Chronic obstructive pulmonary disease (COPD) is a common lung disease, and quantitative CT-based bronchial phenotypes are of increasing interest as a means of exploring COPD sub-phenotypes, establishing disease progression, and evaluating intervention outcomes. Reliable, fully automated, and accurate segmentation of pulmonary airway trees is critical to such exploration. We present a novel approach of multi-parametric freeze-and-grow (FG) propagation which starts with a conservative segmentation parameter and captures finer details through iterative parameter relaxation. First, a CT intensity-based FG algorithm is developed and applied for airway tree segmentation. A more efficient version is produced using deep learning methods generating airway lumen likelihood maps from CT images, which are input to the FG algorithm. Both CT intensity- and deep learning-based algorithms are fully automated, and their performance, in terms of repeat scan reproducibility, accuracy, and leakages, is evaluated and compared with results from several state-of-the-art methods including an industry-standard one, where segmentation results were manually reviewed and corrected. Both new algorithms show a reproducibility of 95% or higher for total lung capacity (TLC) repeat CT scans. Experiments on TLC CT scans from different imaging sites at standard and low radiation dosages show that both new algorithms outperform the other methods in terms of leakages and branch-level accuracy. Considering the performance and execution times, the deep learning-based FG algorithm is a fully automated option for large multi-site studies.

In this paper, we present a semi-supervised deep learning approach to accurately recover high-resolution (HR) CT images from low-resolution (LR) counterparts. Specifically, with the generative adversarial network (GAN) as the building block, we enforce the cycle-consistency in terms of the Wasserstein distance to establish a nonlinear end-to-end mapping from noisy LR input images to denoised and deblurred HR outputs. We also include the joint constraints in the loss function to facilitate structural preservation. In this process, we incorporate deep convolutional neural network (CNN), residual learning, and network in network techniques for feature extraction and restoration. In contrast to the current trend of increasing network depth and complexity to boost the imaging performance, we apply a parallel 1×1 CNN to compress the output of the hidden layer and optimize the number of layers and the number of filters for each convolutional layer. The quantitative and qualitative evaluative results demonstrate that our proposed model is accurate, efficient and robust for super-resolution (SR) image restoration from noisy LR input images. In particular, we validate our composite SR networks on three large-scale CT datasets, and obtain promising results as compared to the other state-of-the-art methods.

Numerous multi-center studies related to chronic obstructive pulmonary disease use computed tomography (CT) based characterization of the lung parenchyma and bronchial tree to understand the disease's status and progression. To our knowledge, there are no fully automated methods for airway tree segmentation that don't require post-segmentation manual revision and intervention. In this paper, we present a novel CT-based airway tree segmentation algorithm using topological leakage detection and volume freezing. The method is fully automated requiring no manual inputs or post-segmentation editing. It uses intensity-based connectivity and novel approaches of leakage detection and volume freezing to iteratively grow an airway tree starting from an initial seed inside the trachea. It begins with a conservative threshold and then, iteratively shifts toward generous threshold values. The method was applied on chest CT scans of ten non-smoking healthy subjects at total lung capacity, and the results were highly promising with no visual segmentation leakages.

Skeletonization offers a compact representation of an object while preserving important topological and geometrical features. Literature on skeletonization of binary objects is quite mature. However, challenges involved with skeletonization of fuzzy objects are mostly unanswered. This paper presents a new theory and algorithm of skeletonization for fuzzy objects, evaluates its performance, and demonstrates its applications. A formulation of fuzzy grassfire propagation is introduced; its relationships with fuzzy distance functions, level sets, and geodesics are discussed; and several new theoretical results are presented in the continuous space. A notion of collision-impact of fire-fronts at skeletal points is introduced, and its role in filtering noisy skeletal points is demonstrated. A fuzzy object skeletonization algorithm is developed using new notions of surface- and curve-skeletal voxels, digital collision-impact, filtering of noisy skeletal voxels, and continuity of skeletal surfaces. A skeletal noise pruning algorithm is presented using branch-level significance. Accuracy and robustness of the new algorithm are examined on computer-generated phantoms and micro- and conventional CT imaging of trabecular bone specimens. An application of fuzzy object skeletonization to compute structure-width at a low image resolution is demonstrated, and its ability to predict bone strength is examined. Finally, the performance of the new fuzzy object skeletonization algorithm is compared with two binary object skeletonization methods.

