Curvature Entropy Research Articles

The curvature entropy inequalities of convex bodies

Abstract There are many entropy inequalities in geometry, some of them can be seen as the Minkowski inequalities in the form of entropy, which play important roles in convex geometry. In this article, let P P and Q Q be convex bodies, we introduce the mixed curvature entropy M f ( P , Q ) {M}_{f}\left(P,Q) and the curvature entropy E f ( P , Q ) {E}_{f}\left(P,Q) with respect to a continuous function f f . Moreover, we establish the log \log -Minkowski inequalities of the mixed curvature entropy and the curvature entropy for two classes of three-dimensional convex bodies.

Inspiring Designers’ Innovative Thinking: An Evolutionary Design Method for Product Forms

The product form serves as a crucial information carrier for expressing design concepts and encompasses significant valuable references. During the product iteration process, changes in design subjects, such as designers and decision-makers, result in substantial variability and uncertainty in the direction of product form evolution. To address these issues, an evolutionary design method for product forms based on the gray Markov model and an evolutionary algorithm is proposed in this study. Firstly, quadratic curvature entropy is utilized to quantify historical form features of product evolution. Subsequently, the original data on product form feature evolution are fitted and predicted using the gray Markov model, thereby obtaining the predicted value of the latest generation of product form features, which is determined to be 0.14586. Finally, this study uses this predicted value to construct a fitness function in the framework of an evolutionary algorithm, which in turn identifies next-generation product forms that can stimulate designers’ creative thinking. The method’s application is illustrated using the side outer contour of the Audi A4 automobile as an example. The research findings demonstrate that combining the gray Markov model with an evolutionary algorithm can effectively simulate designers’ understanding of previous generations’ design concepts and achieve stable inheritance of these design concepts during product iteration. This approach mitigates the risk of abrupt changes in design concepts caused by designers and decision-makers due to personal cognitive biases, thereby enhancing product development efficiency.

A cardiac MRI motion artifact reduction method based on edge enhancement network

Cardiac magnetic resonance imaging (MRI) usually requires a long acquisition time. The movement of the patients during MRI acquisition will produce image artifacts. Previous studies have shown that clear MR image texture edges are of great significance for pathological diagnosis. In this paper, a motion artifact reduction method for cardiac MRI based on edge enhancement network is proposed. Firstly, the four-plane normal vector adaptive fractional differential mask is applied to extract the edge features of blurred images. The four-plane normal vector method can reduce the noise information in the edge feature maps. The adaptive fractional order is selected according to the normal mean gradient and the local Gaussian curvature entropy of the images. Secondly, the extracted edge feature maps and blurred images are input into the de-artifact network. In this network, the edge fusion feature extraction network and the edge fusion transformer network are specially designed. The former combines the edge feature maps with the fuzzy feature maps to extract the edge feature information. The latter combines the edge attention network and the fuzzy attention network, which can focus on the blurred image edges. Finally, extensive experiments show that the proposed method can obtain higher peak signal-to-noise ratio and structural similarity index measure compared to state-of-art methods. The de-artifact images have clear texture edges.

The log-Minkowski inequality of curvature entropy for convex bodies

The log-Minkowski inequality of curvature entropy for convex bodies

Deep Learning-Driven Real-Time Liver Tumor Localization via Optical Surface Imaging and Biomechanical Modeling

