Minimum Hausdorff Distance Research Articles

LatticeOpt: An automatic tool for planning optimisation of spatially fractionated stereotactic body radiotherapy

PurposeLattice radiotherapy (LRT) is a three dimensional (3D) implementation of spatially fractionated radiation therapy, based on regular spatial distribution of high dose spheres (vertices) inside the target. Due to tumour shape heterogeneity, finding the best lattice arrangement is not trivial.The aim of this study was to develop the LatticeOpt tool to generate the best lattice structures on clinical cases for treatment planning. MethodsDeveloped in MATLAB, LatticeOpt finds the 3D-spatial configurations that maximize the number of vertices within the gross target volume (GTV). If organs at risk (OARs) are considered, it chooses the solution that minimizes the overlapping volume histograms (OVH). Otherwise, the lattice structure with the minimum Hausdorff distance between vertices and GTV boundary is chosen to avoid unpopulated regions.Different lattice structures were created for 20 patients, with (OVHopt) and without (OVHunopt) OVH minimization. Imported into TPS (Eclipse, Varian), corresponding plans were generated and evaluated in terms of OAR mean and maximum doses, GTV vertex coverage and dose gradients, as well as pre-clinical plan dosimetry. ResultsPlans based on an optimized lattice structure (OVHopt, OVHunopt) had similar dose distributions in terms of vertex coverage and gradient index score. OAR sparing was observed in all patients, with a 4 % and 9 % difference for mean and max dose (both p-values <0.01), respectively. The best vertices dimensions and their relative distances were patient dependent. ConclusionsLatticeOpt was able to reduce the time-consuming procedures of LRT, as well as to achieve standardized and reproducible results, useful for multicentre studies.

Semi-automated detection of aortic root in human heart MSCT images using nonlinear filtering and unsupervised clustering

A semiautomatic technique to detect the aortic root in three-dimensional multi-slice computerised tomography images is proposed. Three steps are considered: conditioning, filtering, and detection. The conditioning is based on multi-planar reconstruction and it is required for reformatting the information to orthogonal planes to the aortic root. During the filtering, three nonlinear filters based on similarity enhancement, median and weighted median are considered to reduce noise and enhance the reformatted images. In the detection, the filtered volumes are processed with a clustering technique. Dice score, the point-to-mesh and the Hausdorff distances are used to compare the obtained results with respect to ground truth traced by a cardiologist. A clinical dataset of 90 volumes from 45 patients is used to validate the technique. The maximum Dice score (0.92), the minimum average point-to-mesh distance (0.96 mm) and the minimum average Hausdorff distance (4.80 mm) are obtained during preprocessed volumes segmentation using similarity enhancement.

Extended ResNet and Label Feature Vector Based Chromosome Classification

Human chromosome classification is essential to the clinical diagnosis of cytogenetical diseases such as genetic disorders and cancer. This process, however, is time-consuming and requires specialist knowledge. Considerable efforts have been made to automat the process. Recently, methods based on Convolutional Neural Networks achieved state-of-the-art results on the chromosome classification task. Many studies used karyotype images in performance evaluation, few studies have reported the results of human chromosome classification on microscopical images. This paper proposes a novel method to classify single chromosome images into one of 24 types. In the proposed method an extended ResNet was first devised to extract features of single chromosome images. A label feature vector was then extracted for each of 24 chromosome types based on a validation dataset. Hausdorff distance between feature vector of an input image and each of 24 label feature vectors were calculated, and the label feature vector that has minimum hausdorff distance to the feature vector of the input image was selected as the potential label of the input image. To finally allocate the single chromosomes from a same microscopical image into one of 24 types, a Label Redistribution strategy was used to shrink the label space and to increase the efficiency of chromosome classification. Experiments were implemented with 90,624 single chromosome images, 644 of which were randomly picked to form a testing set in advance. The classification accuracy on microscopical images using our proposed method achieved an accuracy of 94.72%.

Strip Adjustment of Airborne LiDAR Data in Urban Scenes Using Planar Features by the Minimum Hausdorff Distance.

