Geodesic Analysis Research Articles

Gender discriminating models from facial surface normals

In this paper, we show how to use facial shape information to construct discriminating models for gender classification. We represent facial shapes using 2.5D fields of facial surface normals, and investigate three different methods to improve the gender discriminating capacity of the model constructed using the standard eigenspace method. The three methods are novel variants of principal geodesic analysis (PGA) namely (a) weighted PGA, (b) supervised weighted PGA, and (c) supervised PGA. Our starting point is to define a weight map over the facial surface that indicates the importance of different locations in discriminating gender. We show how to compute the relevant weights and how to incorporate the weights into the 2.5D model construction. We evaluate the performance of the alternative methods using facial surface normals extracted from 3D range images or recovered from brightness images. Experimental results demonstrate the effectiveness of our methods. Moreover, the classification accuracy, which is as high as 97%, demonstrates the effectiveness of using facial shape information for gender classification. ► We propose three strategies to incorporate gender weights into feature extraction. ► The proposed methods improve the discriminating capacity of leading eigen-features. ► The supervised weighted PGA method significantly outperforms the other two. ► The supervised weighted PGA method competes with the state-of-the-art methods.

Strengthening of the beds and foundations of the Tilla-Kori mosque of the Registan ensemble in Samarkand

Causes of the deformation of the universally known 17th century Tilla-Kori Mosque of the Registan ensemble in Samarkand are examined. Results of geomechanical investigations and geodesic observations, analysis, construction, and design of elements for the strengthening of the bed and foundation and their implementation are described.

Generation of an atlas of the proximal femur and its application to trabecular bone analysis.

Automatic placement of anatomically corresponding volumes of interest and comparison of parameters against a standard of reference are essential components in studies of trabecular bone. Only recently, in vivo MR images of the proximal femur, an important fracture site, could be acquired with high-spatial resolution. The purpose of this MRI trabecular bone study was two-fold: (1) to generate an atlas of the proximal femur to automatically place anatomically corresponding volumes of interest in a population study and (2) to demonstrate how mean models of geodesic topological analysis parameters can be generated to be used as potential standard of reference. Ten females were used to generate the atlas and geodesic topological analysis models, and 10 females were used to demonstrate the atlas-based trabecular bone analysis. All alignments were based on three-dimensional (3D) multiresolution affine transformations followed by 3D multiresolution free-form deformations. Mean distances less than 1 mm between aligned femora, and sharp edges in the atlas and in fused gray-level images of registered femora indicated that the anatomical variability was well accommodated and explained by the free-form deformations.

A nonparametric Riemannian framework for processing high angular resolution diffusion images and its applications to ODF-based morphometry

High angular resolution diffusion imaging (HARDI) has become an important technique for imaging complex oriented structures in the brain and other anatomical tissues. This has motivated the recent development of several methods for computing the orientation probability density function (PDF) at each voxel. However, much less work has been done on developing techniques for filtering, interpolation, averaging and principal geodesic analysis of orientation PDF fields. In this paper, we present a Riemannian framework for performing such operations. The proposed framework does not require that the orientation PDFs be represented by any fixed parameterization, such as a mixture of von Mises–Fisher distributions or a spherical harmonic expansion. Instead, we use a nonparametric representation of the orientation PDF. We exploit the fact that under the square-root re-parameterization, the space of orientation PDFs forms a Riemannian manifold: the positive orthant of the unit Hilbert sphere. We show that various orientation PDF processing operations, such as filtering, interpolation, averaging and principal geodesic analysis, may be posed as optimization problems on the Hilbert sphere, and can be solved using Riemannian gradient descent. We illustrate these concepts with numerous experiments on synthetic, phantom and real datasets. We show their application to studying left/right brain asymmetries.