<h3>Purpose/Objective(s)</h3> To track liver tumors in real time via a deep learning-based framework, by exploring patient-specific correlations between patients' body surface maps and internal anatomical motion. <h3>Materials/Methods</h3> Real-time liver tumor localization was achieved via deformable registration-based motion estimation, in a two-step framework: (a) liver boundary motion estimation via deep features learnt from optical body surface imaging; and (b) intra-liver motion propagation via deep learning-based biomechanical modeling from liver boundary motion. In step (a), a patient-specific, fully-connected convolutional neural network (SurfCNN) predicts the motion of nodes residing on the boundary of a patient-specific reference liver mesh, by extracting saliency features from curvature entropy maps derived out of real-time optical body surface images. Subsequently, step (b) uses the liver boundary motion solved in (a) to infer intra-liver tumor motion, using a U-Net-style model (UNet-Bio) inspired by biomechanical modeling via finite element analysis (FEA). The cascaded framework was evaluated using a dataset of 8 liver cancer patients from our institute. Each patient had a 10-phase, contrast-enhanced 4D-CT set with liver and liver tumor contoured. To generate sufficient motion variations to train and test the patient-specific SurfCNN model, we augmented the dataset of each patient by simulating varying real-time motion patterns and magnitudes. The augmentation was achieved via a principal component analysis-based statistical model, expanding the nine inter-phase deformation-vector-fields (DVFs) of each 4D-CT into 1,728 different motion states including both rigid and deformable motion. Optical surface imaging was simulated from each motion state via extracted body surface maps, while corresponding liver boundary motion was derived via the augmented DVFs to supervise the training of SurfCNN. UNet-Bio, on the other hand, could be trained as a population-based model using supervisions from intra-liver DVFs solved by FEA-based biomechanical modeling. The liver tumor localization accuracy was assessed through Dice similarity coefficient (DSC), Hausdorff distance (HD), and center-of-mass-error (COME), by comparing the augmentation DVF-propagated ‘ground-truth' liver tumor volumes against ones deformed by the cascaded framework. <h3>Results</h3> Tested using 576 unseen scenarios for each patient case, the cascaded SurfCNN and UNet-Bio scheme can localize liver tumors to 0.791(mean) ± 0.141 (s.d.) in DSC, 3.6 ± 1.9 mm in HD, and 2.3 ± 1.7 mm in COME. In comparison, the prior reference image without deformable registration yielded 0.534 ± 0.277 in DSC, 8.3 ± 6.1 mm in HD and 6.8 ± 5.8 mm in COME. The whole model inference time was less than 100 milliseconds, satisfying the temporal constraints of real-time imaging. <h3>Conclusion</h3> The deep learning-based framework allows accurate 3D liver tumor tracking in real-time via non-ionizing, marker-less, and high frame-rate optical surface imaging.

Regional importance detection of 3D mesh via fusion of local color difference and curvature entropy

Predicting relative visual salient regions on three-dimensional (3D) meshes benefits many computer graphics applications. Most computation models for mesh saliency focus on geometrical information alone. Nevertheless, the ignored texture, lighting and material also provide more detailed appearance information, especially in the context of static scene rendering. In this paper, we propose a mesh saliency detection algorithm considering both geometrical and colorimetric information to address this challenge. Our model first computes the local curvature entropy at multi-scale to capture the geometrical details. Second, a set of images are projected onto the screen at several viewpoints with specified material and lighting model. Potentially salient regions on the rendered images are detected by fusion of multiple color difference maps measured with an approximated multi-scale Laplacian of Gaussian filter. A Gaussian distribution-based central bias model is applied to the image saliency map to emphasize the global rarity of salient regions. Third, the saliency maps of rendered images are projected back to the 3D mesh via the ray casting method. In the end, both saliency maps are combined linearly as the saliency map of 3D mesh. Experiment on the human fixation database demonstrates the performance of our method compared to the classic methods in terms of linear correlation coefficient and AUC.

Cross-modal Effect between Taste and Shape Controlled by Curvature Entropy

Recently, cross-modal effects in which perceptions interact with each other have been drawing attention. In the case of the cross-modal effect between vision and taste, the effect of the angularity of shapes on taste has been widely studied. However, there has been little research on the other features of shapes. Also, previous studies have shown that the emotional valence arisen from visual perception causes the cross-modal effect. Therefore, this study focuses on the complexity of shapes, which is said to influence emotional valence, as a visual stimulus and aims to confirm the cross-modal effect induced by its sensation. First, by using particle swarm optimization algorithm, closed curve shapes were generated based on curvature entropy, a quantitative index of the complexity of shapes. Second, cup holders, which had these closed curve shapes on their sides, were created by using a 3D printer. Third, by comparing the tastes of orange juice in these cup holders, the effect of the complexity of shapes on the perception of sweetness was confirmed. Finally, we propose index of Angularity derived from curvature, which has been suggested to be relevant in previous studies, and additionally test for cross-modal effects on this index. The results suggest that the complexity of shapes controlled by curvature entropy and the Angularity of shapes controlled by index of Angularity derived from curvature weaken the perception of sweetness. This finding can be used for reducing sugar intake in bottle packaging.