In Airborne Light Detection and Ranging (LiDAR) data acquisition practice, discrepancies exist between adjacent strips even though careful system calibrations have been performed. A strip adjustment method using planar features acquired by the Minimum Hausdorff Distance (MHD) is proposed to eliminate these discrepancies. First, semi-suppressed fuzzy C-means and restricted region growing algorithms are used to extract buildings. Second, a binary image is generated from the minimum bounding rectangle that covers overlapping regions. Then, connected components labeling algorithm is applied to process the binary image to extract individual buildings. After that, building matching is performed based on MHD. Third, a coarse-to-fine approach is used to segment building roof planes. Then, plane matching is conducted under the constraints of MHD and normal vectors similarity. The last step is the calculation of the parameters based on Euclidean distance minimization between matched planes. Two different types of datasets, one of which was acquired by a dual-channel LiDAR system Trimble AX80, were selected to verify the proposed method. Experimental results show that the corresponding planar features that meet adjustment requirements can be successfully detected without any manual operations or auxiliary data or transformation of raw data, while the discrepancies between strips can be effectively eliminated. Although adjustment results of the proposed method slightly outperform the comparison alternative, the proposed method also has the advantage of processing the adjustment in a more automatic manner than the comparison method.

基于最小Hausdorff距离和NSST的遥感图像融合

基于最小Hausdorff距离和NSST的遥感图像融合

DOCUMENT IMAGE REGISTRATION FOR IMPOSED LAYER EXTRACTION

Extraction of filled-in information from document images in the presence of template poses challenges due to geometrical distortion. Filled-in document image consists of null background, general information foreground and vital information imposed layer. Template document image consists of null background and general information foreground layer. In this paper a novel document image registration technique has been proposed to extract imposed layer from input document image. A convex polygon is constructed around the content of the input and the template image using convex hull. The vertices of the convex polygons of input and template are paired based on minimum Euclidean distance. Each vertex of the input convex polygon is subjected to transformation for the permutable combinations of rotation and scaling. Translation is handled by tight crop. For every transformation of the input vertices, Minimum Hausdorff distance (MHD) is computed. Minimum Hausdorff distance identifies the rotation and scaling values by which the input image should be transformed to align it to the template. Since transformation is an estimation process, the components in the input image do not overlay exactly on the components in the template, therefore connected component technique is applied to extract contour boxes at word level to identify partially overlapping components. Geometrical features such as density, area and degree of overlapping are extracted and compared between partially overlapping components to identify and eliminate components common to input image and template image. The residue constitutes imposed layer. Experimental results indicate the efficacy of the proposed model with computational complexity. Experiment has been conducted on variety of filled-in forms, applications and bank cheques. Data sets have been generated as test sets for comparative analysis.

Optimization of the Hausdorff distance between sets in Euclidean space

Set approximation problems play an important role in many areas of mathematics and mechanics. For example, approximation problems for solvability sets and reachable sets of control systems are intensively studied in differential game theory and optimal control theory. In N.N. Krasovskii and A.I. Subbotin’s investigations devoted to positional differential games, one of the key problems was the problem of identification of solvability sets, which are maximal stable bridges. Since this problem can be solved exactly in rare cases only, the question of the approximate calculation of solvability sets arises. In papers by A.B. Kurzhanskii and F.L. Chernous’ko and their colleagues, reachable sets were approximated by ellipsoids and parallelepipeds. In the present paper, we consider problems in which it is required to approximate a given set by arbitrary polytopes. Two sets, polytopes A and B, are given in Euclidean space. It is required to find a position of the polytopes that provides a minimum Hausdorff distance between them. Though the statement of the problem is geometric, methods of convex and nonsmooth analysis are used for its investigation. One of the approaches to dealing with planar sets in control theory is their approximation by families of disks of equal radii. A basic component of constructing such families is best n-nets and their generalizations, which were described, in particular, by A.L. Garkavi. The authors designed an algorithm for constructing best nets based on decomposing a given set into subsets and calculating their Chebyshev centers. Qualitative estimates for the deviation of sets from their best n-nets as n grows to infinity were given in the general case by A.N. Kolmogorov. We derive a numerical estimate for the Hausdorff deviation of one class of sets. Examples of constructing best n-nets are given.

