Krzysztof Ciesielski

Iterative relative fuzzy connectedness for multiple objects with multiple seeds

In this paper we present a new theory and an algorithm for image segmentation based on a strength of connectedness between every pair of image elements. The object definition used in the segmentation algorithm utilizes the notion of iterative relative fuzzy connectedness, IRFC. In previously published research, the IRFC theory was developed only for the case when the segmentation was involved with just two segments, an object and a background, and each of the segments was indicated by a single seed. (See Udupa, Saha, Lotufo [15] and Saha, Udupa [14].) Our theory, which solves a problem of Udupa and Saha from [13], allows simultaneous segmentation involving an arbitrary number of objects. Moreover, each segment can be indicated by more than one seed, which is often more natural and easier than a single seed object identification.The first iteration step of the IRFC algorithm gives a segmentation known as relative fuzzy connectedness, RFC, segmentation. Thus, the IRFC technique is an extension of the RFC method. Although the RFC theory, due to Saha and Udupa [19], is developed in the multi object/multi seed framework, the theoretical results presented here are considerably more delicate in nature and do not use the results from [19]. On the other hand, the theoretical results from [19] are immediate consequences of the results presented here. Moreover, the new framework not only subsumes previous fuzzy connectedness descriptions but also sheds new light on them. Thus, there are fundamental theoretical advances made in this paper.We present examples of segmentations obtained via our IRFC based algorithm in the multi object/multi seed environment, and compare it with the results obtained with the RFC based algorithm. Our results indicate that, in many situations, IRFC outperforms RFC, but there also exist instances where the gain in performance is negligible.

Fuzzy Connectedness Image Segmentation in Graph Cut Formulation: A Linear-Time Algorithm and a Comparative Analysis

A deep theoretical analysis of the graph cut image segmentation framework presented in this paper simultaneously translates into important contributions in several directions.The most important practical contribution of this work is a full theoretical description, and implementation, of a novel powerful segmentation algorithm, GCmax. The output of GCmax coincides with a version of a segmentation algorithm known as Iterative Relative Fuzzy Connectedness, IRFC. However, GCmax is considerably faster than the classic IRFC algorithm, which we prove theoretically and show experimentally. Specifically, we prove that, in the worst case scenario, the GCmax algorithm runs in linear time with respect to the variable M=|C|+|Z|, where |C| is the image scene size and |Z| is the size of the allowable range, Z, of the associated weight/affinity function. For most implementations, Z is identical to the set of allowable image intensity values, and its size can be treated as small with respect to |C|, meaning that O(M)=O(|C|). In such a situation, GCmax runs in linear time with respect to the image size |C|.We show that the output of GCmax constitutes a solution of a graph cut energy minimization problem, in which the energy is defined as the ℓ∞ norm ∥FP∥∞ of the map FP that associates, with every element e from the boundary of an object P, its weight w(e). This formulation brings IRFC algorithms to the realm of the graph cut energy minimizers, with energy functions ∥FP∥q for q∈[1,∞]. Of these, the best known minimization problem is for the energy ∥FP∥1, which is solved by the classic min-cut/max-flow algorithm, referred to often as the Graph Cut algorithm.We notice that a minimization problem for ∥FP∥q, q∈[1,∞), is identical to that for ∥FP∥1, when the original weight function w is replaced by wq. Thus, any algorithm GCsum solving the ∥FP∥1 minimization problem, solves also one for ∥FP∥q with q∈[1,∞), so just two algorithms, GCsum and GCmax, are enough to solve all ∥FP∥q-minimization problems. We also show that, for any fixed weight assignment, the solutions of the ∥FP∥q-minimization problems converge to a solution of the ∥FP∥∞-minimization problem (∥FP∥∞=limq→∞∥FP∥q is not enough to deduce that).An experimental comparison of the performance of GCmax and GCsum algorithms is included. This concentrates on comparing the actual (as opposed to provable worst scenario) algorithms’ running time, as well as the influence of the choice of the seeds on the output.