Longitudinal evaluation of the effects of alendronate on MRI bone microarchitecture in postmenopausal osteopenic women

We evaluated longitudinal effects of alendronate on MRI-based trabecular bone structure parameters derived from dual thresholding and fuzzy clustering (BE-FCM) trabecular bone segmentation. Treatment effects were observed in the distal tibia after 24 months. The BE-FCM method increased correlations to HR-pQCT-based parameters. Introduction High-resolution magnetic resonance imaging (MRI) allows for non-invasive bone microarchitecture analysis. The goal of this study was to examine the potential of MRI-based trabecular bone structure parameters to monitor effects of alendronate in humans in vivo, and to compare the results to HR-pQCT and DXA measurements. Materials and methods Postmenopausal osteopenic women were divided into alendronate treatment and control groups, and imaged at baseline, 12 months, and 24 months ( n = 52 at baseline) using 3 T MRI, HR-pQCT, and DXA. Image acquisition sites included distal tibia (MRI and HR-pQCT), distal radius (MRI, DXA, and HR-pQCT), and the proximal femur (MRI and DXA). Two different regions of interest were evaluated. One contained the trabecular bone region within the entire MRI acquisition, and the second contained a subregion matched to the region contained in the HR-pQCT acquisition. The trabecular bone was segmented using two different methods; dual thresholding and BE-FCM. Trabecular bone structure parameters included bone volume fraction (BV/TV), number (Tb.N), spacing (Tb.Sp), and thickness (Tb.Th), along with seven geodesic topological analysis (GTA) parameters. Longitudinal changes and correlations to HR-pQCT and DXA measurements were evaluated. Results Apparent Tb.N and four GTA parameters showed treatment effects ( p < 0.05) in the distal tibia after 24 months in the entire MRI region using BE-FCM, as well as Tb.N using dual thresholding. No treatment effects after 24 months were observed in the HR-pQCT or in MRI analysis for the HR-pQCT-matched regions. Apparent BV/TV and Tb.N from BE-FCM had significantly higher correlations to HR-pQCT values compared to those derived from thresholding. Conclusions This study demonstrates the influence of computational methods and region of interest definitions on measurements of trabecular bone structure, and the feasibility of MRI-based quantification of longitudinal changes in bone microarchitecture due to bisphosphonate therapy. The results suggest that there may be a need to reevaluate the current standard HR-pQCT region definition for increased treatment sensitivity.

Supervised relevance maps for increasing the distinctiveness of facial images

This paper shows how to improve holistic face analysis by assigning importance factors to different facial regions (termed as face relevance maps). We propose a novel supervised learning algorithm for generating face relevance maps to improve the discriminating capability of existing methods. We have successfully applied the developed technique to face identification based on the Eigenfaces and Fisherfaces methods, and also to gender classification based on principal geodesic analysis (PGA). We demonstrate how to iteratively learn the face relevance map using labelled data. Experimental results confirm the effectiveness of the developed approach.

Facial gender classification using shape-from-shading

The aim in this paper is to show how to use the 2.5D facial surface normals (needle-maps) recovered using shape-from-shading (SFS) to perform gender classification. We use principal geodesic analysis (PGA) to model the distribution of facial surface normals which reside on a Remannian manifold. We incorporate PGA into shape-from-shading, and develop a principal geodesic shape-from-shading (PGSFS) method. This method guarantees that the recovered needle-maps exhibit realistic facial shape by satisfying a statistical model. Moreover, because the recovered facial needle-maps satisfy the data-closeness constraint as a hard constraint, they not only encode facial shape but also implicitly encode image intensity. Experiments explore the gender classification performance using the recovered facial needle-maps on two databases (Notre Dame and FERET), and compare the results with those obtained using intensity images. The results demonstrate the feasibility of gender classification using the recovered facial shape information.

Visual Tracking via Particle Filtering on the Affine Group

We present a particle filtering algorithm for visual tracking, in which the state equations for the object motion evolve on the two-dimensional affine group. We first formulate, in a coordinate-invariant and geometrically meaningful way, particle filtering on the affine group that allows for combined state—covariance estimation. Measurement likelihoods are also calculated from the image covariance descriptors using incremental principal geodesic analysis, a generalization of principal component analysis to curved spaces. Comparative visual tracking studies demonstrate the increased robustness of our tracking algorithm.

Scattering of massless scalar waves by Reissner-Nordström black holes

We present a study of scattering of massless planar scalar waves by a charged non-rotating black hole. Partial wave methods are applied to compute scattering and absorption cross sections, for a range of incident wavelengths. We compare our numerical results with semi-classical approximations from a geodesic analysis, and find excellent agreement. The glory in the backward direction is studied, and its properties are shown to be related to the properties of the photon orbit. The effects of black hole charge upon scattering and absorption are examined in detail. As the charge of the black hole is increased, we find that the absorption cross section decreases, and the angular width of the interference fringes of the scattering cross section at large angles increases. In particular, the glory spot in the backward direction becomes wider. We interpret these effects under the light of our geodesic analysis.

