Punam K. Saha

Scale-based fuzzy connected image segmentation: theory, algorithms, and validation

This paper extends a previously reported theory and algorithms for object definition based on fuzzy connectedness. In this approach, a strength of connectedness is determined between every pair of image elements. This is done by considering all possible connecting paths between the two elements in each pair. The strength assigned to a particular path is defined as the weakest affinity between successive pairs of elements along the path. Affinity specifies the degree to which elements hang together locally in the image. Although the theory allowed any neighborhood size for affinity definition, it did not indicate how this was to be selected. By bringing object scale into the framework in this paper, not only the size of the neighborhood is specified but also it is allowed to change in different parts of the image. This paper argues that scale-based affinity, and hence connectedness, is natural in object definition and demonstrates that this leads to more effective object segmentation.The approach presented here considers affinity to consist of two components. The homogeneity-based component indicates the degree of affinity between image elements based on the homogeneity of their intensity properties. The object-feature-based component captures the degree of closeness of their intensity properties to some expected values of those properties for the object. A family of non-scale-based and scale-based affinity relations are constructed dictated by how we envisage the two components to characterize objects. A simple and effective method for giving a rough estimate of scale at different locations in the image is presented. The original theoretical and algorithmic framework remains more-or-less the same but considerably improved segmentations result. The method has been tested in several applications qualitatively. A quantitative statistical comparison between the non-scale-based and the scale-based methods was made based on 250 phantom images. These were generated from 10 patient MR brain studies by first segmenting the objects, then setting up appropriate intensity levels for the object and the background, and then by adding five different levels for each of noise and blurring and a fixed slow varying background component. Both the statistical and the subjective tests clearly indicate that the scale-based method is superior to the non-scale-based method in capturing details and in robustness to noise. It is also shown, based on these phantom images, that any (global) optimum threshold selection method will perform inferior to the fuzzy connectedness methods described in this paper.

Digital topological analysis of in vivo magnetic resonance microimages of trabecular bone reveals structural implications of osteoporosis.

Osteoporosis is a disease characterized by bone volume loss and architectural deterioration. The majority of work aimed at evaluating the structural implications of the disease has been performed based on stereologic analysis of histomorphometric sections. Only recently noninvasive imaging methods have emerged that provide sufficient resolution to resolve individual trabeculae. In this article, we apply digital topological analysis (DTA) to magnetic resonance microimages (μ‐MRI) of the radius obtained at 137 × 137 × 350 μm3 voxel size in a cohort of 79 women of widely varying bone mineral density (BMD) and vertebral deformity status. DTA is a new method that allows unambiguous determination of the three‐dimensional (3D) topology of each voxel in a trabecular bone network. The analysis involves generation of a bone volume fraction map, which is subjected to subvoxel processing to alleviate partial volume blurring, followed by thresholding and skeletonization. The skeletonized images contain only surfaces, profiles, curves, and their mutual junctions as the remnants of trabecular plates and rods after skeletonization. DTA parameters were compared with integral BMD in the lumbar spine and femur as well as MR‐derived bone volume fraction (BV/TV). Vertebral deformities were determined based on sagittal MRIs of the spine with a semiautomatic method and the number of deformities counted after threshold setting. DTA structural indices were found the strongest discriminators of subjects with deformities from those without deformities. Subjects with deformities (n = 29) had lower topological surface (SURF) density (p < 0.0005) and surface‐to‐curve ratio (SCR; a measure of the ratio of platelike to rodlike trabeculae; p < 0.0005) than those without. Profile interior (PI) density, a measure of intact trabecular rods, was also lower in the deformity group (p < 0.0001). These data provide the first in vivo evidence for the structural implications inherent in postmenopausal osteoporosis accompanying bone loss, that is, the conversion of trabecular plates to rods and disruption of rods due to repeated osteoclastic resorption.

Detection of 3-D simple points for topology preserving transformations with application to thinning

The problems of 3-D digital topology preservation under binary transformations and 3-D object thinning are considered in this correspondence. First, the authors establish the conditions under which transformation of an object voxel to a non-object voxel, or its inverse does not affect the image topology. An efficient algorithm to detect a simple point has been proposed on the basis of those conditions. In this connection, some other interesting properties of 3-D digital geometry are also discussed. Using these properties and the simple point detection algorithm, the authors have proposed an algorithm to generate a surface-skeleton so that the topology of the original image is preserved, the shape of the image is maintained as much as possible, and the results are less affected by noise. >

3D Digital Topology under Binary Transformation with Applications

In this paper we study 3D digital topology under the transformation of an object point to a nonobject point and vice versa. As a result of such a transformation, an object component in the 3 × 3 × 3 neighborhood of the affected point may vanish or split into two or more components or more than one object components may merge into one. Also, cavities or tunnels in the 3 × 3 × 3 neighborhood may be destroyed or created. One of the goals of this paper is to develop an efficient algorithm (topo_para) to compute the change in the numbers of object components, tunnels and cavities in the 3 × 3 × 3 neighborhood of the transformed point. Another important contribution is the classification of different types of points (e.g., arc inner point, arc edge point, surface inner point, surface edge point) and detection of different types of junction points (e.g., junction between arcs, junction between surfaces and arcs, junction between surfaces) on the surface skeleton representation of a 3D digital image. Using these junction points it is possible to segment a 3D digital surface topologically into meaningful parts. Also, we describe an efficient algorithm for computing the Euler number of a 3D digital image using the topological parameters computed bytopo_para.