ℐ-density continuous functions

The I-density topology is a generalization of the ordinary density topology to the setting of category instead of measure. This work involves functions which are continuous when combinations of the I-density, deep-I-density, density and ordinary topology are used on the domain and range. In the process of examining these functions, the I-density and deep-I-density topologies are deeply explored and the properties of these function classes as semigroups are considered.

Affinity functions in fuzzy connectedness based image segmentation II: Defining and recognizing truly novel affinities

Affinity functions - the measure of how strongly pairs of adjacent spels in the image hang together - represent the core aspect (main variability parameter) of the fuzzy connectedness (FC) algorithms, an important class of image segmentation schemas. In this paper, we present the first ever theoretical analysis of the two standard affinities, homogeneity and object-feature, the way they can be combined, and which combined versions are truly distinct from each other. The analysis is based on the notion of equivalent affinities, the theory of which comes from a companion Part I of this paper (Ciesielski and Udupa, in this issue) [11]. We demonstrate that the homogeneity based and object feature based affinities are equivalent, respectively, to the difference quotient of the intensity function and Rosenfelds degree of connectivity. We also show that many parameters used in the definitions of these two affinities are redundant in the sense that changing their values lead to equivalent affinities. We finish with an analysis of possible ways of combining different component affinities that result in non-equivalent affinities. In particular, we investigate which of these methods, when applied to homogeneity based and object-feature based components lead to truly novel (non-equivalent) affinities, and how this is affected by different choices of parameters. Since the main goal of the paper is to identify, by formal mathematical arguments, the affinity functions that are equivalent, extensive experimental confirmations are not needed - they show completely identical FC segmentations - and as such, only relevant examples of the theoretical results are provided. Instead, we focus mainly on theoretical results within a perspective of the fuzzy connectedness segmentation literature.

Body-Wide Hierarchical Fuzzy Modeling, Recognition, and Delineation of Anatomy in Medical Images

To make Quantitative Radiology (QR) a reality in radiological practice, computerized body-wide Automatic Anatomy Recognition (AAR) becomes essential. With the goal of building a general AAR system that is not tied to any specific organ system, body region, or image modality, this paper presents an AAR methodology for localizing and delineating all major organs in different body regions based on fuzzy modeling ideas and a tight integration of fuzzy models with an Iterative Relative Fuzzy Connectedness (IRFC) delineation algorithm. The methodology consists of five main steps: (a) gathering image data for both building models and testing the AAR algorithms from patient image sets existing in our health system; (b) formulating precise definitions of each body region and organ and delineating them following these definitions; (c) building hierarchical fuzzy anatomy models of organs for each body region; (d) recognizing and locating organs in given images by employing the hierarchical models; and (e) delineating the organs following the hierarchy. In Step (c), we explicitly encode object size and positional relationships into the hierarchy and subsequently exploit this information in object recognition in Step (d) and delineation in Step (e). Modality-independent and dependent aspects are carefully separated in model encoding. At the model building stage, a learning process is carried out for rehearsing an optimal threshold-based object recognition method. The recognition process in Step (d) starts from large, well-defined objects and proceeds down the hierarchy in a global to local manner. A fuzzy model-based version of the IRFC algorithm is created by naturally integrating the fuzzy model constraints into the delineation algorithm. The AAR system is tested on three body regions - thorax (on CT), abdomen (on CT and MRI), and neck (on MRI and CT) - involving a total of over 35 organs and 130 data sets (the total used for model building and testing). The training and testing data sets are divided into equal size in all cases except for the neck. Overall the AAR method achieves a mean accuracy of about 2 voxels in localizing non-sparse blob-like objects and most sparse tubular objects. The delineation accuracy in terms of mean false positive and negative volume fractions is 2% and 8%, respectively, for non-sparse objects, and 5% and 15%, respectively, for sparse objects. The two object groups achieve mean boundary distance relative to ground truth of 0.9 and 1.5 voxels, respectively. Some sparse objects - venous system (in the thorax on CT), inferior vena cava (in the abdomen on CT), and mandible and naso-pharynx (in neck on MRI, but not on CT) - pose challenges at all levels, leading to poor recognition and/or delineation results. The AAR method fares quite favorably when compared with methods from the recent literature for liver, kidneys, and spleen on CT images. We conclude that separation of modality-independent from dependent aspects, organization of objects in a hierarchy, encoding of object relationship information explicitly into the hierarchy, optimal threshold-based recognition learning, and fuzzy model-based IRFC are effective concepts which allowed us to demonstrate the feasibility of a general AAR system that works in different body regions on a variety of organs and on different modalities.