Geodesic topological analysis of trabecular bone microarchitecture from high‐spatial resolution magnetic resonance images

In vivo assessment of trabecular bone microarchitecture could improve the prediction of fracture risk and the efficacy of osteoporosis treatment and prevention. Geodesic topological analysis (GTA) is introduced as a novel technique to quantify the trabecular bone microarchitecture from high-spatial resolution magnetic resonance (MR) images. Trabecular bone parameters that quantify the scale, topology, and anisotropy of the trabecular bone network in terms of its junctions are the result of GTA. The reproducibility of GTA was tested with in vivo images of human distal tibiae and radii (n = 6) at 1.5 Tesla; and its ability to discriminate between subjects with and without vertebral fracture was assessed with ex vivo images of human calcanei at 1.5 and 3.0 Tesla (n = 30). GTA parameters yielded an average reproducibility of 4.8%, and their individual areas under the curve (AUC) of the receiver operating characteristic curve analysis for fracture discrimination performed better at 3.0 than at 1.5 Tesla reaching values of up to 0.78 (p < 0.001). Logistic regression analysis demonstrated that fracture discrimination was improved by combining GTA parameters, and that GTA combined with bone mineral density (BMD) allow for better discrimination than BMD alone (AUC = 0.95; p < 0.001). Results indicate that GTA can substantially contribute in studies of osteoporosis involving imaging of the trabecular bone microarchitecture.

Forensic genotypes and genetic landscapes of extant Argentinean population

Modern forensic studies provide well-established neutral markers, suitable to explore genetic composition of extant populations. Argentinean population integrates indigenous and primarily European groups, mixed in different proportions in the distinct geographical regions. Fine-scale pattern of genetic variation of extant Argentinean population is spatially assessed based on Y-STR haplotype groups. Population genetic measurements are combined with geodesic analysis to detect spatial patterns of genetic variation: the genetic landscapes. A Genetic Geographical Information System (GenGIS) sets the framework to model and to analyze genetic landscapes. The observed northwest-rest gradient of genetic variation was quantified with transect profiles and confirmed with short-term Reynolds’ genetic distance. GenGIS assessment of genetic landscapes provides an insight into the spatial composition of extant, highly admixed Argentinean population.

Automatic shape model building based on principal geodesic analysis bootstrapping

We present a novel method for automatic shape model building from a collection of training shapes. The result is a shape model consisting of the mean model and the major modes of variation with a dense correspondence map between individual shapes. The framework consists of iterations where a medial shape representation is deformed into the training shapes followed by computation of the shape mean and modes of shape variation. In the first iteration, a generic shape model is used as starting point - in the following iterations in the bootstrap method, the resulting mean and modes from the previous iteration are used. Thereby, we gradually capture the shape variation in the training collection better and better. Convergence of the method is explicitly enforced. The method is evaluated on collections of artificial training shapes where the expected shape mean and modes of variation are known by design. Furthermore, collections of real prostates and cartilage sheets are used in the evaluation. The evaluation shows that the method is able to capture the training shapes close to the attainable accuracy already in the first iteration. Furthermore, the correspondence properties measured by generality, specificity, and compactness are improved during the shape model building iterations.

Facial Shape-from-shading and Recognition Using Principal Geodesic Analysis and Robust Statistics

The aim in this paper is to use principal geodesic analysis to model the statistical variations for sets of facial needle maps. We commence by showing how to represent the distribution of surface normals using the exponential map. Shape deformations are described using principal geodesic analysis on the exponential map. Using ideas from robust statistics we show how this deformable model may be fitted to facial images in which there is significant self-shadowing. Moreover, we demonstrate that the resulting shape-from-shading algorithm can be used to recover accurate facial shape and albedo from real world images. In particular, the algorithm can effectively fill-in the facial surface when more than 30% of its area is subject to self-shadowing. To investigate the utility of the shape parameters delivered by the method, we conduct experiments with illumination insensitive face recognition. We present a novel recognition strategy in which similarity is measured in the space of the principal geodesic parameters. We also use the recovered shape information to generate illumination normalized prototype images on which recognition can be performed. Finally we show that, from a single input image, we are able to generate the basis images employed by a number of well known illumination-insensitive recognition algorithms. We also demonstrate that the principal geodesics provide an efficient parameterization of the space of harmonic basis images.