Optimum image thresholding via class uncertainty and region homogeneity

Thresholding is a popular image segmentation method that converts a gray-level image into a binary image. The selection of optimum thresholds has remained a challenge over decades. Besides being a segmentation tool on its own, often it is also a step in many advanced image segmentation techniques in spaces other than the image space. We introduce a thresholding method that accounts for both intensity-based class uncertainty-a histogram-based property-and region homogeneity-an image morphology-based property. A scale-based formulation is used for region homogeneity computation. At any threshold, intensity-based class uncertainty is computed by fitting a Gaussian to the intensity distribution of each of the two regions segmented at that threshold. The theory of the optimum thresholding method is based on the postulate that objects manifest themselves with fuzzy boundaries in any digital image acquired by an imaging device. The main idea here is to select that threshold at which pixels with high class uncertainty accumulate mostly around object boundaries. To achieve this, a threshold energy criterion is formulated using class-uncertainty and region homogeneity such that, at any image location, a high energy is created when both class uncertainty and region homogeneity are high or both are low. Finally, the method selects that threshold which corresponds to the minimum overall energy. The method has been compared to a maximum segmented image information method. Superiority of the proposed method was observed both qualitatively on clinical medical images as well as quantitatively on 250 realistic phantom images generated by adding different degrees of blurring, noise, and background variation to real objects segmented from clinical images.

A new shape preserving parallel thinning algorithm for 3D digital images

This paper is concerned with a new parallel thinning algorithm for three-dimensional digital images that preserves the topology and maintains their shape. We introduce an approach of selecting shape points and outer-layer used for erosion during each iteration. The approach produces good skeleton for different types of corners. The concept of using two image versions in thinning is introduced and its necessity in parallel thinning is justified. The robustness of the algorithm under pseudo-random noise as well as rotation with respect to shape properties is studied and the results are found to be satisfactory.

Fuzzy connectedness and image segmentation

Image segmentation-the process of defining objects in images-remains the most challenging problem in image processing despite decades of research. Many general methodologies have been proposed to date to tackle this problem. An emerging framework that has shown considerable promise recently is that of fuzzy connectedness. Images are by nature fuzzy. Object regions manifest themselves in images with a heterogeneity of image intensities owing to the inherent object material heterogeneity, and artifacts such as blurring, noise and background variation introduced by the imaging device. In spite of this gradation of intensities, knowledgeable observers can perceive object regions as a gestalt. The fuzzy connectedness framework aims at capturing this notion via a fuzzy topological notion called fuzzy connectedness which defines how the image elements hang together spatially in spite of their gradation of intensities. In defining objects in a given image, the strength of connectedness between every pair of image elements is considered, which in turn is determined by considering all possible connecting paths between the pair. In spite of a high combinatorial complexity, theoretical advances in fuzzy connectedness have made it possible to delineate objects via dynamic programming at close to interactive speeds on modern PCs. This paper gives a tutorial review of the fuzzy connectedness framework delineating the various advances that have been made. These are illustrated with several medical applications in the areas of Multiple Sclerosis of the brain, magnetic resonance (MR) and computer tomographic (CT) angiography, brain tumor, mammography, upper airway disorders in children, and colonography.

Complete Volumetric Decomposition of Individual Trabecular Plates and Rods and Its Morphological Correlations With Anisotropic Elastic Moduli in Human Trabecular Bone

Trabecular plates and rods are important microarchitectural features in determining mechanical properties of trabecular bone. A complete volumetric decomposition of individual trabecular plates and rods was used to assess the orientation and morphology of 71 human trabecular bone samples. The ITS‐based morphological analyses better characterize microarchitecture and help predict anisotropic mechanical properties of trabecular bone.

Quantification of the roles of trabecular microarchitecture and trabecular type in determining the elastic modulus of human trabecular bone.

The roles of microarchitecture and types of trabeculae in determining elastic modulus of trabecular bone have been studied in μCT images of 29 trabecular bone samples by comparing their Youngs moduli calculated by finite element analysis (FEA) with different trabecular type‐specific reconstructions. The results suggest that trabecular plates play an essential role in determining elastic properties of trabecular bone.

Topological analysis of trabecular bone MR images

Recently, imaging techniques have become available which permit nondestructive analysis of the three-dimensional (3-D) architecture of trabecular bone (TB), which forms a network of interconnected plates and rods. Most osteoporotic fractures occur at locations rich in TB, which has spurred the search for architectural parameters as determinants of bone strength. Here, the authors present a new approach to quantitative characterization of the 3-D microarchitecture of TB, based on digital topology. The method classifies each voxel of the 3-D structure based on the connectivity information of neighboring voxels. Following conversion of the 3-D digital image to a skeletonized surface representation containing only one-dimensional (1-D) and two-dimensional (2-D) structures, each voxel is classified as a curve, surface, or junction. The method has been validated by means of synthesized images and has subsequently been applied to TB images from the human wrist. The topological parameters were found to predict Youngs modulus (YM) for uniaxial loading, specifically, the surface-to-curve ratio was found to be the single strongest predictor of YM (r/sup 2/=0.69). Finally, the method has been applied to TB images from a group of patients showing very large variations in topological parameters that parallel much smaller changes in bone volume fraction (BVF).