Theoretical properties of a multi-scale opening (MSO) algorithm for two conjoined fuzzy objects are established, and its extension to separating two conjoined fuzzy objects with different intensity properties is introduced. Also, its applications to artery/vein (A/V) separation in pulmonary CT imaging and carotid vessel segmentation in CT angiograms (CTAs) of patients with intracranial aneurysms are presented. The new algorithm accounts for distinct intensity properties of individual conjoined objects by combining fuzzy distance transform (FDT), a morphologic feature, with fuzzy connectivity, a topologic feature. The algorithm iteratively opens the two conjoined objects starting at large scales and progressing toward finer scales. Results of application of the method in separating arteries and veins in a physical cast phantom of a pig lung are presented. Accuracy of the algorithm is quantitatively evaluated in terms of sensitivity and specificity on patients' CTA data sets and its performance is compared with existing methods. Reproducibility of the algorithm is examined in terms of volumetric agreement between two users' carotid vessel segmentation results. Experimental results using this algorithm on patients' CTA data demonstrate a high average accuracy of 96.3% with 95.1% sensitivity and 97.5% specificity and a high reproducibility of 94.2% average agreement between segmentation results from two mutually independent users. Approximately, twenty-five to thirty-five user-specified seeds/separators are needed for each CTA data through a custom designed graphical interface requiring an average of thirty minutes to complete carotid vascular segmentation in a patient's CTA data set.

Digital topology and geometry refers to the use of topologic and geometric properties and features for images defined in digital grids. Such methods have been widely used in many medical imaging applications, including image segmentation, visualization, manipulation, interpolation, registration, surface-tracking, object representation, correction, quantitative morphometry etc. Digital topology and geometry play important roles in medical imaging research by enriching the scope of target outcomes and by adding strong theoretical foundations with enhanced stability, fidelity, and efficiency. This paper presents a comprehensive yet compact survey on results, principles, and insights of methods related to digital topology and geometry with strong emphasis on understanding their roles in various medical imaging applications. Specifically, this paper reviews methods related to distance analysis and path propagation, connectivity, surface-tracking, image segmentation, boundary and centerline detection, topology preservation and local topological properties, skeletonization, and object representation, correction, and quantitative morphometry. A common thread among the topics reviewed in this paper is that their theory and algorithms use the principle of digital path connectivity, path propagation, and neighborhood analysis.

Traditional arc skeletonization algorithms using the principle of Blum's transform, often, produce unwanted spurious branches due to boundary irregularities and digital effects on objects and other artifacts. This paper presents a new robust approach of extracting arc skeletons for three-dimensional (3-D) elongated fuzzy objects, which avoids spurious branches without requiring post-pruning. Starting from a root voxel, the method iteratively expands the skeleton by adding a new branch in each iteration that connects the farthest voxel to the current skeleton using a minimum-cost geodesic path. The path-cost function is formulated using a novel measure of local significance factor defined by fuzzy distance transform field, which forces the path to stick to the centerline of the object. The algorithm terminates when dilated skeletal branches fill the entire object volume or the current farthest voxel fails to generate a meaningful branch. Accuracy of the algorithm has been evaluated using computer-generated blurred and noisy phantoms with known skeletons. Performance of the method in terms of false and missing skeletal branches, as defined by human expert, has been examined using in vivo CT imaging of human intrathoracic airways. Experimental results from both experiments have established the superiority of the new method as compared to a widely used conventional method in terms of accuracy of medialness as well as robustness of true and false skeletal branches.

A novel multiscale topomorphologic approach for opening of two isointensity objects fused at different locations and scales is presented and applied to separating arterial and venous trees in 3-D pulmonary multidetector X-ray computed tomography (CT) images. Initialized with seeds, the two isointensity objects (arteries and veins) grow iteratively while maintaining their spatial exclusiveness and eventually form two mutually disjoint objects at convergence. The method is intended to solve the following two fundamental challenges: how to find local size of morphological operators and how to trace continuity of locally separated regions. These challenges are met by combining fuzzy distance transform (FDT), a morphologic feature with a topologic fuzzy connectivity, and a new morphological reconstruction step to iteratively open finer and finer details starting at large scales and progressing toward smaller scales. The method employs efficient user intervention at locations where local morphological separability assumption does not hold due to imaging ambiguities or any other reason. The approach has been validated on mathematically generated tubular objects and applied to clinical pulmonary noncontrast CT data for separating arteries and veins. The tradeoff between accuracy and the required user intervention for the method has been quantitatively examined by comparing with manual outlining. The experimental study, based on a blind seed selection strategy, has demonstrated that above 95% accuracy may be achieved using 25-40 seeds for each of arteries and veins. Our method is very promising for semiautomated separation of arteries and veins in pulmonary CT images even when there is no object-specific intensity variation at conjoining locations.

