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Image Processing on Optimal Volume Sampling Lattices Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1314 Image processing on optimal volume sampling lattices Thinking outside the box ELISABETH SCHOLD LINNÉR ACTA UNIVERSITATIS UPSALIENSIS ISSN 1651-6214 ISBN 978-91-554-9406-3 UPPSALA urn:nbn:se:uu:diva-265340 2015 Dissertation presented at Uppsala University to be publicly examined in Pol2447, Informationsteknologiskt centrum (ITC), Lägerhyddsvägen 2, hus 2, Uppsala, Friday, 18 December 2015 at 10:00 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Professor Alexandre Falcão (Institute of Computing, University of Campinas, Brazil). Abstract Schold Linnér, E. 2015. Image processing on optimal volume sampling lattices. Thinking outside the box. (Bildbehandling på optimala samplingsgitter. Att tänka utanför ramen). Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1314. 98 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-9406-3. This thesis summarizes a series of studies of how image quality is affected by the choice of sampling pattern in 3D. Our comparison includes the Cartesian cubic (CC) lattice, the body- centered cubic (BCC) lattice, and the face-centered cubic (FCC) lattice. Our studies of the lattice Brillouin zones of lattices of equal density show that, while the CC lattice is suitable for functions with elongated spectra, the FCC lattice offers the least variation in resolution with respect to direction. The BCC lattice, however, offers the highest global cutoff frequency. The difference in behavior between the BCC and FCC lattices is negligible for a natural spectrum. We also present a study of pre-aliasing errors on anisotropic versions of the CC, BCC, and FCC sampling lattices, revealing that the optimal choice of sampling lattice is highly dependent on lattice orientation and anisotropy. We suggest a new reference function for studies of aliasing errors on alternative sampling lattices. This function has a spherical spectrum, and a frequency content proportional to the distance from the origin, facilitating studies of pre-aliasing in spatial domain. The accuracy of anti-aliased Euclidean distance transform is improved by application of more sofisticated methods for computing the sub-spel precision term. We find that both accuracy and precision are higher on the BCC and FCC lattices than on the CC lattice. We compare the performance of several intensity-weighted distance transforms on MRI data, and find that the derived segmentation result, with respect to relative error in segmented volume, depends neither on the sampling lattice, nor on the sampling density. Lastly, we present LatticeLibrary, a open source C++ library for processing of sampled data, supporting a number of common image processing methods for CC, BCC, and FCC lattices. We also introduce BccFccRaycaster, a tool for visualizing data sampled on CC, BCC, and FCC lattices. We believe that the work summarized in this thesis provide both the motivation and the tools for continuing research on application of the BCC and FCC lattices in image processing and analysis. Keywords: BCC, FCC, aliasing, distance transform, segmentation Elisabeth Schold Linnér, Department of Information Technology, Division of Visual Information and Interaction, Box 337, Uppsala University, SE-751 05 Uppsala, Sweden. © Elisabeth Schold Linnér 2015 ISSN 1651-6214 ISBN 978-91-554-9406-3 urn:nbn:se:uu:diva-265340 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-265340) Till min stora förvåning. List of papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I E. Linnér and R. Strand. Aliasing Properties of Voxels in Three-Dimensional Sampling Lattices. In Large Scale Scientific Computing, Sozopol, Bulgaria, I. Lirkov, S. Margenov, and J. Wasinewski´ (Eds.), Lecture Notes in Computer Science, 7116: pp 507–514, Jun. 2011. c 2012, Springer-Verlag Berlin Heidelberg II E. Linnér and R. Strand. A Graph-Based Implementation of the Anti-Aliased Euclidean Distance Transform. In 22nd International Conference on Pattern Recognition (ICPR), Stockholm, Sweden: pp 1025-1030, Aug. 2014. c 2014 IEEE III E. Linnér and R. Strand. Anti-Aliased Euclidean Distance Transform on 3D Sampling Lattices. In Discrete Geometry for Computer Imagery, E. Barcucci, A. Frosini, and S. Rinaldi (Eds.), Siena, Italy, Lecture Notes in Computer Science, 8668: pp 88-98, Sep. 2014. c 2014, Springer International Publishing Switzerland IV E. Schold Linnér and R. Strand. Pre-aliasing and anisotropy on the CC, BCC, and FCC sampling lattices. Submitted for journal publication. V E. Schold Linnér, J. Kullberg, and R. Strand. Fuzzy Segmentation of Synthetic and MRI Volume Data sampled on Optimal Lattices. Submitted journal for publication. VI E. Schold Linnér, M. Morén, K.