SU-E-J-85: The Effect of Different Imaging Modalities On the Delineation of the True Spinal Cord for Spinal Stereotactic Body Radiation Therapy

Purpose: SBRT allows the delivery of high dose radiation treatments to localized tumors while minimizing dose to surrounding tissues. Due to the large doses delivered, accurate contouring of organs at risk is essential. In this study, differences between the true spinal cord as seen using MRI and CT myelogram (CTM) have been assessed in patients with spinal metastases treated using SBRT. Methods: Ten patients were identified that have both a CTM and a MRI. Using rigid registration tools, the MRI was fused to the CTM. The thecal sac and true cord were contoured using each imaging modality. Images were exported and analyzed for similarity by computing the Dice similarity coefficient and the modified Hausdorff distance (greatest distance from a point in one set to the closest point in the other set). Results: The Dice coefficient was calculated for the thecal sac (0.81 ±0.06) and true cord (0.63 ±0.13). These two measures are correlated; however, some points show a low true cord overlap despite a high overlap for the thecal sac. The Hausdorff distance for structure comparisons was also calculated. For thecal sac structures, the average value, 1.6mm (±1.1), indicates good overlap. For true cord comparison, the average value, 0.3mm (±0.16),more » indicates very good overlap. The minimum Hausdorff distance between the true cord and thecal sac was on average 1.6mm (±0.9) Conclusion: The true cord position as seen in MRI and CTM is fairly constant, although care should be taken as large differences can be seen in individual patients. Avoidning the true cord in spine SBRT is critical, so the ability to visualize the true cord before performing SBRT to the vertebrae is essential. Here, CT myelogram appears an excellent, robust option, that can be obtained the day of treatment planning and is unaffected by uncertainties in image fusion.« less

Computerised segmentation of suspicious lesions in the digital mammograms

In this paper, a robust marker-controlled watershed method is proposed to yield more accurate segmentation results to delineate the masses in mammograms. The proposed method consists of three main steps: pre-processing, marker extraction and the final segmentation. In the first step, pre-processing algorithm is developed using top-hat morphological filter, wavelet transform followed by a noise smoothing anisotropic diffusion filter. In the second step, a novel technique is developed for the extraction of robust markers to locate the accurate position of the suspicious lesions. Finally, the extracted markers are used within the watershed algorithm to allow the reliable segmentation and quantification of masses in mammograms. The developed computer method was quantitatively evaluated using the area overlap metric (AOM), average minimum Euclidean distance (AMED) and Hausdorff distance (HD). The mean ± standard deviation values of AOM, AMED and HD for our method are 0.83 ± 0.10, 1.49 ± 1.20 mm and 4.62 ± 0.80 mm. We compared our method with previously developed marker-controlled watershed algorithm with respect to the manual segmentation performed by an expert radiologist. Experimental results demonstrate that our method has a strong potential to be used as an aid to radiologists in the interpretation of screening mammograms.

An Efficient Trajectory Clustering Framework Based Relative Distance

along with more and more trajectory dataset being collected into application servers, the research in trajectory clustering has become increasingly important topic. This paper proposes a new mobile object trajectory Clustering algorithm (Trajectory Clustering based Improved Minimum Hausdorff Distance under Translation, TraClustMHD). In this framework, improved Minimum Hausdorff Distance under Translation is presented to measure the similarity between sub-segments. In additional, R-Tree is employed to improve the efficiency. The experimental results showed that this algorithm better than based on Hausdorff distance and based on line Hausdorff distance has good trajectory clustering performance.