Trabecular bone (TB) is a complex quasi-random network of interconnected plates and rods. TB constantly remodels to adapt to the stresses to which it is subjected (Wolff's Law). In osteoporosis, this dynamic equilibrium between bone formation and resorption is perturbed, leading to bone loss and structural deterioration. Both bone loss and structural deterioration increase fracture risk. Bone's mechanical behavior can only be partially explained by variations in bone mineral density, which led to the notion of bone structural quality. Previously, we developed digital topological analysis (DTA) which classifies plates, rods, profiles, edges, and junctions in a TB skeletal representation. Although the method has become quite popular, a major limitation of DTA is that it provides only hard classifications of different topological entities, failing to distinguish between narrow and wide plates. Here, we present a new method called volumetric topological analysis (VTA) for regional quantification of TB topology. At each TB location, the method uniquely classifies its topology on the continuum between perfect plates and perfect rods, facilitating early detections of TB alterations from plates to rods according to the known etiology of osteoporotic bone loss. Several new ideas, including manifold distance transform, manifold scale, and feature propagation have been introduced here and combined with existing DTA and distance transform methods, leading to the new VTA technology. This method has been applied to multidetector computed tomography (CT) and micro-computed tomography ( μCT) images of four cadaveric distal tibia and five distal radius specimens. Both intra- and inter-modality reproducibility of the method has been examined using repeat CT and μCT scans of distal tibia specimens. Also, the method's ability to predict experimental biomechanical properties of TB via CT imaging under in vivo conditions has been quantitatively examined and the results found are very encouraging.

In tomographic imagery, partial volume effects (PVEs) cause several artifacts in volume renditions. In X-ray computed tomography (CT), for example, soft-tissue-like pseudo structures appear in bone-to-air and bone-to-fat interfaces. Further, skin, which is identical to soft tissue in terms of CT number, obscures the rendition of the latter. The purpose of this paper is to demonstrate these phenomena and to provide effective solutions that yield significantly improved renditions. We introduce two methods that detect and classify voxels with PVE in X-ray CT. Further, a method is described to automatically peel skin so that PVE-resolved renditions of bone and soft tissue reveal considerably more detail. In the first method to address PVE, called the fraction measure (FM) method, the fraction of each tissue material in each voxel v is estimated by taking into account the intensities of the voxels neighboring v. The second method, called uncertainty principle (UP) method, is based on the following postulate (Saha and Udupa, 2001): In any acquired image, voxels with the highest uncertainty occur in the vicinity of object boundaries. The removal of skin is achieved by means of mathematical morphology. Volume renditions have been created before and after applying the methods for several patient CT datasets. A mathematical phantom experiment involving different levels of PVE has been conducted by adding different degrees of noise and blurring. A quantitative evaluation is done utilizing the mathematical phantom and clinical CT data wherein an operator carefully masked out voxels with PVE in the segmented images. All results have demonstrated the enhanced quality of display of bone and soft tissue after applying the proposed methods. The quantitative evaluations indicate that more than 98% of the voxels with PVE are removed by the two methods and the second method performs slightly better than the first. Further, skin peeling vividly reveals fine details in the soft tissue structures.

In many medical imaging applications, due to the limited field of view of imaging devices, acquired images often include only a part of a structure. In such situations, it is impossible to guarantee that the images will contain exactly the same physical extent of the structure at different scans, which leads to difficulties in registration and in many other tasks, such as the analysis of the morphology, architecture, and kinematics of the structures. To facilitate such analysis, we developed a general method, referred to as iso-shaping, that generates structures of the same shape from segmented image sequences. The basis for this method is to automatically find a set of key points, called shape centers, in the segmented partial anatomic structure such that these points are present in all images and that they represent the same physical location in the object, and then trim the structure using these points as reference. The application area considered here is the analysis of the morphology, architecture, and kinematics of the joints of the foot from magnetic resonance images acquired at different joint positions and load conditions. The accuracy of the method is analyzed by utilizing ten data sets for iso-shaping the tibia and the fibula via four evaluative experiments. The analysis indicates that iso-shaping produces results as predicted by the theoretical framework.

