Existence, uniqueness and regularity of the projection onto differentiable manifolds.

We investigate the maximal open domain E ( M ) on which the orthogonal projection map p onto a subset M ⊆ R d can be defined and study essential properties of p. We prove that if M is a C 1 submanifold of R d satisfying a Lipschitz condition on the tangent spaces, then E ( M ) can be described by a lower semi-continuous function, named frontier function. We show that this frontier function is continuous if M is C 2 or if the topological skeleton of M c is closed and we provide an example showing that the frontier function need not be continuous in general. We demonstrate that, for a C k -submanifold M with k ≥ 2 , the projection map is C k - 1 on E ( M ) , and we obtain a differentiation formula for the projection map which is used to discuss boundedness of its higher order differentials on tubular neighborhoods. A sufficient condition for the inclusion M ⊆ E ( M ) is that M is a C 1 submanifold whose tangent spaces satisfy a local Lipschitz condition. We prove in a new way that this condition is also necessary. More precisely, if M is a topological submanifold with M ⊆ E ( M ) , then M must be C 1 and its tangent spaces satisfy the same local Lipschitz condition. A final section is devoted to highlighting some relations between E ( M ) and the topological skeleton of M c .

Shape coding in occipito-temporal cortex relies on object silhouette, curvature, and medial axis.

Object recognition relies on different transformations of the retinal input, carried out by the visual system, that range from local contrast to object shape and category. While some of those transformations are thought to occur at specific stages of the visual hierarchy, the features they represent are correlated (e.g., object shape and identity) and selectivity for the same feature overlaps in many brain regions. This may be explained either by collinearity across representations or may instead reflect the coding of multiple dimensions by the same cortical population. Moreover, orthogonal and shared components may differently impact distinctive stages of the visual hierarchy. We recorded functional MRI activity while participants passively attended to object images and employed a statistical approach that partitioned orthogonal and shared object representations to reveal their relative impact on brain processing. Orthogonal shape representations (silhouette, curvature, and medial axis) independently explained distinct and overlapping clusters of selectivity in the occitotemporal and parietal cortex. Moreover, we show that the relevance of shared representations linearly increases moving from posterior to anterior regions. These results indicate that the visual cortex encodes shared relations between different features in a topographic fashion and that object shape is encoded along different dimensions, each representing orthogonal features.NEW & NOTEWORTHY There are several possible ways of characterizing the shape of an object. Which shape description better describes our brain responses while we passively perceive objects? Here, we employed three competing shape models to explain brain representations when viewing real objects. We found that object shape is encoded in a multidimensional fashion and thus defined by the interaction of multiple features.

Automatic Fiber Length Measurements with a Multi-Stencil Fast Marching Method on Microscopy Images.

Fiber length has a strong impact on the mechanical properties of composite materials. It is one of the most important quantitative features in characterizing microstructures for understanding the material performance. Studies conducted to determine fiber length distribution have primarily focused on sample preparation and fiber dispersion. However, the subsequent image analysis is frequently performed manually or semi-automatically, which either requires careful sample preparation or manual intervention in the image analysis and processing. In this article, an image processing and analysis method has been developed based on medial axis transformation via the multi-stencil fast marching method for fiber length measurements on acquired microscopy images. The developed method can be implemented fully automatically and without any user induced delays. This method offers high efficiency, sub-pixel accuracy, and excellent statistical representativity.

Open-source graphical user interface for the creation of synthetic skeletons for medical image analysis.

Purpose: Deformable medial modeling is an inverse skeletonization approach to representing anatomy in medical images, which can be used for statistical shape analysis and assessment of patient-specific anatomical features such as locally varying thickness. It involves deforming a pre-defined synthetic skeleton, or template, to anatomical structures of the same class. The lack of software for creating such skeletons has been a limitation to more widespread use of deformable medial modeling. Therefore, the objective of this work is to present an open-source user interface (UI) for the creation of synthetic skeletons for a range of medial modeling applications in medical imaging. Approach: A UI for interactive design of synthetic skeletons was implemented in 3D Slicer, an open-source medical image analysis application. The steps in synthetic skeleton design include importation and skeletonization of a 3D segmentation, followed by interactive 3D point placement and triangulation of the medial surface such that the desired branching configuration of the anatomical structure's medial axis is achieved. Synthetic skeleton design was evaluated in five clinical applications. Compatibility of the synthetic skeletons with open-source software for deformable medial modeling was tested, and representational accuracy of the deformed medial models was evaluated. Results: Three users designed synthetic skeletons of anatomies with various topologies: the placenta, aortic root wall, mitral valve, cardiac ventricles, and the uterus. The skeletons were compatible with skeleton-first and boundary-first software for deformable medial modeling. The fitted medial models achieved good representational accuracy with respect to the 3D segmentations from which the synthetic skeletons were generated. Conclusions: Synthetic skeleton design has been a practical challenge in leveraging deformable medial modeling for new clinical applications. This work demonstrates an open-source UI for user-friendly design of synthetic skeletons for anatomies with a wide range of topologies.

