Skeletonization of Volumetric Angiograms for Display

The display of three-dimensional angiograms can benefit from the knowledge of quantitative shape features such as tangent and curvature of the centerline of vessels. These can be obtained from a curve-like skeleton representation. If connectivity and topology are preserved, and if geometrical constraints such as smoothness and centeredness are satisfied, it is possible to estimate length, orientation, curvature, and torsion. It is also required that no part of the original object be left unrepresented. An efficient method for the identification of such shape components is developed. First, a suitable representation is obtained using a voxel coding approach to yield connected and labeled unit-thick paths. The desired features are estimated from a smoothed version of the skeleton produced by a moving average filter. The computational cost is linear, of the order of N_object, the total number of object voxels contained in the binary volumetric data. The method is also shown to be robust to boundary noise. Examples are discussed.     

Visual shape representation with geometrically characterized contour partitions

This paper proposes a biologically plausible matching method to recognize general shapes based on contour curvature information. The human visual system recognizes general shapes flexibly in real-world scenes through the ventral pathway. The pathway is typically modeled using artificial neural networks. These network models, however, do not construct a shape representation that satisfies the following required constraints: (1) The original shape should be represented by a group of partitioned contours in order to retrieve the whole shape (global information) from the partial contours (local information). (2) Coarse and fine structures of the original shapes should be individually represented in order for the visual system to respond to shapes as quickly as possible based on the least number of their features, and to discriminate between shapes based on detailed information. (3) The shape recognition realized with an artificial visual system should be invariant to geometric transformation such as expansion, rotation, or shear. In this paper, we propose a visual shape representation with geometrically characterized contour partitions described on multiple spatial scales.     

A method for the characterization of foldings in protein ribbon models

The ribbon model of chain macromolecules is a useful tool for analyzing some of the targe-scale shape features of these complex systems. Up to now, the ribbon model has been used mostly to produce graphical displays, which are usually analyzed by visual inspection. In this work we suggest a computational method for characterizing automatically, in a concise and algebraic fashion, some of the important shape features of these ribbon models. The procedure is based on a graph-theoretical and knot-theoretical characterization of three well-defined projections of a space curve associated with the ribbon. The labeled graphs can be characterized by the handedness of the crossovers in the ribbon that are the vertices of the graph. The method can be used to provide a fully algebraic representation of the changes occurring when a molecule, such as a protein, undergoes conformational rearrangements (folding), as well as to provide a shape comparison for a pair of related molecular ribbons. This algebraic representation is well suited for easy storage, retrieval, and computer manipulation of the information on the ribbon's shape. Illustrative examples of the method are provided.     

Micron-scale plasma membrane curvature is recognized by the septin cytoskeleton

Cells change shape in response to diverse environmental and developmental conditions, creating topologies with micron-scale features. Although individual proteins can sense nanometer-scale membrane curvature, it is unclear if a cell could also use nanometer-scale components to sense micron-scale contours, such as the cytokinetic furrow and base of neuronal branches. Septins are filament-forming proteins that serve as signaling platforms and are frequently associated with areas of the plasma membrane where there is micron-scale curvature, including the cytokinetic furrow and the base of cell protrusions. We report here that fungal and human septins are able to distinguish between different degrees of micron-scale curvature in cells. By preparing supported lipid bilayers on beads of different curvature, we reconstitute and measure the intrinsic septin curvature preference. We conclude that micron-scale curvature recognition is a fundamental property of the septin cytoskeleton that provides the cell with a mechanism to know its local shape.     

Approximation and characterization of molecular surfaces

The representation and characterization of molecular surfaces are important in many areas of molecular modeling. Parametric representations of protein molecular surfaces are a compact way to describe a surface, and are useful for the evaluation of surface properties such as the normal vector, principal curvatures, and principal curvature directions. Simplified representations of molecular surfaces are useful for efficient rendering and for the display of large-scale surface features. Several techniques for representing surfaces by expansions of spherical harmonic functions have been reported, but these techniques require that the radius function is single valued, that is, each ray from an origin inside the surface intersects the surface at one and only one point. A new technique is described that removes this limitation and can be used to compute surface shape properties. © 1993 John Wiley & Sons, Inc.     