Trabecular or cancellous bone, the type of bone found in the vertebrae and near the joints of long bones, consists of a network of plates and struts. Accurate measurement of trabecular thickness is of significant interest, for example, to assess the effectiveness of anabolic (bone forming) agents of patients with osteoporosis. Here, we introduce a new fuzzy distance transform (FDT)-based thickness computation method that obviates binary segmentation and that can effectively deal with images acquired at a voxel size comparable to the typical trabecular bone thickness. The method's robustness is shown on the basis of micro-CT images of human trabecular bone, resampled at progressively coarser resolution and after application of rotation and addition of noise as a means to simulate the in vivo situation. Reproducibility of the method is demonstrated with micro-CT images by comparing histograms of thickness within and between data sets and with micro-MRI volume data sets of human volunteers imaged repeatedly. Finally, with in vivo micro-MR images from a prior study in rabbits subjected to corticosteroid exposure, it is demonstrated that short-term treatment resulting in trabecular thinning can be quantified with the new method.

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.

The mechanical competence of trabecular bone, the type of bone constituting the vertebrae and ends of the long bones, is largely determined by the bone's volume fraction and architectural make-up. Experimental and clinical evidence suggests that up to 50% of trabecular bone strength is determined by parameters characterizing the network's architecture. Although rarely clinically indicated because of its invasiveness, a bone biopsy can provide detailed insight into the bone's structural arrangement. Recent advances in magnetic resonance micro-imaging (/spl mu/MRI), in conjunction with new image processing and feature extraction approaches, now allow detailed structural information to be obtained by what we refer to as "virtual bone biopsy" (VBB). Governed by signal-to-noise, the voxel size achievable in vivo is of the same order as trabecular thickness (100-150 /spl mu/m). Therefore, methods had to be conceived to deal with the inherently fuzzy nature of the objects to be extracted. It is shown that a cascade of image processing steps, starting with noise deconvolution of the images to yield bone volume fraction maps, followed by what we refer to as "subvoxel processing" for resolution enhancement, are essential for evaluating topology and scale of the trabecular network. Digital topological analysis of the binarized and skeletonized images can provide a detailed picture of network connectivity and the nature of the structural elements (plate and strut architecture). In addition to its complicated topological make-up, trabecular bone is highly anisotropic since bone adapts to the stresses to which it is subjected (Wolff's law). Therefore, evaluation of structural orientation or fabric is of interest as well. Several methods are discussed to quantify structural anisotropy, including digital topological analysis. Although connectivity is an important determinant of the bone's mechanical behavior, the thickness of the structural elements is equally relevant. Here again, the limited spatial resolution achievable in vivo precludes the use of the classical histomorphometric approaches for thickness measurement. A method conceived recently in the authors' laboratory is the fuzzy distance transform, which obviates the need for binarization and which is shown to provide accurate measurements of structural thickness. Excerpts are provided from applications of the VBB to the study of postmenopausal osteoporosis, male hypogonadism, and secondary hyperparathyroidism. Finally, the serial reproducibility achievable suggests the VBB to be suited for assessing treatment efficacy longitudinally.

We present a new class of approaches for rigid-body registration and their evaluation in studying multiple sclerosis (MS) via multiprotocol magnetic resonance imaging (MRI). Three pairs of rigid-body registration algorithms were implemented, using cross-correlation and mutual information (MI), operating on original gray-level images, and utilizing the intermediate images resulting from our new scale-based method. In the scale image, every voxel has the local "scale" value assigned to it, defined as the radius of the largest ball centered at the voxel with homogeneous intensities. Three-dimensional image data of the head were acquired from ten MS patients for each of six MRI protocols. Images in some of the protocols were acquired in registration. The registered pairs were used as ground truth. Accuracy and consistency of the six registration methods were measured within and between protocols for known amounts of misregistrations. Our analysis indicates that there is no "best" method. For medium misregistration, the method using MI, for small add large misregistration the method using normalized cross-correlation performs best. For high-resolution data the correlation method and for low-resolution data the MI method, both using the original gray-level images, are the most consistent. We have previously demonstrated the use of local scale information in fuzzy connectedness segmentation and image filtering. Scale may also have potential for image registration as suggested by this work.

