Foundation Phase students, followed up by interviews to explore errors made when responding to the test items. The errors were then mapped to the design of a Van Hiele phase-based instructional programme.Results: The results revealed that the students had several misconceptions with geometric translations. The misconceptions were delineated into the errors that the students displayed and these were classified under two themes. The first theme was incorrect properties of transformation and under this theme, the errors were coded as confusing translation with rotation, wrong translation method, incorrect interpretation of coordinates and confusing the x and y axis. The second theme was errors involving basic mathematics operations including wrong diagrammatic representation of coordinates and incorrect calculations.Conclusion: The study showed that if the students’ misconceptions and the resulting errors are mapped to specific instructional approaches, their faulty reasoning in geometric transformations is addressed and effective learning is enhanced.

Utilization of CBCT to improve the delivery accuracy of Gamma Knife radiosurgery with G-frame.

PurposeTo quantify the G‐frame based stereotactic coordinate definition accuracy of Leksell coordinate G‐frame‐based Gamma Knife radiosurgery (GKRS) by the on‐board cone‐beam CT (CBCT) and establish remedial action rules to minimize the delivery errors.MethodsWe analyzed the data of 108 patients (a total of 201 tumors) treated by GKRS with G‐frame for head fixation. After co‐registering the CBCT images and plan reference images, the Leksell GammaPlan (LGP) treatment planning system provided the amount of geometric translation and rotation required to minimize the position difference between the plan and treatment. The software also calculated maximum displacement, which characterizes the position shift more clearly. We studied how much these predicted dosimetric quantities changed if the treatment was delivered without correcting the patient's position.ResultsThe maximum displacement of the patient position obtained from the co‐registration of CBCT and plan reference images was 0.81 ± 0.38 mm (0.24–2.03 mm). The target coverage decreased by 3.3 ± 7.0% on average (−48.5% to +35.7%). The decrease of the target coverage, however, became smaller as the target volume increased. In particular, if the volume was greater than 2 cm3, the %change in target coverage was always less than −5%.ConclusionsThe position differences reported by the registration module of LGP were within the accuracy limit of image registration for most clinical cases, but the errors could be larger in some cases. Therefore, we propose the following decision process. We do not advise position adjustment for G‐frame based GKRS if the maximum displacement is less than 1 mm. When this limit is exceeded, however, another criterion should be applied to the decision making by considering the tumor size (or the treatment volume) together with the acceptable change of the tumor coverage.

Reproduction of patterns in melanocytic proliferations by agent-based simulation and geometric modeling.

Spatio-temporal patterns of melanocytic proliferations observed in vivo are important for diagnosis but the mechanisms that produce them are poorly understood. Here we present an agent-based model for simulating the emergence of the main biologic patterns found in melanocytic proliferations. Our model portrays the extracellular matrix of the dermo-epidermal junction as a two-dimensional manifold and we simulate cellular migration in terms of geometric translations driven by adhesive, repulsive and random forces. ed cellular functions and melanocyte-matrix interactions are modeled as stochastic events. For identification and validation we use visual renderings of simulated cell populations in a horizontal perspective that reproduce growth patterns observed in vivo by sequential dermatoscopy and corresponding vertical views that reproduce the arrangement of melanocytes observed in histopathologic sections. Our results show that a balanced interplay of proliferation and migration produces the typical reticular pattern of nevi, whereas the globular pattern involves additional cellular mechanisms. We further demonstrate that slight variations in the three basic cellular properties proliferation, migration, and adhesion are sufficient to produce a large variety of morphological appearances of nevi. We anticipate our model to be a starting point for the reproduction of more complex scenarios that will help to establish functional connections between abstracted microscopic behavior and macroscopic patterns in all types of melanocytic proliferations including melanoma.