Multi-scale keypoints in V1 and beyond: object segregation, scale selection, saliency maps and face detection

End-stopped cells in cortical area V1, which combine outputs of complex cells tuned to different orientations, serve to detect line and edge crossings, singularities and points with large curvature. These cells can be used to construct retinotopic keypoint maps at different spatial scales (level-of-detail). The importance of the multi-scale keypoint representation is studied in this paper. It is shown that this representation provides very important information for object recognition and face detection. Different grouping operators can be used for object segregation and automatic scale selection. Saliency maps for focus-of-attention can be constructed. Such maps can be employed for face detection by grouping facial landmarks at eyes, nose and mouth. Although a face detector can be based on processing within area V1, it is argued that such an operator must be embedded into dorsal and ventral data streams, to and from higher cortical areas, for obtaining translation-, rotation- and scale-invariant detection.     

Learning to use illumination gradients as an unambiguous cue to three dimensional shape

The luminance and colour gradients across an image are the result of complex interactions between object shape, material and illumination. Using such variations to infer object shape or surface colour is therefore a difficult problem for the visual system. We know that changes to the shape of an object can affect its perceived colour, and that shading gradients confer a sense of shape. Here we investigate if the visual system is able to effectively utilise these gradients as a cue to shape perception, even when additional cues are not available. We tested shape perception of a folded card object that contained illumination gradients in the form of shading and more subtle effects such as inter-reflections. Our results suggest that observers are able to use the gradients to make consistent shape judgements. In order to do this, observers must be given the opportunity to learn suitable assumptions about the lighting and scene. Using a variety of different training conditions, we demonstrate that learning can occur quickly and requires only coarse information. We also establish that learning does not deliver a trivial mapping between gradient and shape; rather learning leads to the acquisition of assumptions about lighting and scene parameters that subsequently allow for gradients to be used as a shape cue. The perceived shape is shown to be consistent for convex and concave versions of the object that exhibit very different shading, and also similar to that delivered by outline, a largely unrelated cue to shape. Overall our results indicate that, although gradients are less reliable than some other cues, the relationship between gradients and shape can be quickly assessed and the gradients therefore used effectively as a visual shape cue.     

Medial axis shape coding in macaque inferotemporal cortex

The basic, still unanswered question about visual object representation is this: what specific information is encoded by neural signals? Theorists have long predicted that neurons would encode medial axis or skeletal object shape, yet recent studies reveal instead neural coding of boundary or surface shape. Here, we addressed this theoretical/experimental disconnect, using adaptive shape sampling to demonstrate explicit coding of medial axis shape in high-level object cortex (macaque monkey inferotemporal cortex or IT). Our metric shape analyses revealed a coding continuum, along which most neurons represent a configuration of both medial axis and surface components. Thus, IT response functions embody a rich basis set for simultaneously representing skeletal and external shape of complex objects. This would be especially useful for representing biological shapes, which are often characterized by both complex, articulated skeletal structure and specific surface features.     

On the use of size functions for shape analysis

According to a recent mathematical theory a shape can be represented by size functions, which convey information on both the topological and metric properties of the viewed shape. In this paper the relevance of the theory of size functions to visual perception is investigated. An algorithm for the computation of the size functions is presented, and many theoretical properties of the theory are demonstrated on real images. It is shown that the representation of shape in terms of size functions (1) can be tailored to suit the invariance of the problem at hand and (2) is stable against small qualitative and quantitative changes of the viewed shape. A distance between size functions is used as a measure of similarity between the representations of two different shapes. The results obtained indicate that size functions are likely to be very useful for object recognition. In particular, they seem to be well suited for the recognition of natural and articulated objects.     

An intuitive approach to measuring protein surface curvature

A natural way to measure protein surface curvature is to generate the least squares fitted (LSF) sphere to a surface patch and use the radius as the curvature measure. While the concept is simple, the sphere-fitting problem is not trivial and known means of protein surface curvature measurement use alternative schemes that are arguably less straightforward to interpret. We have developed an approach to solve the LSF sphere problem by turning the sphere-fitting problem into a solvable plane-fitting problem using a transformation known as geometric inversion. The approach works on any arbitrary surface patch, and returns a radius of curvature that has direct physical interpretation. Additionally, it is flexible in its ability to find the curvature of an arbitrary surface patch, and the "resolution" can be adjusted to highlight atomic features or larger features such as peptide binding sites. We include examples of applying the method to visualization of peptide recognition pockets and protein conformational change, as well as a comparison with a commonly used solid-angle curvature method showing that the LSF method produces more pronounced curvature results.     

Spatial visualization of DNA in solution.