The notion of fuzzy connectedness captures the idea of "hanging-togetherness" of image elements in an object by assigning a strength of connectedness to every possible path between every possible pair of image elements. This concept leads to powerful image segmentation algorithms based on dynamic programming whose effectiveness has been demonstrated on 1,000s of images in a variety of applications. In the previous framework, a fuzzy connected object is defined with a threshold on the strength of connectedness. We introduce the notion of relative connectedness that overcomes the need for a threshold and that leads to more effective segmentations. The central idea is that an object gets defined in an image because of the presence of other co-objects. Each object is initialized by a seed element. An image element c is considered to belong to that object with respect to whose reference image element c has the highest strength of connectedness. In this fashion, objects compete among each other utilizing fuzzy connectedness to grab membership of image elements. We present a theoretical and algorithmic framework for defining objects via relative connectedness and demonstrate utilizing the theory that the objects defined are independent of reference elements chosen as long as they are not in the fuzzy boundary between objects. An iterative strategy is also introduced wherein the strongest relative connected core parts are first defined and iteratively relaxed to conservatively capture the more fuzzy parts subsequently. Examples from medical imaging are presented to illustrate visually the effectiveness of relative fuzzy connectedness. A quantitative mathematical phantom study involving 160 images is conducted to demonstrate objectively the effectiveness of relative fuzzy connectedness.DisclaimerA claim of priority in research and publication appeared on page 1486 in the paper "Relative Fuzzy Connectedness and Object Definition: Theory, Algorithms, and Applications in Image Segmentation" by J.K. Udupa, P.K. Saha, and' R.A. Lotufo (IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 24, no. 11, pp. 1485-1500,Nov. 2002) with respect to the paper "Multiseeded Segmentation Using Fuzzy Connectedness by G.T. Herman and B.M. Carvalho" (IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 23, no. 5, pp. 460-474, May 2001). Furthermore, the wording of this claim suggests professional misconduct on the part of G.T. Herman and B.M. Carvalho. Responsibility for the content of published papers rests with the authors. The peer review process is intended to determine the overall significance of the technical contribution of a manuscript. The peer review process does not provide a way to validate every statement in a manuscript. In particular, the IEEE has not validated the claim referred to in the first paragraph above. The IEEE regrets publishing this unauthenticated statement and the pain that such publication may have caused to G.T. Herman and B.M. Carvalho.

mm , and for the clear layer, and mm , mm , and for highly scattering layers. The dotted line in the figure shows the backward scattering pulse profile of the homogeneous slab without the clear layer at the 20 mm away from laser illumination points. The profile changes by adding the clear layer especially in the trailing edge of the pulse retaining the peak amplitude. When absorption coefficient in the third layer was increased from mm to mm , the decay of the reflectance is also increased as shown in the figure. This means that the absorption changes inside the clear layer are estimated by the time profile changes of the reflectance. The sensitivity for the absorption changes becomes maximum when source detector separation is 20 mm. The mean path lengths are 200, 660, 825, and 870 ps for the source detector spacing 10, 20, 30, and 37.5 mm, respectively. The mean path length saturate for the spacing larger than 25 mm, which are coincident with results reported by Monte Carlo and FEM simulation [9]. V. C ONCLUSION FDTD method has been successfully formulated for analysis of pulse responses in biological tissues. Boundary conditions have been defined using fluence rate at the scattering and no scattering material interfaces. The conditions to give stabilities for numerical solutions have been become clear in terms of scattering coefficients and mean cosine of scattering angles. Using the formulation, the reflectance of three-layered slabs containing a clear layer has been calculated. As a result, it has been become clear that absorption coefficient changes of the highly scattering media beyond the clear layer are estimated from the time profiles of the reflectance. The analysis is easily extended to calculate pulse responses in tissues with 3-D inhomogeneous optical parameters, which is effective for image reconstruction in practical diffused optical tomography.

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.

This paper presents a near-automatic process for separating vessels from background and other clutter as well as for separating arteries and veins in contrast-enhanced magnetic resonance angiographic (CE-MRA) image data, and an optimal method for three-dimensional visualization of vascular structures. The separation process utilizes fuzzy connected object delineation principles and algorithms. The first step of this separation process is the segmentation of the entire vessel structure from the background and other clutter via absolute fuzzy connectedness. The second step is to separate artery from vein within this entire vessel structure via iterative relative fuzzy connectedness. After seed voxels are specified inside artery and vein in the CE-MRA image, the small regions of the bigger aspects of artery and vein are separated in the initial iterations, and further detailed aspects of artery and vein are included in later iterations. At each iteration, the artery and vein compete among themselves to grab membership of each voxel in the vessel structure based on the relative strength of connectedness of the voxel in the artery and vein. This approach has been implemented in a software package for routine use in a clinical setting and tested on 133 CE-MRA studies of the pelvic region and two studies of the carotid system from six different hospitals. In all studies, unified parameter settings produced correct artery-vein separation. When compared with manual segmentation/separation, our algorithms were able to separate higher order branches, and therefore produced vastly more details in the segmented vascular structure. The total operator and computer time taken per study is on the average about 4.5 min. To date, this technique seems to be the only image processing approach that can be routinely applied for artery and vein separation.

