Andrzej Szymczak

Standardized evaluation methodology and reference database for evaluating coronary artery centerline extraction algorithms.

Efficiently obtaining a reliable coronary artery centerline from computed tomography angiography data is relevant in clinical practice. Whereas numerous methods have been presented for this purpose, up to now no standardized evaluation methodology has been published to reliably evaluate and compare the performance of the existing or newly developed coronary artery centerline extraction algorithms. This paper describes a standardized evaluation methodology and reference database for the quantitative evaluation of coronary artery centerline extraction algorithms. The contribution of this work is fourfold: (1) a method is described to create a consensus centerline with multiple observers, (2) well-defined measures are presented for the evaluation of coronary artery centerline extraction algorithms, (3) a database containing 32 cardiac CTA datasets with corresponding reference standard is described and made available, and (4) 13 coronary artery centerline extraction algorithms, implemented by different research groups, are quantitatively evaluated and compared. The presented evaluation framework is made available to the medical imaging community for benchmarking existing or newly developed coronary centerline extraction algorithms.

WRAP&Zip decompression of the connectivity of triangle meshes compressed with edgebreaker

Abstract The Edgebreaker compression (Rossignac, 1999; King and Rossignac, 1999) is guaranteed to encode any unlabeled triangulated planar graph of t triangles with at most 1.84t bits. It stores the graph as a CLERS string—a sequence of t symbols from the set { C,L,E,R,S} , each represented by a 1, 2 or 3 bit code. We show here that, in practice, the string can be further compressed to between 0.91t and 1.26t bits using an entropy code. These results improve over the 2.3t bits code proposed by Keeler and Westbrook (1995) and over the various 3D triangle mesh compression techniques published recently (Gumhold and Strasser, 1998; Itai and Rodeh, 1982; Naor, 1990; Touma and Gotsman, 1988; Turan, 1984), which exhibit either larger constants or cannot guarantee a linear worst case storage complexity. The decompression proposed by Rossignac (1999) is complicated and exhibits a non-linear time complexity. The main contribution reported here is a simpler and efficient decompression algorithm, called WrapZ King and Rossignac, 1999) yields a simple and effective solution for compressing the connectivity information of large and small triangle meshes that must be downloaded over the Internet. The CLERS stream may be interleaved with an encoding of the vertex coordinates and photometric attributes enabling inline decompression. The availability of local incidence information permits to use, during decompression, the location and attributes of neighboring vertices to predict new ones, and thus supports most of the recently proposed vertex compression techniques (Deering, 1995; Gumhold and Strasser, 1999; Taubin and Rossignac, 1998; Touma and Gotsman, 1998).

An edgebreaker-based efficient compression scheme for regular meshes

Abstract One of the most natural measures of regularity of a triangular mesh homeomorphic to the two-dimensional sphere is the fraction of its vertices having degree 6. We construct a linear-time connectivity compression scheme build upon Edgebreaker which explicitly takes advantage of regularity and prove rigorously that, for sufficiently large and regular meshes, it produces encodings not longer than 0.811 bits per triangle: 50% below the information-theoretic lower bound for the class of all meshes. Our method uses predictive techniques enabled by the Spirale Reversi decoding algorithm.

Out-of-core compression and decompression of large n-dimensional scalar fields

We present a simple method for compressing very large and regularly sampled scalar fields. Our method is particularlyattractive when the entire data set does not fit in memory and when the sampling rate is high relative to thefeature size of the scalar field in all dimensions. Although we report results for and data sets, the proposedapproach may be applied to higher dimensions. The method is based on the new Lorenzo predictor, introducedhere, which estimates the value of the scalar field at each sample from the values at processed neighbors. The predictedvalues are exact when the n‐dimensional scalar field is an implicit polynomial of degree n− 1. Surprisingly,when the residuals (differences between the actual and predicted values) are encoded using arithmetic coding,the proposed method often outperforms wavelet compression in anL∞sense. The proposed approach may beused both for lossy and lossless compression and is well suited for out‐of‐core compression and decompression,because a trivial implementation, which sweeps through the data set reading it once, requires maintaining only asmall buffer in core memory, whose size barely exceeds a single (n−1)‐dimensional slice of the data.