Neontological evaluation of pelvic skeletal attributes purported to reflect sex in extinct non-avian archosaurs

ABSTRACT Sex in non-avian archosaurs has been inferred using a variety of osteological attributes. However, little quantitative data have been presented showing that these phenotypes truly exist. In this study, testing for the presence of pelvic osteological correlates of sex in extant archosaurs was conducted, using skeletons of wild-caught A. mississippiensis as a neontological model. For outgroup comparison, the squamate Iguana iguana is included. A sample of 16 females and 19 males of A. mississippiensis, and 18 females and 10 males of I. iguana were examined. Measurements included pelvic canal area, dorsoventral depth, and mediolateral width of the pelvic canal, mediolateral width between the dorsal edge of each ilium, and ischium orientation. These data were analyzed using analyses of covariance, a t-test, and a recently developed geodesic distance shape analysis. Results indicate that there is sexual dimorphism in the proportions of the pelvic canal in A. mississippiensis, with females typically having deeper pelvic canals than males. This dimorphism might be synapomorphic for Archosauria. No dimorphism was found in I. iguana. The detection of dimorphism in A. mississipiensis required large sample sizes owing to substantial overlap between sexes. Thus, sexing isolated specimens using this metric is tenuous at best. Assuming similar variance in the relative pelvic depth versus width in other non-avian archosaurs, this criterion would also produce imprecise determinations of sex for these taxa.

A geometric approach to quantum circuit lower bounds

What is the minimal size quantum circuit required to exactly implement a specified n-qubit unitary operation, U, without the use of ancilla qubits? We show that a lower bound on the minimal size is provided by the length of the minimal geodesic between U and the identity, I, where length is defined by a suitable Finsler metric on the manifold SU(2^n). The geodesic curves on these manifolds have the striking property that once an initial position and velocity are set, the remainder of the geodesic is completely determined by a second order differential equation known as the geodesic equation. This is in contrast with the usual case in circuit design, either classical or quantum, where being given part of an optimal circuit does not obviously assist in the design of the rest of the circuit. Geodesic analysis thus offers a potentially powerful approach to the problem of proving quantum circuit lower bounds. In this paper we construct several Finsler metrics whose minimal length geodesics provide lower bounds on quantum circuit size. For each Finsler metric we give a procedure to compute the corresponding geodesic equation. We also construct a large class of solutions to the geodesic equation, which we call \emph{Pauli geodesics}, since they arise from isometries generated by the Pauli group. For any unitary U diagonal in the computational basis, we show that: (a) provided the minimal length geodesic is unique, it must be a Pauli geodesic; (b) finding the length of the minimal Pauli geodesic passing from I to U is equivalent to solving an exponential size instance of the closest vector in a lattice problem (CVP); and (c) all but a doubly exponentially small fraction of such unitaries have minimal Pauli geodesics of exponential length.

WE‐C‐I‐609‐07: On Constructing Priors and Likelihoods for Deformable Shape Models

Purpose: Explicit deformable shape models (DSMs) can be used in a Bayesian statistical framework to provide a priori information for posterior optimization to match the DSM against a target image for automatic segmentation. In this approach a DSM is initialized in the target image and undergoes a series of deformations to closely match the target object. Deformation is driven by optimizing an objective function with terms for geometric typicality (prior) and model‐to‐image match (likelihood). The purpose of this work was to develop strategy, methodology, and tools for constructing the geometric prior and intensity likelihood for a particular form of DSM called m‐reps. Method and Materials: Geometric truth is defined for an object of interest by a statistically significant collection of expert human segmentations of training images. M‐reps are fit to the human drawn contours by minimizing the distance between the surfaces of the m‐rep and the contours under added conditions that lead to positional correspondence across training cases. The geometry of the resulting set of training m‐reps is analyzed in non‐Euclidean space using an approach called principal geodesic analysis (PGA) to yield a set of eigenmodes that define the geometric prior. The intensity likelihood is constructed by registering each training m‐rep with the corresponding gray scale image and collecting regional intensity information that is statistically characterized over all training cases. The intensity information can be in several forms including linear profiles and regional histograms. Results: PGA produces modes that include natural deformations such as local twisting, bending, bulging, and constricting. Unlike analysis in Euclidean space, improper shapes are avoided. The form of the intensity prior can be customized to each object of interest for optimal performance. Conclusion: These methods are powerful, robust and generalizable to other DSMs. Conflict of Interest: The presenting author has a financial interest in Morphormics, Inc.

