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Weighted Laplacians of Grids and Their Application for Inspection of Spectral Graph Clustering Methods

Kłopotek Mieczysław, Wierzchoń Sławomir, Kłopotek Robert

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2021

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Vol. 25, No 3

329--353

EN

This paper investigates the relationship between various types of spectral clustering methods and their kinship to relaxed versions of graph cut methods. This predominantly analytical study exploits the closed (or nearly closed) form of eigenvalues and eigenvectors of unnormalized (combinatorial), normalized, and random walk Laplacians of multidimensional weighted and unweighted grids. We demonstrate that spectral methods can be compared to (normalized) graph cut clustering only if the cut is performed to minimize the sum of the weight square roots (and not the sum of weights) of the removed edges. We demonstrate also that the spectrogram of the regular grid graph can be derived from the composition of spectrograms of path graphs into which such a graph can be decomposed, only for combinatorial Laplacians. It is impossible to do so both for normalized and random-walk Laplacians. We investigate the in-the-limit behavior of combinatorial and normalized Laplacians demonstrating that the eigenvalues of both Laplacians converge to one another with an increase in the number of nodes while their eigenvectors do not. Lastly, we show that the distribution of eigenvalues is not uniform in the limit, violating a fundamental assumption of the compact spectral clustering method.

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On cerebrospinal fluid segmentation from CT brain scans using interactive graph cuts

Węgliński T., Fabijańska A.

Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska

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2012

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nr 4b

7-9

EN

Inaccuracy of the manual assessment of brain diseases forces medicine to look for a new solutions. The key factor in the diagnosis of many brain lesions is an accumulation, volume and pressure of the cerebrospinal fluid (CSF) in ventricles and cavities of the brain. In this paper, the problem of segmentation of the CSF is regarded. Specifically, the min-cut/max-flow algorithm is investigated and applied to several CT scans. The results reveals that this approach may provide a basis for further quantitative analysis of brain lesions.

PL

Niedoskonałość manualnych metod diagnostycznych w ocenie zmian chorobowych w obszarze mózgu sprawia, że współczesna medycyna poszukuje nowych rozwiązań. Jednym z kluczowych wskaźników postępu choroby jest nagromadzenie, objętość i ciśnienie płynu mózgowo-rdzeniowego (PMR). Artykuł rozważa problem segmentacji PMR z obrazów tomograficznych. Prezentowane podejście bazuje na interaktywnym algorytmie segmentacji opartym na grafach, którego skuteczność daje podstawy do późniejszej, wiarygodnej analizy ilościowej danego schorzenia.

3

An Improved Interactive Color Image Segmentation Using Region-Based Graph Cuts

Yuan T., Cheng W., Tao G., Lijun L.

Machine Graphics and Vision

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2009

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Vol. 18, No. 4

477-488

EN

The problem of efficient interactive extraction of a foreground object in a complex environment is the primary one in image processing and computer vision. The segmentation method based on graph cuts has been studied over the recent years. There are two main drawbacks of these studies: decrease in performance when the foreground and the background have similar colors, and long computing time when the image is large. In this paper, we present a new foreground objects extraction method using a region-based graph cuts algorithm. The image is pre-segmented into a large number of small partitions using the mean shift (MS) method. We use the partitions to represent the nodes in the graph instead of pixels. This approach can reduce the optimization time, which is closely related to the number of nodes and edges in the graph. Compared with the pixel-based method, our method can yield an excellent performance and exhibit a faster speed.