-O. Smed, J. Nysjö, and R. Strand. LatticeLibrary and BccFccRaycaster: Software for processing and viewing 3D data on optimal sampling lattices. Submitted for journal publication. Reprints were made with permission from the publishers. All papers were researched and written under the supervision of R. Strand. The frequency domain study in Paper I was performed by R. Strand. In Paper V, planning and implementation were the work of E. Linnér, and data acquisi- tion was the joint work of E. Linnér and J. Kullberg. LatticeLibrary, published in Paper VI is the work of E. Linnér, while BccFccRaycaster is implemented by M. Morén and K.-O. Smed, under the supervision of E. Linnér and J. Nysjö. Related work In addition to the papers included in this thesis, the author has also written or contributed to the following publications: • E. Linnér and R. Strand. Comparison of normalized convolution on square and Hexagonal grids. In Proceedings SSBA’11 Symposium on Image Analysis, Stockholm, Sweden, 2011. • E. Linnér and R. Strand. Comparison of restoration quality on square and hexagonal grids using normalized convolution. In 21st International Conference on Pattern Recognition (ICPR), Tsukuba, Japan: pp 3046- 3049, Nov. 2012. • R. Strand, F. Malmberg, P. K. Saha, and E. Linnér. The Minimum Barrier Distance - Stability to Seed Point Position. In Discrete Geometry for Computer Imagery, E. Barcucci, A. Frosini, and S. Rinaldi (Eds.), Siena, Italy, Lecture Notes in Computer Science, 8668: pp 111-121, Sep. 2014. Contents 1 Initial remarks ............................................................................................ 17 2 Introduction to computerized image processing and analysis ................ 19 2.1 Image pre-processing ..................................................................... 21 2.1.1 Convolutional filtering .................................................... 21 2.1.2 Filtering in frequency domain ........................................ 21 2.2 Image processing ............................................................................ 23 2.2.1 Image segmentation ......................................................... 23 2.2.2 Distance transformation .................................................. 25 2.2.3 Registration ...................................................................... 28 2.3 Image analysis ................................................................................ 28 3 Sampling theory ......................................................................................... 30 3.1 One-dimensional point-sampling .................................................. 30 3.2 Interpolation and super-sampling .................................................. 33 3.3 Sampling lattices ............................................................................ 35 3.4 Non-band-limited functions .......................................................... 39 3.4.1 Studies in the frequency domain .................................... 40 3.4.2 Studies in the spatial domain .......................................... 45 3.5 Anisotropic sampling ..................................................................... 48 4 Image acquisition on optimal sampling lattices ...................................... 52 4.1 Acquisition of 2D images .............................................................. 52 4.1.1 Photography ..................................................................... 52 4.1.2 Sonography ...................................................................... 53 4.2 Acquisition of volume images ....................................................... 54 4.2.1 Computed tomography .................................................... 54 4.2.2 Emission Tomography ..................................................... 57 4.2.3 Magnetic Resonance Imaging ......................................... 58 4.2.4 3D sonography ................................................................. 60 5 Image processing and analysis on optimal sampling lattices ................. 61 5.1 Raster scanning .............................................................................. 62 5.2 Surface area of fuzzy binary objects ............................................. 63 5.3 Distance transforms and segmentation ......................................... 64 5.3.1 Wave-front propagation ................................................... 65 5.3.2 Improving the anti-aliased Euclidean distance transform .......................................................................... 66 5.3.3 Seeded segmentation on optimal lattices ....................... 69 5.4 Fourier transform ..........................................................................
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