A Product Shape Design and Evaluation Model Based on Morphology Preference and Macroscopic Shape Information.

A product form design and evaluation model are proposed. In this method, the forms can be evaluated by three sub-evaluation models which can help designers to grasp consumers’ preferences. In the process, the overall shape is first disassembled into several shape elements, and a morphological chart is constructed. Then, the priority of shape elements is analyzed through the fuzzy analytical hierarchy process, and several required combinations are selected from a morphological chart. Moreover, the fuzzy comprehensive evaluation (I), preference questionnaire (II), and quadratic curvature entropy (III) of the selected combinations are analyzed. Finally, the results of I, II, and III are compared. In conclusion, the three sub-evaluation systems are consistent, which confirms the applicability of the proposed model.

Point cloud simplification algorithm based on the feature of adaptive curvature entropy

Considering the diversity of point cloud features, the simplification effects of traditional point cloud simplification algorithms (such as curvature, random and isometry simplification algorithms) are poor. To overcome these drawbacks, a point cloud simplification method based on adaptive curvature entropy is proposed. Points with large curvatures are extracted to construct the initial point cloud boundary by defining a given proportion. The point cloud is clustered using the dichotomy clustering method. Subsequently, a preliminary simplification based on an adaptive random algorithm is performed for each clustered point cloud to reduce the point cloud capacity. The curvature entropy of each clustered point cloud is calculated to remove redundant points and preserve feature points so that the simplified point cloud is eventually obtained. The extracted initial point cloud boundary and simplified point cloud constitute the final simplified result. The classic Stanford rabbit model is introduced to verify the effect of the proposed approach. Experimental results show that the proposed algorithm can effectively reflect the details of the point cloud despite a simplification proportion of up to 90%. Compared to traditional curvature simplification algorithms, the proposed method has the lowest deviation and highest accuracy at the same simplicity level, as numerous feature points are preserved, which facilitates the point cloud processing.

Gaussian Curvature Entropy for Curved Surface Shape Generation.

The overall shape features that emerge from combinations of shape elements, such as “complexity” and “order”, are important in designing shapes of industrial products. However, controlling the features of shapes is difficult and depends on the experience and intuition of designers. Among these features, “complexity” is said to have an influence on the “beauty” and “preference” of shapes. This research proposed a Gaussian curvature entropy as a “complexity” index of a curved surface shape. The proposed index is calculated based on Gaussian curvature, which is obtained by the sampling and quantization of a curved surface shape and validated by the sensory evaluation experiment while using two types of sample shapes. The result indicates the correspondence of the index to perceived “complexity” (the determination coefficient is greater than 0.8). Additionally, this research constructed a shape generation method that was based on the index as a car design supporting apparatus, in which the designers can refer many shapes generated by controlling “complexity”. The applicability of the proposed method was confirmed by the experiment while using the generated shapes.

Fingertip detection and tracking for recognition of air-writing in videos

Air-writing is the process of writing characters or words in free space using finger or hand movements without the aid of any hand-held device. In this work, we address the problem of mid-air finger writing using web-cam video as input. In spite of recent advances in object detection and tracking, accurate and robust detection and tracking of the fingertip remains a challenging task, primarily due to small dimension of the fingertip. Moreover, the initialization and termination of mid-air finger writing is also challenging due to the absence of any standard delimiting criterion. To solve these problems, we propose a new writing hand pose detection algorithm for initialization of air-writing using the Faster R-CNN framework for accurate hand detection followed by hand segmentation and finally counting the number of raised fingers based on geometrical properties of the hand. Further, we propose a robust fingertip detection and tracking approach using a new signature function called distance-weighted curvature entropy. Finally, a fingertip velocity-based termination criterion is used as a delimiter to mark the completion of the air-writing gesture. Experiments show the superiority of the proposed fingertip detection and tracking algorithm over state-of-the-art approaches giving a mean precision of 73.1% while achieving real-time performance at 18.5 fps, a condition which is of vital importance to air-writing. Character recognition experiments give a mean accuracy of 96.11% using the proposed air-writing system, a result which is comparable to that of existing handwritten character recognition systems.