A novel procedure for medial axis reconstruction of vessels from Medical Imaging segmentation.

A procedure for reconstructing the central axis from diagnostic image processing is presented here, capable of solving the widespread problem of stepped shape effect that characterizes the most common algorithmic tools for processing the central axis for diagnostic imaging applications through the development of an algorithm correcting the spatial coordinates of each point belonging to the axis from the use of a common discrete image skeleton algorithm. The procedure is applied to the central axis traversing the vascular branch of the cerebral system, appropriately reconstructed from the processing of diagnostic images, using investigations of the local intensity values identified in adjacent voxels. The percentage intensity of the degree of adherence to a specific anatomical tissue acts as an attraction pole in the identification of the spatial center on which to place each point of the skeleton crossing the investigated anatomical structure. The results were shown in terms of the number of vessels identified overall compared to the original reference model. The procedure demonstrates high accuracy margin in the correction of the local coordinates of the central points that permits to allocate precise dimensional measurement of the anatomy under examination. The reconstruction of a central axis effectively centered in the region under examination represents a fundamental starting point in deducing, with a high margin of accuracy, key informations of a geometric and dimensional nature that favours the recognition of phenomena of shape alterations ascribable to the presence of clinical pathologies.

Variable Offset Computation Space for Automatic Cooling Dimensioning.

The injection mold is one of the most important elements for the part precision of this important mass production process. The thermal mold design is realized by cooling channels around the cavity and poses as a decisive factor for the part quality. Thus, the objective but specific design of the cooling channel layout is crucial for a reproducible part with high-dimensional accuracy in production. Consequently, knowledge-based and automated methods are used to create the optimal heat management in the mold. One of these methods is the inverse thermal mold design, which uses a specific calculation space. The geometric boundary conditions of the optimization algorithm influence the resulting thermal balance within the mold. As the calculation area in the form of an offset around the molded part is one of these boundary conditions, its influence on the optimization result is determined. The thermal optimizations show a dependency on different offset shapes due to the offset thickness and coalescence of concave geometries. An algorithm is developed to generate an offset for this thermal mold design methodology considering the identified influences. Hence, a reproducible and adaptive offset is generated automatically for a complex geometry, and the quality function result improves by 43% in this example.

Skeletal representations of shape in the human visual cortex.

Shape perception is crucial for object recognition. However, it remains unknown exactly how shape information is represented and used by the visual system. Here, we tested the hypothesis that the visual system represents object shape via a skeletal structure. Using functional magnetic resonance imaging (fMRI) and representational similarity analysis (RSA), we found that a model of skeletal similarity explained significant unique variance in the response profiles of V3 and LO. Moreover, the skeletal model remained predictive in these regions even when controlling for other models of visual similarity that approximate low-to high-level visual features (i.e., Gabor-jet, GIST, HMAX, and AlexNet), and across different surface forms, a manipulation that altered object contours while preserving the underlying skeleton. Together, these findings shed light on shape processing in human vision, as well as the computational properties of V3 and LO. We discuss how these regions may support two putative roles of shape skeletons: namely, perceptual organization and object recognition.

Perception of an object's global shape is best described by a model of skeletal structure in human infants.

Categorization of everyday objects requires that humans form representations of shape that are tolerant to variations among exemplars. Yet, how such invariant shape representations develop remains poorly understood. By comparing human infants (6-12 months; N=82) to computational models of vision using comparable procedures, we shed light on the origins and mechanisms underlying object perception. Following habituation to a never-before-seen object, infants classified other novel objects across variations in their component parts. Comparisons to several computational models of vision, including models of high-level and low-level vision, revealed that infants' performance was best described by a model of shape based on the skeletal structure. Interestingly, infants outperformed a range of artificial neural network models, selected for their massive object experience and biological plausibility, under the same conditions. Altogether, these findings suggest that robust representations of shape can be formed with little language or object experience by relying on the perceptually invariant skeletal structure.