An image analysis method is presented which allows for the reconstruction of the three-dimensional path of filamentous objects from two of their projections. Starting with stereo pairs, this method is used to trace the trajectory of DNA molecules embedded in vitreous ice and leads to a faithful representation of their three-dimensional shape in solution. This computer-aided reconstruction is superior to the subjective three-dimensional impression generated by observation of stereo pairs of micrographs because it enables one to look at the reconstructed molecules from any chosen direction and distance and allows quantitative analysis such as determination of distances, curvature, persistence length, and writhe of DNA molecules in solution.     

利用指骨弯曲程度推断化石人猿超科成员移动行为方式的新方法:以禄丰古猿为例

长久以来,指骨弯曲程度都被用来推断化石灵长类移动行为方式。此前已有一些方法被提出并用于定量化比较指骨弯曲程度,包括半径弯曲程度法（radius of curvature）、夹角法（included angle, IA）、标准化矩臂弯曲程度法（normalized curvature moment arm, NCMA）以及高精度多项式曲线拟合法（high-resolution polynomial curve fitting, HR-PCF）。然而,在对指骨弯曲程度进行定量化的过程中,这些方法都显示出了理论或技术上的局限性。因此,在运用这些方法之前,应当谨慎考虑其适用性和精确程度对分析结果所产生的影响。鉴于此,为了避免先前方法中存在的问题并更加精确地定量描述指骨弯曲程度,本文介绍了一种新方法作为替代。该方法基于对指骨侧视图背侧轮廓曲线几何形态测量学标志点数据的四阶多项式曲线拟合,称为几何形态测量学—多项式曲线拟合法（4th order polynomial curve fitting on geometric morphometric landmark data, GM-PCF）。它以标准化指骨曲线高度（normalized phalangeal curve height, NPCH）作为指骨弯曲程度的定量指标,并且可以将平均标准化指骨曲线进行可视化以用于其弯曲程度的直观对比。此外,它还可以提供在解释指骨弯曲程度的功能意义中非常关键的指骨（背侧轮廓曲线）长度比例信息。GM-PCF还能够分析化石中常见的不完整的指骨。为了检验新方法的适用性,我们从现生类人猿（anthropoids）中选取了15个涵盖灵长类大部分移动行为方式的类群作为参考样本,采用GM-PCF方法对其指骨弯曲程度进行了定量分析,结果表明标准化指骨曲线高度（NPCH）对灵长类移动行为方式有很好的指示意义,配合指骨曲线长度,还可以更进一步了解树栖四足行走（arboreal quadrupedalism）、悬垂（suspension）与摆荡（brachiation）等行为与灵长类体型大小的关系。作为个案,我们采用新方法对中国中新世的两种禄丰古猿（禄丰禄丰古猿Lufengpithecus lufengensis和蝴蝶禄丰古猿Lufengpithecus hudienensis）的指骨弯曲程度与参考样本进行了对比,并根据对比结果对其最为可能的移动行为方式偏好进行了推断。     

A machine learning approach to investigate the relationship between shape features and numerically predicted risk of ascending aortic aneurysm

Geometric features of the aorta are linked to patient risk of rupture in the clinical decision to electively repair an ascending aortic aneurysm (AsAA). Previous approaches have focused on relationship between intuitive geometric features (e.g., diameter and curvature) and wall stress. This work investigates the feasibility of a machine learning approach to establish the linkages between shape features and FEA-predicted AsAA rupture risk, and it may serve as a faster surrogate for FEA associated with long simulation time and numerical convergence issues. This method consists of four main steps: (1) constructing a statistical shape model (SSM) from clinical 3D CT images of AsAA patients; (2) generating a dataset of representative aneurysm shapes and obtaining FEA-predicted risk scores defined as systolic pressure divided by rupture pressure (rupture is determined by a threshold criterion); (3) establishing relationship between shape features and risk by using classifiers and regressors; and (4) evaluating such relationship in cross-validation. The results show that SSM parameters can be used as strong shape features to make predictions of risk scores consistent with FEA, which lead to an average risk classification accuracy of 95.58% by using support vector machine and an average regression error of 0.0332 by using support vector regression, while intuitive geometric features have relatively weak performance. Compared to FEA, this machine learning approach is magnitudes faster. In our future studies, material properties and inhomogeneous thickness will be incorporated into the models and learning algorithms, which may lead to a practical system for clinical applications.     

On the role of medial geometry in human vision.