Deep-learning based surface region selection for deep inspiration breath hold (DIBH) monitoring in left breast cancer radiotherapy

Deep inspiration breath hold (DIBH) with surface supervising is a common technique for cardiac dose reduction in left breast cancer radiotherapy. Surface supervision accuracy relies on the characteristics of surface region. In this study, a convolutional neural network (CNN) based automatic region-of-interest (ROI) selection method was proposed to select an optimal surface ROI for DIBH surface monitoring. The curvature entropy and the normal of each vertex on the breast cancer patient surface were calculated and formed as representative maps for ROI selection learning. 900 ROIs were randomly extracted from each patient’s surface representative map, and the corresponding rigid ROI registration errors (REs) were calculated. The VGG-16 (a 16-layer network structure developed by Visual Geometry Group(VGG) from University of Oxford) pre-trained on a large natural image database ImageNet were fine-tuned using 27 thousand extracted ROIs and the corresponding REs from thirty patients. The RE prediction accuracy of the trained model was validated on additional ten patients. Satisfactory RE predictive accuracies were achieved with the root mean square error (RMSE)/mean absolute error (MAE) smaller than 1 mm/0.7 mm in translations and 0.45°/0.35° in rotations, respectively. The REs of the model selected ROIs on ten testing cases is close to the minimal predicted RE with mean RE differences <1 mm and <0.5° for translation and rotation, respectively. The proposed RE predictive model can be utilized for selecting a quasi-optimal ROI in left breast cancer DIBH radiotherapy (DIBH-RT).

Local curvature entropy-based 3D terrain representation using a comprehensive Quadtree

Large scale 3D digital terrain modeling is a crucial part of many real-time applications in geoinformatics. In recent years, the improved speed and precision in spatial data collection make the original terrain data more complex and bigger, which poses challenges for data management, visualization and analysis. In this work, we presented an effective and comprehensive 3D terrain representation based on local curvature entropy and a dynamic Quadtree. The Level-of-detail (LOD) models of significant terrain features were employed to generate hierarchical terrain surfaces. In order to reduce the radical changes of grid density between adjacent LODs, local entropy of terrain curvature was regarded as a measure of subdividing terrain grid cells. Then, an efficient approach was presented to eliminate the cracks among the different LODs by directly updating the Quadtree due to an edge-based structure proposed in this work. Furthermore, we utilized a threshold of local entropy stored in each parent node of this Quadtree to flexibly control the depth of the Quadtree and dynamically schedule large-scale LOD terrain. Several experiments were implemented to test the performance of the proposed method. The results demonstrate that our method can be applied to construct LOD 3D terrain models with good performance in terms of computational cost and the maintenance of terrain features. Our method has already been deployed in a geographic information system (GIS) for practical uses, and it is able to support the real-time dynamic scheduling of large scale terrain models more easily and efficiently.

Optimal estimate for the gradient of Green’s function on degenerating surfaces and applications

In this paper we prove a uniform estimate for the gradient of the Green function on a closed Riemann surface, independent of its conformal class, and we derive compactness results for immersions with L 2 -bounded second fundamental form and for riemannian surfaces of uniformly bounded gaussian curvature entropy.