Comparison of human and automatic segmentations of kidneys from CT images

A controlled observer study was conducted to compare a method for automatic image segmentation with conventional user-guided segmentation of right and left kidneys from planning computerized tomographic (CT) images. Deformable shape models called m-reps were used to automatically segment right and left kidneys from 12 target CT images, and the results were compared with careful manual segmentations performed by two human experts. M-rep models were trained based on manual segmentations from a collection of images that did not include the targets. Segmentation using m-reps began with interactive initialization to position the kidney model over the target kidney in the image data. Fully automatic segmentation proceeded through two stages at successively smaller spatial scales. At the first stage, a global similarity transformation of the kidney model was computed to position the model closer to the target kidney. The similarity transformation was followed by large-scale deformations based on principal geodesic analysis (PGA). During the second stage, the medial atoms comprising the m-rep model were deformed one by one. This procedure was iterated until no changes were observed. The transformations and deformations at both stages were driven by optimizing an objective function with two terms. One term penalized the currently deformed m-rep by an amount proportional to its deviation from the mean m-rep derived from PGA of the training segmentations. The second term computed a model-to-image match term based on the goodness of match of the trained intensity template for the currently deformed m-rep with the corresponding intensity data in the target image. Human and m-rep segmentations were compared using quantitative metrics provided in a toolset called Valmet. Metrics reported in this article include (1) percent volume overlap; (2) mean surface distance between two segmentations; and (3) maximum surface separation (Hausdorff distance). Averaged over all kidneys the mean surface separation was 0.12 cm, the mean Hausdorff distance was 0.99 cm, and the mean volume overlap for human segmentations was 88.8%. Between human and m-rep segmentations the mean surface separation was 0.18-0.19 cm, the mean Hausdorff distance was 1.14-1.25 cm, and the mean volume overlap was 82-83%. Overall in this study, the best m-rep kidney segmentations were at least as good as careful manual slice-by-slice segmentations performed by two experienced humans, and the worst performance was no worse than typical segmentations from our clinical setting. The mean surface separations for human-m-rep segmentations were slightly larger than for human-human segmentations but still in the subvoxel range, and volume overlap and maximum surface separation were slightly better for human-human comparisons. These results were expected because of experimental factors that favored comparison of the human-human segmentations. In particular, m-rep agreement with humans appears to have been limited largely by fundamental differences between manual slice-by-slice and true three-dimensional segmentation, imaging artifacts, image voxel dimensions, and the use of an m-rep model that produced a smooth surface across the renal pelvis.

Principal Geodesic Analysis for the Study of Nonlinear Statistics of Shape

A primary goal of statistical shape analysis is to describe the variability of a population of geometric objects. A standard technique for computing such descriptions is principal component analysis. However, principal component analysis is limited in that it only works for data lying in a Euclidean vector space. While this is certainly sufficient for geometric models that are parameterized by a set of landmarks or a dense collection of boundary points, it does not handle more complex representations of shape. We have been developing representations of geometry based on the medial axis description or m-rep. While the medial representation provides a rich language for variability in terms of bending, twisting, and widening, the medial parameters are not elements of a Euclidean vector space. They are in fact elements of a nonlinear Riemannian symmetric space. In this paper, we develop the method of principal geodesic analysis, a generalization of principal component analysis to the manifold setting. We demonstrate its use in describing the variability of medially-defined anatomical objects. Results of applying this framework on a population of hippocampi in a schizophrenia study are presented.

Geodesic optical waveguide lens analysis: Comment

Get PDF Email Share Share with Facebook Tweet This Post on reddit Share with LinkedIn Add to CiteULike Add to Mendeley Add to BibSonomy Get Citation Copy Citation Text E. Marom and O. G. Ramer, "Geodesic optical waveguide lens analysis: Comment," J. Opt. Soc. Am. 69, 791-792 (1979) Export Citation BibTex Endnote (RIS) HTML Plain Text Citation alert Save article