A key challenge underlying theories of vision is how the spatially restricted, retinotopically represented feature analysis can be integrated to form abstract, coordinate-free object models. A resolution likely depends on the use of intermediate-level representations which can on the one hand be populated by local features and on the other hand be used as atomic units underlying the formation of, and interaction with, object hypotheses. The precise structure of this intermediate representation derives from the varied requirements of a range of visual tasks which motivate a significant role for incorporating a geometry of visual form. The need to integrate input from features capturing surface properties such as texture, shading, motion, color, etc., as well as from features capturing surface discontinuities such as silhouettes, T-junctions, etc., implies a geometry which captures both regional and boundary aspects. Curves, as a geometric model of boundaries, have been extensively used as an intermediate representation in computational, perceptual, and physiological studies, while the use of the medial axis (MA) has been popular mainly in computer vision as a geometric region-based model of the interior of closed boundaries. We extend the traditional model of the MA to represent images, where each MA segment represents a region of the image which we call a visual fragment. We present a unified theory of perceptual grouping and object recognition where through various sequences of transformations of the MA representation, visual fragments are grouped in various configurations to form object hypotheses, and are related to stored models. The mechanisms underlying both the computation and the transformation of the MA is a lateral wave propagation model. Recent psychophysical experiments depicting contrast sensitivity map peaks at the medial axes of stimuli, and experiments on perceptual filling-in, and brightness induction and modulation, are consistent with both the use of an MA representation and a propagation-based scheme. Also, recent neurophysiological recordings in V1 correlate with the MA hypothesis and a horizontal propagation scheme. This evidence supports a geometric computational paradigm for processing sensory data where both dynamic in-plane propagation and feedforward-feedback connections play an integral role.     

Role of the skeleton in determination of the branching points in hydroid colonies

Colonial hydroids of the suborder Thecaphora have rigid outer skeleton that possesses species-specific shape of the colony elements. Organisation of the elements within a colony shows strict spatial patterning. The points of branching (emergence of the new growing tip) within shoots are strictly determined and show positive correlation with the place of the most pronounced curvature of the elements skeleton. As it was shown earlier, the shoot growing tip, after emergence, performs its program of functioning independently from the colony condition. Several modifications of experiments with grafting of the growing tip together with rotation around its longitudinal axis were fulfilled. As the result of such grafting the normal orientation of the formed skeleton was altered. In overwhelming cases of further tip growth and shoot development the new growing tip emerged in correspondence with new orientation of the formed skeleton. The orientation of the hydranth of the maternal shoot element had no effect upon the orientation of the new tip emergence after grafting. It is supposed that the place of the new tip emergence is regulated hierarchically. At first level, the interaction of the soft tissues with the skeleton has priority. Anisotropy of mechanical tensions within cell layers due to interaction with the skeleton at the point of its most curvature serves as a primary stimulus in the chain of events leading to the initiation of the new growing tip. If this does not work the intrinsic tissue property (polarity) determines the place of the tip emergence. Such two-level mechanism of determination of the place of the new tip emergence in sympodial shoots stabilises spatial organisation of the entire shoot and decreases the probability of epigenetic mistakes.     

水果形状的傅里叶描述子研究

水果的形状是水果分级的重要指标之一。本文研究了不规则物体形状的数学描述方法,认为在水果的分级过程中采用曲线拟合的方法来描述水果的形状是不合适的;提出了仅需利用物体的边界信息求物体的形心坐标和描述果形的新方法;发现用Fourier描述子的前4个谐波分量的变化特性就能较好地代表水果的形状,用前15个谐波分量来描述形状则可以达到相当高的精度。而且傅立叶描述子可以进行平移、旋转和缩放,并具有很强的水果外形重建功能,是一描述水果形状的非常有效的方法。     

Computational analysis of looping of a large family of highly bent DNA by LacI

Sequence-dependent intrinsic curvature of DNA influences looping by regulatory proteins such as LacI and NtrC. Curvature can enhance stability and control shape, as observed in LacI loops formed with three designed sequences with operators bracketing an A-tract bend. We explore geometric, topological, and energetic effects of curvature with an analysis of a family of highly bent sequences, using the elastic rod model from previous work. A unifying straight-helical-straight representation uses two phasing parameters to describe sequences composed of two straight segments that flank a common helically supercoiled segment. We exercise the rod model over this two-dimensional space of phasing parameters to evaluate looping behaviors. This design space is found to comprise two subspaces that prefer parallel versus anti-parallel binding topologies. The energetic cost of looping varies from 4 to 12 kT. Molecules can be designed to yield distinct binding topologies as well as hyperstable or hypostable loops and potentially loops that can switch conformations. Loop switching could be a mechanism for control of gene expression. Model predictions for linking numbers and sizes of LacI-DNA loops can be tested using multiple experimental approaches, which coupled with theory could address whether proteins or DNA provide the observed flexibility of protein-DNA loops.