Set Theoretic Real Analysis

Abstract This article is a survey of the recent results that concern real functions (from into ℝ) and whose solutions or statements involve the use of set theory. The choice of the topics follows the authors personal interest in the subject, and there are probably some important results in this area that did not make it to this survey. Most of the results presented here are left without proofs.

Affinity functions in fuzzy connectedness based image segmentation I: Equivalence of affinities

Fuzzy connectedness (FC) constitutes an important class of image segmentation schemas. Although affinity functions represent the core aspect (main variability parameter) of FC algorithms, they have not been studied systematically in the literature. In this paper, we began filling this gap by introducing and studying the notion of equivalent affinities: if any two equivalent affinities are used in the same FC schema to produce two versions of the algorithm, then these algorithms are equivalent in the sense that they lead to identical segmentations. We give a complete and elegant characterization of the affinity equivalence. We also demonstrate that any segmentation obtained via a relative fuzzy connectedness (RFC) algorithm can be viewed as segmentation obtained via absolute fuzzy connectedness (AFC) algorithm with an automatic and adaptive threshold detection. Since the main goal of the paper is to identify, by formal mathematical arguments, the affinity functions that are equivalent, extensive experimental confirmations are not needed - they show completely identical segmentations - and as such, only relevant examples of the theoretical results are provided.

Darboux-like functions within the classes of Baire one, Baire two, and additive functions

Abstract In the paper we present an exhaustive discussion of the relations between Darboux-like functions within the classes of Baire one, Baire two, Borel, and additive functions from R n into R . In particular we construct an additive extendable discontinuous function f : R → R , answering a question of Gibson and Natkaniec (1996–97, p. 499), and show that there is no similar function from R 2 into R . We also describe a Baire class two almost continuous function f : R → R which is not extendable. This gives a negative answer to a problem of Brown, Humke, and Laczkovich (1988, Problem 1). (See also Problem 3.21 of Gibson and Natkaniec (1996–97).)

Efficient algorithm for finding the exact minimum barrier distance

Abstract The minimum barrier distance, MBD, introduced recently in [1] , is a pseudo-metric defined on a compact subset D of the Euclidean space R n and whose values depend on a fixed map (an image) f from D into R . The MBD is defined as the minimal value of the barrier strength of a path between the points, which constitutes the length of the smallest interval containing all values of f along the path. In this paper we present a polynomial time algorithm, that provably calculates the exact values of MBD for the digital images. We compare this new algorithm, theoretically and experimentally, with the algorithm presented in [1] , which computes the approximate values of the MBD. Moreover, we notice that every generalized distance function can be naturally translated to an image segmentation algorithm. The algorithms that fall under such category include: Relative Fuzzy Connectedness, and those associated with the minimum barrier, fuzzy distance, and geodesic distance functions. In particular, we compare experimentally these four algorithms on the 2D and 3D natural and medical images with known ground truth and at varying level of noise, blur, and inhomogeneity.

Fuzzy object modeling

To make Quantitative Radiology (QR) a reality in routine clinical practice, computerized automatic anatomy recognition (AAR) becomes essential. As part of this larger goal, we present in this paper a novel fuzzy strategy for building bodywide group-wise anatomic models. They have the potential to handle uncertainties and variability in anatomy naturally and to be integrated with the fuzzy connectedness framework for image segmentation. Our approach is to build a family of models, called the Virtual Quantitative Human, representing normal adult subjects at a chosen resolution of the population variables (gender, age). Models are represented hierarchically, the descendents representing organs contained in parent organs. Based on an index of fuzziness of the models, 32 thorax data sets, and 10 organs defined in them, we found that the hierarchical approach to modeling can effectively handle the non-linear relationships in position, scale, and orientation that exist among organs in different patients.