Image acquisition techniques often suffer from low signal-to-noise ratio (SNR) and/or contrast-to-noise ratio (CNR). Although many acquisition techniques are available to minimize these, post acquisition filtering is a major off-line image processing technique commonly used to improve the SNR and CNR. A major drawback of filtering is that it often diffuses/blurs important structures along with noise. In this paper, we introduce two scale-based filtering methods that use local structure size or "object scale" information to arrest smoothing around fine structures and across even low-gradient boundaries. The first of these methods uses a weighted average over a scale-dependent neighborhood while the other employs scale-dependent diffusion conductance to perform filtering. Both methods adaptively modify the degree of filtering at any image location depending on local object scale. Object scale allows us to accurately use a restricted homogeneity parameter for filtering in regions with fine details and in the vicinity of boundaries while a generous parameter in the interiors of homogeneous regions. Qualitative experiments based on both phantoms and patient magnetic resonance images show significant improvements using the scale-based methods over the extant anisotropic diffusive filtering method in preserving fine details and sharpness of object boundaries. Quantitative analyses utilizing 25 phantom images generated under a range of conditions of blurring, noise, and background variation confirm the superiority of the new scale-based approaches.

Studies reported in the literature indicate that breast cancer risk is associated with mammographic densities. An objective, repeatable, and a quantitative measure of risk derived from mammographic densities will be of considerable use in recommending alternative screening paradigms and/or preventive measures. However, image processing efforts toward this goal seem to be sparse in the literature, and automatic and efficient methods do not seem to exist. In this paper, we describe and validate an automatic and reproducible method to segment dense tissue regions from fat within breasts from digitized mammograms using scale-based fuzzy connectivity methods. Different measures for characterizing mammographic density are computed from the segmented regions and their robustness in terms of their linear correlation across two different projections--cranio-caudal and medio-lateral-oblique--are studied. The accuracy of the method is studied by computing the area of mismatch of segmented dense regions using the proposed method and using manual outlining. A comparison between the mammographic density parameter taking into account the original intensities and that just considering the segmented area indicates that the former may have some advantages over the latter.

Classical digital geometry deals with sets of cubical voxels (or square pixels) that can share faces, edges, or vertices, but basic parts of digital geometry can be generalized to sets S of convex voxels (or pixels) that can have arbitrary intersections. In particular, it can be shown that if each voxel P of S has only finitely many neighbors (voxels of S that intersect P), and if any nonempty intersection of neighbors of P intersects P, then the neighborhood N(P) of every voxel P is simply connected and without cavities, and if the topology of N(P) does not change when P is deleted (i.e., P is a “simple” voxel), then deletion of P does not change the topology of S.

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.

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. In this paper, we 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 Young's modulus (YM) for uniaxial loading, specifically, the surface-to-curve ratio was found to be the single strongest predictor of YM (r2 = 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).

A set P of nondegenerate convex polygons P in R 2 , or polyhedra P in R 3 , will be called normal if the intersection of any two of the P s of P is a face (in the case of polyhedra), an edge, a vertex or empty. P is called strongly normal (SN) if it is normal and, for all P , P 1 , …, P n , if each P i intersects P and I = P 1 ∩⋯∩ P n is nonempty, then I intersects P . The union of the P i ∈ P that intersect P ∈ P is called the neighborhood of P in P , and is denoted by N P ( P ). We prove that P is SN iff for any P ′ ⊆ P and P ∈ P ′, N P ′ ( P ) is simply connected. Thus SN characterizes sets P of polyhedra (or polygons) in which the neighborhood of any polyhedron, relative to any subset P ′ of P , is simply connected. Tessellations of R 2 or R 3 into convex polygons or polyhedra are normal, but they may not be SN; for example, the square and hexagonal regular tessellations of R 2 are SN, but the triangular regular tessellation is not.

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.>