Knowledge Graph Embedding Based on Multi-View Clustering Framework

Knowledge representation is one of the critical problems in knowledge engineering and artificial intelligence, while knowledge embedding as a knowledge representation methodology indicates entities and relations in knowledge graph as low-dimensional, continuous vectors. In this way, knowledge graph is compatible with numerical machine learning models. Major knowledge embedding methods employ geometric translation to design score function, which is weak-semantic for natural language processing. To overcome this disadvantage, in this paper, we propose our model based on multi-view clustering framework, which could generate semantic representations of knowledge elements (i.e., entities/relations ). With our semantic model, we also present an empowered solution to entity retrieval with entity description. Extensive experiments show that our model achieves substantial improvements against baselines on the task of knowledge graph completion, triple classification, entity classification, and entity retrieval.

Solomon-Tukachinsky’s Versus Welschinger’s Open Gromov-Witten Invariants of Symplectic Six-Folds

AbstractOur previous paper describes a geometric translation of the construction of open Gromov–Witten invariants by Solomon and Tukachinsky from a perspective of $A_{\infty }$-algebras of differential forms. We now use this geometric perspective to show that these invariants reduce to Welschinger’s open Gromov–Witten invariants in dimension 6, inline with their and Tian’s expectations. As an immediate corollary, we obtain a translation of Solomon–Tukachinsky’s open WDVV equations into relations for Welschinger’s invariants.

Enhancing knowledge graph embedding with relational constraints

Knowledge graph embedding is studied to embed entities and relations of a knowledge graph into continuous vector spaces, which benefits a variety of real-world applications. Among existing solutions, translational models, which employ geometric translation to design score function, have drawn much attention. However, these models primarily concentrate on evidence from observing whether the triplets are plausible, and ignore the fact that the relation also implies certain semantic constraints on its subject or object entity. In this paper, we present a general framework for enhancing knowledge graph embedding with relational constraints (KRC). Specifically, we elaborately design the score function by encoding regularities between a relation and its arguments into the translational embedding space. Additionally, we propose a soft margin-based ranking loss for effectively training the KRC model, which characterizes different semantic distances between negative and positive triplets. Furthermore, we combine regularities with distributional representations to predict the missing triplets, which can possess certain robust guarantee. We evaluate our method on the tasks of knowledge graph completion and entity classification. Extensive experiments show that KRC achieves a better, or comparable performance against state-of-the-art methods. Besides, KRC makes a great improvement when dealing with long-tail entities, which have few instances in the knowledge graph.

Planning for Flexible Surgical Robots via Bézier Spline Translation

In a minimally invasive surgery, new flexible instruments enable safer and easier access to difficult-to-reach anatomical regions. However, their introduction into the clinical workflow requires robust replanning because navigation errors during surgery render initially planned trajectories infeasible. Such replanning requires to regularly solve an expensive two-point boundary value problem (BVP) that connects the target pose of the instrument with the currently measured one. We propose a hybrid planning scheme that features both robust and safe replanning. This two-step approach first solves the BVP and then transforms the result to circular arcs that fit the motion of our instruments’ models. We exploit implicitly defined Bezier splines as a robust method for interpolation in the first step. A novel geometric translation of these splines, then, provides a convenient solution for movement along circular arcs. We consider two example applications: 1) planning for a drilling unit in temporal bone surgery; and 2) guidewires in catheter insertion. Evaluation on real patient data of both temporal bone and aorta show that our proposed hybrid two-step approach achieves, on average, ${\text{55}\%}$ higher replanning rates and provides ${\text{31}\%}$ larger clearance to risk structures, thus improving trajectory quality with regard to clinical safety.