The best [formula omitted] cubic and [formula omitted] quartic Bézier approximations of circular arcs

We obtain cubic and quartic Bézier approximations of circular arcs that respectively satisfy G1 and G2 end-point interpolation conditions. We identify the necessary and sufficient conditions for such approximations to be the best, in the sense that they have the minimum Hausdorff distance to the circular arc. We then establish the existence and uniqueness of these best approximations and present practical methods to calculate them, which are verified by examples.

Minimum Hausdorff distance under rigid motions and comparison of protein structures

Hausdorff distance between two compact sets, defined as the maximum distance from a point of one set to another set, has many application in computer science. It is a good measure for the similarity of two sets. This paper proves that the shape distance between two compact sets in Rn defined by minimum Hausdorff distance under rigid motions is a distance. The authors introduce similarity comparison problems in protein science, and propose that this measure may have good application to comparison of protein structure as well. For calculation of this distance, the authors give one dimensional formulas for problems (2, n), (3, 3), and (3, 4). These formulas can reduce time needed for solving these problems. The authors did some numerical experiments for (2, n). On these sets of data, this formula can reduce time needed to one fifteenth of the best algorithms known on average. As n increases, it would save more time.

Finding the best conic approximation to the convolution curve of two compatible conics based on Hausdorff distance

We consider the convolution of two compatible conic segments. First, we find an exact parametric expression for the convolution curve, which is not rational in general, and then we find the conic approximation to the convolution curve with the minimum error. The error is expressed as a Hausdorff distance which measures the square of the maximal collinear normal distance between the approximation and the exact convolution curve. For this purpose, we identify the necessary and sufficient conditions for the conic approximation to have the minimum Haudorff distance from the convolution curve. Then we use an iterative process to generate a sequence of weights for the rational quadratic Bézier curves which we use to represent conic approximations. This sequence converges to the weight of the rational quadratic Bézier curve with the minimum Hausdorff distance, within a given tolerance. We verify our method with several examples.

A new algorithm for computing the minimum Hausdorff distance between two point sets on a line under translation

To determine the similarity of two point sets is one of the major goals of pattern recognition and computer graphics. One widely studied similarity measure for point sets is the Hausdorff distance. So far, various computational methods have been proposed for computing the minimum Hausdorff distance. In this paper, we propose a new algorithm to compute the minimum Hausdorff distance between two point sets on a line under translation, which outperforms other existing algorithms in terms of efficiency despite its complexity of O ( ( m + n ) lg ( m + n ) ) , where m and n are the sizes of two point sets.

Method of defect edge reconstruction based on region growing algorithm and the Hausdorff distance

Aiming to edge deformation in extracting the de- fects of X-ray weld images, a method of defect edge recon- struction based on region growing and the Hausdorff distance is proposed. Firstly, the complement of the image is computed and the size of structure is chosen by watershed algorithm. Then, the image is filtered by improved adaptive morphology filter and its two valued image comprised of the defects is obtained. On this basis, the seeds are obtained by shrinking the objects to points. The original gray value of seeds serve as the beginning value and the object region grows by combining these selected pixels whose gray values are greater than or equal to the begin- ning value which is renewed by 2 subtracted from the previous. Then, the object region grows iteratively and the Hausdorff distance of the Canny edge image of objects and the original objects are calculated each time. The region growing stop when the minimum Hausdorff distance is achieved. When all objectsregion growing are completed, the defect edge is reconstructed by region growing. The method is verified by experiments and the results prove that the method is avail to renew the original shape of defect and obtain good results in defect edge recon- struction.

WE‐C‐330A‐08: An Efficient Morphometric Skull Atlas for Image Guided Radiotherapy