Shape design of magnetic circuit of the magnetic lens based on medial axis transform.

This paper proposes a method of designing a magnetic lens circuit based on axial transform theory. A unified mathematical description of the magnetic circuit profile is given within a medial axial transform in differential geometry. The objective constraint and geometric constraint equations of the magnetic circuit form a closed set of differential equations. Lastly, the optimized magnetic circuit is obtained using the solutions of the equations. The optical performance of the magnetic lens is significantly improved.

Real-Time Thinning Algorithms for 2D and 3D Images using GPU processors.

The skeletonization of binary images is a common task in many image processing and machine learning applications. Some of these applications require very fast image processing. We propose novel techniques for efficient 2D and 3D thinning of binary images using GPU processors. The algorithms use bit-encoded binary images to process multiple points simultaneously in each thread. The simpleness of a point is determined based on Boolean algebra using only bitwise logical operators. This avoids computationally expensive decoding and encoding steps and allows for additional parallelization. The 2D algorithm is evaluated using a dataset of handwritten characters images. It required an average computation time of 3.53 ns for 32×32 pixels and 0.25 ms for 1024×1024 pixels. This is 52 to 18,380 times faster than a multi-threaded border-parallel algorithm. The 3D algorithm was evaluated based on clinical images of the human vasculature and required computation times of 0.27 ms for 128×128×128 voxels and 20.32 ms for 512×512×512 voxels, which is 32 to 46 times faster than the compared border-sequential algorithm using the same GPU processor. The proposed techniques enable efficient real-time 2D and 3D skeletonization of binary images, which could improve the performance of many existing machine learning applications.

Shape discrimination along morph-spaces.

We investigated the dimensions defining mental shape space, by measuring shape discrimination thresholds along "morph-spaces" defined by pairs of shapes. Given any two shapes, one can construct a morph-space by taking weighted averages of their boundary vertices (after normalization), creating a continuum of shapes ranging from the first shape to the second. Previous studies of morphs between highly familiar shape categories (e.g. truck and turkey) have shown elevated discrimination at category boundaries, reflecting a kind of "categorical perception" in shape space. Here, we use this technique to explore the underlying representation of unfamiliar shapes. Subjects were shown two shapes at nearby points along a morph-space, and asked to judge whether they were the same or different, with an adaptive procedure used to estimate discrimination thresholds at each point along the morph-space. We targeted several potentially important categorical distinctions, such one- vs. two-part shapes, two- vs. three-part shapes, changes in symmetry structure, and other potentially important distinctions. Observed discrimination thresholds showed substantial and systematic deviations from uniformity at different points along each shape continuum, meaning that subjects were consistently better at discriminating at certain points along each morph-space than at others. We introduce a shape similarity measure, based on Bayesian skeletal shape representations, which gives a good account of the observed variations in shape sensitivity.

Use of medial axis for reorientation by the Clark's nutcracker (Nucifraga columbiana).

Many animals are challenged with the task of reorientation. Considerable research over the years has shown a diversity of species extract geometric information (e.g., distance and direction) from continuous surfaces or boundaries to reorient. How this information is extracted from the environment is less understood. Three encoding strategies that have received the most study are the use of principal axes, medial axis or local geometric cues. We used a modeling approach to investigate which of these three general strategies best fit the spatial search data of a highly-spatial corvid, the Clark's nutcracker (Nucifraga columbiana). Individual nutcrackers were trained in a rectangular-shaped arena, and once accurately locating a hidden goal, received non-reinforced tests in an L-shaped arena. The specific shape of this arena allowed us to dissociate among the three general encoding strategies. Furthermore, we reanalyzed existing data from chicks, pigeons and humans using our modeling approach. Overall, we found the most support for the use of the medial axis, although we additionally found that pigeons and humans may have engaged in random guessing. As with our previous studies, we find no support for the use of principal axes.

Seeing structure: Shape skeletons modulate perceived similarity.