The Development of Two High School Students’ Discourses on Geometric Translation in Relation to the Teacher’s Discourse in the Classroom

Research exploring student development on geometric translation in relation to classroom teaching is scarce. We used a qualitative case study design and examined two students’ development of thinking on translation in relation to the teacher’s discourse in a Turkish high school classroom through a discursive framework. We examined the teacher’s discourse through the video-taped classroom session in which he talked about translation. We examined the students’ discursive development through three video-taped task-based interviews. Finally, we compared the students’ discourses on translation with the teacher’s discourse. The teacher’s discourse on translation in the classroom was formal and based on an algebraic realization of the concept as an addition and a geometric transformation. The students adopted particular discursive elements used by the teacher right after instruction but later abandoned some of those and used their more established realizations of translation. The findings revealed the complexity of discursive development as the students continually and actively adjusted their discourses by taking into account the teacher’s discourse while also making it compatible with their own realizations of translation. We conclude that socio-cultural and discursive approaches have the potential to shed additional light on issues regarding learning and teaching of geometric translation.

Determining workspace parameters for a new type of 6RSS parallel manipulator based on structural and geometric models

Workspace geometric modelling of a new type of 6RSS parallel manipulator is described below. In the beginning, the researches undertaken in this area by other authors are highlighted and then a definition of this type of mechanisms is provided. The structural model of the 6RSS manipulator is briefly described. Inverse geometric model and translation subspace methods are used in order to determine the dimensions that define the workspace volume of the parallel manipulator. The reachable workspace is defined as a subset of the whole workspace in relation with the positions achieved by the characteristic point.

Shape similarity measurement model for holed polygons based on position graphs and Fourier descriptors

ABSTRACTIn geographic information retrieval and spatial data mining, similarity is used to resolve shape matching and clustering. Many approaches have been developed to calculate similarity between simple geometric shapes. However, complex spatial objects are common in spatial database systems, spatial query languages and Geographic Information Science (GIS) applications. With holed polygons, many similarity measurement approaches are restricted to address the relationships between holes or between the holes and the entire complex geometric shape. A successful method should remove the restrictions due to these complex relations and retain invariant during geometric translation (rotation, moving and scaling). To overcome these deficiencies, we utilize position graphs to describe the distribution of holes in complex geometric shapes by storing invariants, such as angles and distances. In addition, Fourier descriptors and the position graph-based method are used to measure the similarity between holed polygons. Experiments show that the proposed method takes into account the relationships in an entire complex geometric shape. It can effectively calculate the similarity of holed polygons, even if they contain different numbers of holes.

Evaluation of feature- and pixel-based methods for deflection measurements in temporary structure monitoring

Recent forensic studies have revealed quite a few construction site catastrophes are associated with failure of temporary structures (e.g., collapse of formworks, scaffoldings, etc.). This is particularly crucial for those located in dense urban areas where the failure of a temporary structure not only impacts the site itself but may also damage adjacent structures and cause injuries and casualties of passers-by. In this paper, the work of evaluating the optimum feature detection and matching algorithm is reported, which is the key in realizing a real-time visual sensing-based surveillance method in order to monitor the integrity and safety of on-site temporary structures. A series of experiments are designed and conducted to test three algorithms: population-based intelligent digital image correlation (DIC), David Lowe’s Scale-invariant feature transform, and Hessian matrix-based speeded-up robust features. In these experiments, synthetic images through 2D geometrical translation, rotation, deformation, and illumination changes are generated to provide the sample data and ground truths. As the experiment output, the accuracy and efficiency of these algorithms are measured and compared with each other. Analyses including feasibility of interest point localization with specific regions and the error estimation with respect to real-world distances are conducted. The results show that the DIC algorithm holds the most promise to be implemented in structural monitoring, but several challenges still remain, which call for efforts to further improvement of the technique.

Middle-school students’ concept images of geometric translations

This study explored sixth grade students’ concept images of geometric translations and the possible sources of their conceptions in a non-technological environment. The data were gathered through a written instrument, student and teacher interviews and document analyses. Analyses of student responses revealed two major concept images of geometric translations: (a) translation as translational motion, and (b) translation as both translational and rotational motion. Students who held these conceptions showed various levels of understanding, such as conceiving translations as undefined motion, partially-defined motion, and defined-motion of a single geometric figure on the plane. The findings of the study suggested, in general, consistencies between students’ concept images and their concept definitions. However, most of the students’ concept definitions were inconsistent with the formal concept definition of geometric translations.Data analyses also revealed five interpretations of a translation vector: (a) vector as a reference line, (b) vector as a symmetry line, (c) vector as a direction indicator, (d) vector as a parameter, and (e) vector as an abstract tool. Furthermore, classroom instruction, mathematics and science textbooks, real-life examples and everyday language were the major sources of students’ concept images of geometric translations.