Piecewise regular meshes: construction and compression

We present an algorithm which splits a 3D surface into reliefs, relatively fiat regions that have smooth boundaries. The surface is then resampled in a regular manner within each of the reliefs. As a result, we obtain a piecewise regular mesh (PRM) having a regular structure on large regions. Experimental results show that we are able to approximate the input surface with the mean square error of about 0.01- 0.02% of the diameter of the bounding box without increasing the number of vertices. We introduce a compression scheme tailored to work with our remeshed models and show that it is able to compress them losslessly (after quantizing the vertex locations) without significantly increasing the approximation error using about 4 bits per vertex of the resampled model.

Coronary vessel trees from 3D imagery: a topological approach.

We propose a simple method for reconstructing vascular trees from 3D images. Our algorithm extracts persistent maxima of the intensity on all axis-aligned 2D slices of the input image. The maxima concentrate along 1D intensity ridges, in particular along blood vessels. We build a forest connecting the persistent maxima with short edges. The forest tends to approximate the blood vessels present in the image, but also contains numerous spurious features and often fails to connect segments belonging to one vessel in low contrast areas. We improve the forest by applying simple geometric filters that trim short branches, fill gaps in blood vessels and remove spurious branches from the vascular tree to be extracted. Experiments show that our technique can be applied to extract coronary trees from heart CT scans.

Grow & fold: compression of tetrahedral meshes

Standard representa&,ions of irregular finite element meshes combine vertex data (sample coordina&es and node values) and connectivity (tetrahedron-vertex incidence). Connectivity specifies’ how the samples should be interpolated. It may be encoded as four vertexreferences for each tetrahedron, which requires 128m bits where m is the loumber of tetrahedra in the mesh. Our ‘Grow&Fold’ format reduces the connectivity storage down to 7 bits per tetrahedron: 3 of these are used to encode the presence of children in a tetrahedron spanning tree; the other 4 constrain sequences of ‘folding’ operations, so that they produce the connectivity graph of the original mesh. Additional bits must be used for each handle in the mesh and for each topological ‘lock’ in the tree. However, as our experiments with a prototype implementation show, the increase of the storage cost due to this extra information is typically no more than l-2%. By storing vertex data in an order defined by the tree, we avoid the need to store tetrahedron-vertex references and facilitate variable length coding techniques for the vertex data. We provide the details of simple, loss-less compression and decompression algorithms.

Edgebreaker: a simple compression for surfaces with handles

The Edgebreaker is an efficient scheme for compressing triangulated surfaces. A surprisingly simple implementation of Edgebreaker has been proposed for surfaces homeomorphic to a sphere. It uses the Corner-Table data structure, which represents the connectivity of a triangulated surface by two tables of integers, and encodes them with less than 2 bits per triangle. We extend this simple formulation to deal with triangulated surfaces with handles and present the detailed pseudocode for the encoding and decoding algorithms (which take one page each). We justify the validity of the proposed approach using the mathematical formulation of the Handlebody theory for surfaces, which explains the topological changes that occur when two boundary edges of a portion of a surface are identified.

Robust Morse Decompositions of Piecewise Constant Vector Fields

In this paper, we introduce a new approach to computing a Morse decomposition of a vector field on a triangulated manifold surface. The basic idea is to convert the input vector field to a piecewise constant (PC) vector field, whose trajectories can be computed using simple geometric rules. To overcome the intrinsic difficulty in PC vector fields (in particular, discontinuity along mesh edges), we borrow results from the theory of differential inclusions. The input vector field and its PC variant have similar Morse decompositions. We introduce a robust and efficient algorithm to compute Morse decompositions of a PC vector field. Our approach provides subtriangle precision for Morse sets. In addition, we describe a Morse set classification framework which we use to color code the Morse sets in order to enhance the visualization. We demonstrate the benefits of our approach with three well-known simulation data sets, for which our method has produced Morse decompositions that are similar to or finer than those obtained using existing techniques, and is over an order of magnitude faster.

Extraction of topologically simple isosurfaces from volume datasets

There are numerous algorithms in graphics and visualization whose performance is known to decay as the topological complexity of the input increases. On the other hand, the standard pipeline for 3D geometry acquisition often produces 3D models that are topologically more complex than their real forms. We present a simple and efficient algorithm that allows us to simplify the topology of an isosurface by alternating the values of some number of voxels. Its utility and performance are demonstrated on several examples, including signed distance functions from polygonal models and CT scans.