An intrinsic part of seeing objects is seeing how similar or different they are relative to one another. This experience requires that objects be mentally represented in a common format over which such comparisons can be carried out. What is that representational format? Objects could be compared in terms of their superficial features (e.g., degree of pixel-by-pixel overlap), but a more intriguing possibility is that they are compared on the basis of a deeper structure. One especially promising candidate that has enjoyed success in the computer vision literature is the shape skeleton-a geometric transformation that represents objects according to their inferred underlying organization. Despite several hints that shape skeletons are computed in human vision, it remains unclear how much they actually matter for subsequent performance. Here, we explore the possibility that shape skeletons help mediate the ability to extract visual similarity. Observers completed a same/different task in which two shapes could vary either in their skeletal structure (without changing superficial features such as size, orientation, and internal angular separation) or in large surface-level ways (without changing overall skeletal organization). Discrimination was better for skeletally dissimilar shapes: observers had difficulty appreciating even surprisingly large differences when those differences did not reorganize the underlying skeletons. This pattern also generalized beyond line drawings to 3-D volumes whose skeletons were less readily inferable from the shapes' visible contours. These results show how shape skeletons may influence the perception of similarity-and more generally, how they have important consequences for downstream visual processing.

Medial Axis and Singularities.

This paper is devoted to the study of the medial axes of sets definable in polynomially bounded o-minimal structures, i.e. the sets of points with more than one closest point with respect to the Euclidean distance. Our point of view is that of singularity theory. While trying to make the paper self-contained, we gather here also a large bunch of basic results. Our main interest, however, goes to the characterization of those singular points of a definable, closed set X ⊂ R n , which are reached by the medial axis.

Measurement of smaller colon polyp in CT colonography images using morphological image processing.

PURPOSE: Automated measurement of the size and shape of colon polyps is one of the challenges in Computed tomography colonography (CTC). The objective of this retrospective study was to improve the sensitivity and specificity of smaller polyp measurement in CTC using image processing techniques. METHODS: A domain knowledge-based method has been implemented with hybrid method of colon segmentation, morphological image processing operators for detecting the colonic structures, and the decision-making system for delineating the smaller polyp-based on a priori knowledge. RESULTS: The method was applied on 45 CTC dataset. The key finding was that the smaller polyps were accurately measured. In addition to 6-9 mm range, polyps of even <5 mm were also detected. The results were validated qualitatively and quantitatively using both 2D MPR and 3D view. Implementation was done on a high-performance computer with parallel processing. It takes [Formula: see text] min for measuring the smaller polyp in a dataset of 500 CTC images. With this method, [Formula: see text] and [Formula: see text] were achieved. CONCLUSIONS: The domain-based approach with morphological image processing has given good results. The smaller polyps were measured accurately which helps in making right clinical decisions. Qualitatively and quantitatively the results were acceptable when compared to the ground truth at [Formula: see text].

Image Segmentation and Modeling of the Pediatric Tricuspid Valve in Hypoplastic Left Heart Syndrome.

Hypoplastic left heart syndrome (HLHS) is a single-ventricle congenital heart disease that is fatal if left unpalliated. In HLHS patients, the tricuspid valve is the only functioning atrioventricular valve, and its competence is therefore critical. This work demonstrates the first automated strategy for segmentation, modeling, and morphometry of the tricuspid valve in transthoracic 3D echocardiographic (3DE) images of pediatric patients with HLHS. After initial landmark placement, the automated segmentation step uses multi-atlas label fusion and the modeling approach uses deformable modeling with medial axis representation to produce patient-specific models of the tricuspid valve that can be comprehensively and quantitatively assessed. In a group of 16 pediatric patients, valve segmentation and modeling attains an accuracy (mean boundary displacement) of 0.8 ± 0.2 mm relative to manual tracing and shows consistency in annular and leaflet measurements. In the future, such image-based tools have the potential to improve understanding and evaluation of tricuspid valve morphology in HLHS and guide strategies for patient care.

Spatiotemporal Segmentation and Modeling of the Mitral Valve in Real-Time 3D Echocardiographic Images.