Watermarking Method Resilient to RST and Compression Based on DWT, LPM and Phase Correlation

In this paper, the authors present a digital watermarking method for still image. The hybrid method is based on the application of discrete wavelet transform (DWT), log-polar mapping (LPM) and phase correlation. To be robust against compression, the watermark is embedded in the discrete Fourier transform (DFT) magnitude of the low pass (LL) sub-band obtained by applying the DWT. Moreover, the inverse log-polar mapping (ILPM) approximation, of the positions chosen in the LPM domain of the low-pass sub-band DFT magnitude, is applied to determine where to insert the watermark to ensure the robustness against geometric transformations rotation, scaling and translation (RST). The phase correlation of the LPM of the original image and the LPM of the watermarked image, allows calculating the displacement caused by the geometric transformations (RST). The DWT is used to take advantage of the human visual system (HVS) characteristics and obtaining the optimal place for embedding the watermark, meanwhile the Fourier–Mellin transform (FMT) provides a transformed domain which is invariant to RST. To avoid computing the ILPM for the whole image, the ILPM approximation is applied. In this case, the image quality is preserved and smaller correlation coefficient for under marked images. Experimental results demonstrate that the proposed method is robust against RST, added noise, JPEG compression and the combination of these attacks.

Fast Local Self-Similarity for describing interest regions

Two novel methods for extracting distinctive invariant features from interest regions are presented in this paper. The idea of these methods are associated with that measuring similarity between visual entities from images can be based on matching the internal layout of Local Self-Similarities. The main contributions are two-folds: firstly, two new texture features called Local Self-Similarities (LSS,C) and Fast Local Self-Similarities (FLSS,C) based on Cartesian location grid, are extracted, which are the modified versions of the well-known Local Self-Similarities (LSS,LP) feature based on Log-Polar location grid. To combine the powers of the SIFT and LSS (LP), LSS and FLSS are used as the local features in the SIFT algorithm. Secondly, different from the natural LSS (LP) descriptor that chooses the maximal correlation value in each bucket to get photometric translations invariance, the proposed LSS (C) and FLSS (C) adopt distribution-based representation to achieve more robust geometric translations invariance. In the contexts of image matching and object category classification experiments, the LSS (C) and FLSS (C) both outperform the original LSS (LP), and achieve favorably comparable performance to the SIFT. Furthermore, these descriptors are low computational complexity and simpler than the SIFT.

Prospective middle school mathematics teachers’ preconceptions of geometric translations

This article reports an analysis of 44 prospective middle school mathematics teachers’ pre-existing knowledge of rigid geometric transformations, specifically the geometric translations. The main data source for this study was the participants’ responses to the tasks that were presented during semi-structured clinical interviews. The findings of the study revealed that prospective teachers had difficulties recognizing, describing, executing, and representing geometric translations. The results indicated that teacher candidates held various views about the geometric translations: (1) translation as rotational motion, (2) translation as translational motion, and (3) translation as mapping. The results further revealed various interpretations of the vector that defines translations: (1) vector as a force, (2) vector as a line of symmetry, (3) vector as a direction indicator, and (4) vector as a displacement. Although many of the teacher candidates interviewed knew that a vector has a magnitude and a direction, this knowledge did not generally lead them to conclude that vectors define translations.

Math For Real: Hair Braiding

Many students from various ethnic groups have memories of hair braiding. Braiders use geometric transformations, including dilations, rotations, slides, and translations, as well as reasoning to create the patterns in their designs.