Multi-scale Mesh Saliency with Local Patch Weighted Curvature Entropy

Multi-scale Mesh Saliency with Local Patch Weighted Curvature Entropy

Automobile shape formation and simulation by a computer-aided systematic method

A systematic method which combines the shape curvature entropy and computational fluid dynamics analyses was proposed in this study for creating automobile shapes. Various automobile shapes were evaluated by the proposed method and the parameters including the curvature entropy and the drag coefficient of each configuration were calculated so as to determine the optimal shape design. First, shapes of commercially available car models were collected and their curvature entropy values were determined by the proposed equation. Meanwhile, the drag coefficient of each car model was determined by computational fluid dynamics simulation. Both values were compared and the results indicated a positive correlation between an automobile shape’s curvature entropy value and its drag coefficient. Therefore, the curvature entropy is proven to be a suitable indicator for estimating an automobile shape’s aerodynamic performance. For a new automobile shape design, the required number of numerical analyses can be reduced by utilizing the proposed curvature entropy calculation so that the time spent on designing an automobile shape can be greatly reduced.

Thermodynamics and Lemaitre-Tolman-Bondi void models

It has been argued in the literature that in order to make a matter dominated Friedmann-Lemaitre-Robertson-Walker universe compatible with the generalized second law of thermodynamics, one must invoke dark energy, or modified gravity. In the present article we investigate if in a similar spirit, inhomogeneous cosmological models can be motivated on thermodynamic grounds. We examine a particular minimal void Lemaitre-Tolman-Bondi inhomogeneous model which agrees well with observations. While on the one hand we find that the entropy associated with the apparent horizon is not well-behaved thermodynamically, on the other hand the canonical Weyl curvature entropy shows satisfactory thermodynamic behavior. We suggest that evolution of canonical Weyl curvature entropy might be a useful way to evaluate the thermodynamic viability of inhomogeneous cosmologies.

Curvature Entropy for Curved Profile Generation

In a curved surface design, the overall shape features that emerge from combinations of shape elements are important. However, controlling the features of the overall shape in curved profiles is difficult using conventional microscopic shape information such as dimension. Herein two types of macroscopic shape information, curvature entropy and quadrature curvature entropy, quantitatively represent the features of the overall shape. The curvature entropy is calculated by the curvature distribution, and represents the complexity of a shape (one of the overall shape features). The quadrature curvature entropy is an improvement of the curvature entropy by introducing a Markov process to evaluate the continuity of a curvature and to approximate human cognition of the shape. Additionally, a shape generation method using a genetic algorithm as a calculator and the entropy as a shape generation index is presented. Finally, the applicability of the proposed method is demonstrated using the side view of an automobile as a design example.

Nonanalyticities of thermodynamic functions in finite noninteracting Bose gases within an exact microcanonical ensemble

Within an exact microcanonical (MC) ensemble, we study the nonanalyticities of thermodynamic functions research in finite noninteracting Bose gases in traps. The results show that there exists a rich oscillatory behavior of MC thermodynamical quantities as a function of a system's total energy E (e.g., nonmonotonous temperature, nonanalytic and negative specific heats, and microscopic phase transitions). The origin of these nonanalyticities comes directly from the inverted curvature entropy S(E) with respect to E and the behaviors are different in different trap geometries, boundary conditions, and energy spectrum configurations. Contrary to the usual grandcanonical and canonical results, there exists Bose condensation and the nonanalyticities in the two-dimensional finite noninteracting Bose systems with different traps. We also discuss the critical temperature dependence on the particle number N with different ensembles, traps, and boundary conditions. In large enough N, almost all the results of the thermodynamical quantities become smooth, which are similar to the usual canonical behaviors. We emphasize the finite-size effects on the MC entropy change, which should, in principle, be observable in suitably designed experiments of the small systems.

Entropy-based Mesh Simplification

AbstractComplex mesh models usually affect the performance of algorithms for CAD applications. Therefore, it is often desirable to use simplified models and at the same time preserves the general features of the shape. In this paper, we present an entropy based surface mesh simplification algorithm which can rapidly and efficiently produce high quality approximations of triangle mesh models. The algorithm uses iterative contraction of edges to simplify models. It first exploits the geometric feature of model surface by estimating vertex curvature entropy and then iteratively contracts edges with relatively lower simplification cost. The simplification cost is measured by a combination of vertex curvature and curvature entropy of the two vertices compositing the edge. The goal is to simplify regions that are relatively flat while preserving salient features of the shape. Experiments show promising results of the algorithm.