Transesophageal echocardiography is the primary imaging modality for preoperative assessment of mitral valves with ischemic mitral regurgitation (IMR). While there are well known echocardiographic insights into the 3D morphology of mitral valves with IMR, such as annular dilation and leaflet tethering, less is understood about how quantification of valve dynamics can inform surgical treatment of IMR or predict short-term recurrence of the disease. As a step towards filling this knowledge gap, we present a novel framework for 4D segmentation and geometric modeling of the mitral valve in real-time 3D echocardiography (rt-3DE). The framework integrates multi-atlas label fusion and template-based medial modeling to generate quantitatively descriptive models of valve dynamics. The novelty of this work is that temporal consistency in the rt-3DE segmentations is enforced during both the segmentation and modeling stages with the use of groupwise label fusion and Kalman filtering. The algorithm is evaluated on rt-3DE data series from 10 patients: five with normal mitral valve morphology and five with severe IMR. In these 10 data series that total 207 individual 3DE images, each 3DE segmentation is validated against manual tracing and temporal consistency between segmentations is demonstrated. The ultimate goal is to generate accurate and consistent representations of valve dynamics that can both visually and quantitatively provide insight into normal and pathological valve function.

Post-processing techniques for making reliable measurements from curve-skeletons.

Interconnected 3-D networks occur widely in biology and the geometry of such branched networks can be described by curve-skeletons, allowing parameters such as path lengths, path tortuosities and cross-sectional thicknesses to be quantified. However, curve-skeletons are typically sensitive to small scale surface features which may arise from noise in the imaging data. In this paper, new post-processing techniques for curve-skeletons are presented which ensure that measurements of lengths and thicknesses are less sensitive to these small scale surface features. The techniques achieve sub-voxel accuracy and are based on a minimal sphere-network representation in which the object is modelled as a string of minimally overlapping spheres, and as such samples the object on a scale related to the local thickness. A new measure of cross-sectional dimension termed the modal radius is defined and shown to be more robust in comparison with the standard measure (the internal radius), while retaining the desirable feature of capturing the size of structures in terms of a single measure. The techniques are demonstrated by application to trabecular bone and tumour vascular network case studies where the volumetric data was obtained by high resolution computed tomography.

Segmentation of the Aortic Valve Apparatus in 3D Echocardiographic Images: Deformable Modeling of a Branching Medial Structure.

3D echocardiographic (3DE) imaging is a useful tool for assessing the complex geometry of the aortic valve apparatus. Segmentation of this structure in 3DE images is a challenging task that benefits from shape-guided deformable modeling methods, which enable inter-subject statistical shape comparison. Prior work demonstrates the efficacy of using continuous medial representation (cm-rep) as a shape descriptor for valve leaflets. However, its application to the entire aortic valve apparatus is limited since the structure has a branching medial geometry that cannot be explicitly parameterized in the original cm-rep framework. In this work, we show that the aortic valve apparatus can be accurately segmented using a new branching medial modeling paradigm. The segmentation method achieves a mean boundary displacement of 0.6 ± 0.1 mm (approximately one voxel) relative to manual segmentation on 11 3DE images of normal open aortic valves. This study demonstrates a promising approach for quantitative 3DE analysis of aortic valve morphology.

Medially constrained deformable modeling for segmentation of branching medial structures: Application to aortic valve segmentation and morphometry.

Deformable modeling with medial axis representation is a useful means of segmenting and parametrically describing the shape of anatomical structures in medical images. Continuous medial representation (cm-rep) is a "skeleton-first" approach to deformable medial modeling that explicitly parameterizes an object's medial axis and derives the object's boundary algorithmically. Although cm-rep has effectively been used to segment and model a number of anatomical structures with non-branching medial topologies, the framework is challenging to apply to objects with branching medial geometries since branch curves in the medial axis are difficult to parameterize. In this work, we demonstrate the first clinical application of a new "boundary-first" deformable medial modeling paradigm, wherein an object's boundary is explicitly described and constraints are imposed on boundary geometry to preserve the branching configuration of the medial axis during model deformation. This "boundary-first" framework is leveraged to segment and morphologically analyze the aortic valve apparatus in 3D echocardiographic images. Relative to manual tracing, segmentation with deformable medial modeling achieves a mean boundary error of 0.41 ± 0.10 mm (approximately one voxel) in 22 3DE images of normal aortic valves at systole. Deformable medial modeling is additionally demonstrated on pathological cases, including aortic stenosis, Marfan syndrome, and bicuspid aortic valve disease. This study demonstrates a promising approach for quantitative 3DE analysis of aortic valve morphology.