Purpose: We describe an efficient method to extract the outer skull features from CT transaxial slices for use in IGRT. Method and Materials: A Bezier curve B(t) was fit to four sections of the outer skull on CT slices at 0.3cm spacing. B(t) is a parametric equation defined by two anchor points (A1,A2) and two control points (C1,C2) such that for 0⩽t⩽1. The parameter t is sampled to create a sequence of points at 0.3cm interval. The atlas consists of all the Bezier points and can be transformed by rotation R and translation T such that . Thirty‐two points on the anterior‐posterior and right‐lateral Portal Images are digitized for analytical computation of setup error. The position and orientation (i.e. pose) of the atlas with the minimum Hausdorff Distance (HD) between the PI points and the outer projected atlas contour determines the homogenous transformation. Results: Manual draw‐by‐wire technique can take up to 15 minutes per slice. The total time is reduced by 50% for similar size patients in nearly identical positions. The overall accuracy was determined to be 0.15 cm. A semi‐automatic technique with improved accuracy of 0.1 cm was investigated using simulated annealing. An automatic algorithm using monochromatic bit maps with edge detection was also evaluated. For 30 CT slices, the complete atlas is defined by only 360 points. Setup error for a clinical case was measured and the effects of pure rotation mimicking translation quantified. Conclusion: We have demonstrated that a series of four piecewise continuous Bezier curve segments can accurately extract the outer skull feature from a CT transaxial scan. A patient specific atlas can be quickly morphed from the generic atlas. The atlas has been shown to be clinically useful for 3D analysis of setup error and theoretical quantitative analysis of pure rotation.

Geometric Pattern Matching in d -Dimensional Space

We show that, using the L ∞ metric, the minimum Hausdorff distance under translation between two point sets of cardinality n in d -dimensional space can be computed in time O(n (4d-2)/3 log 2 n) for 3 < d $\leq$ 8, and in time O(n 5d/4 log 2 n) for any d > 8 . Thus we improve the previous time bound of O(n 2d-2 log 2 n) due to Chew and Kedem. For d=3 we obtain a better result of O(n 3 log 2 n) time by exploiting the fact that the union of n axis-parallel unit cubes can be decomposed into O(n) disjoint axis-parallel boxes. We prove that the number of different translations that achieve the minimum Hausdorff distance in d -space is $\Theta(n^{\floor{3d/2}})$ . Furthermore, we present an algorithm which computes the minimum Hausdorff distance under the L 2 metric in d -space in time $O(n^{\ceil{3d/2}+1 +\delta})$ , for any δ > 0.

Getting around a lower bound for the minimum Hausdorff distance

We consider the following geometric pattern matching problem: find the minimum Hausdorff distance between two point sets under translation with L 1 or L ∞ as the underlying metric. Huttenlocher, Kedem and Sharir have shown that this minimum distance can be found by constructing the upper envelope of certain Voronoi surfaces. Further, they show that if the two sets are each of cardinality n then the complexity of the upper envelope of such surfaces is Ω(n 3) . We examine the question of whether one can get around this cubic lower bound, and show that under the L 1 and L ∞ metrics, the time to compute the minimum Hausdorff distance between two point sets is O( n 2log 2 n).

Matching Shapes with a Reference Point

For two given point sets, we present a very simple (almost trivial) algorithm to translate one set so that the Hausdorff distance between the two sets is not larger than a constant factor times the minimum Hausdorff distance which can be achieved in this way. The algorithm just matches the so-called Steiner points of the two sets. The focus of our paper is the general study of reference points (like the Steiner point) and their properties with respect to shape matching. For more general transformations than just translations, our method eliminates several degrees of freedom from the problem and thus yields good matchings with improved time bounds.

Efficient Bitmap Resemblance Under Translations

We present three efficient algorithms for resemblance between two bitmaps. Two of the algorithms are rasterized approximations, based on existing algorithms which compute the exact minimum Hausdorff distance between point sets under translation. The minimum Hausdorff distance is a min – max – min distance. We convert this operation into an and – or – and operation. This conversion, together with the fact that we encode distances into bits in the words of the pixel plane, of standard graphics hardware, contribute to the speed-up of our rasterized algorithms. The third algorithm is faster than the first two rasterized algorithms, and combines speed-up ideas from both. The performance of our rasterized algorithm is compared to an existing rasterized approximation algorithm for bitmap resemblance. We compare runtimes of these algorithms, parametrized by the size of the bitmap, the density of black bits in the bitmap and other parameters. Our results are summarized in tables and show that our algorithm is faster.