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Set operations on linear quadtrees
136
ABSTRACTS OF PAPERS ACCEPTED FOR PUBLICATION
Regions, Hoh, and neir Nesting Level in Time Prqwrtiomd to the Border. H. H. ATKINSON, I. GARGANTINI, AND T. R. S. WALSH. Department of Computer Science, The University of Western Ontario, London, Ontario N6A 5B7, Canada. Received January 15, 1984; revised May 31, 1984.
Counting
Given a 2-dimensional black-and-white digital picture it is sometimes of interest to determine all the black components (together with the number of white holes in each one) and evaluate their level of nesting. Algorithms for solving this problem appear in the literature, but without time-complexity analyses. The algorithm proposed in this paper separates all maximally connected subsets and counts holes, regions, and nesting levels in time proportional to the number of components multiplied by the number of border pixels. The efficiency of this algorithm is due to the use of linear quadtrees to represent pictures. This paper also contains an upper bound for the size of a linear quadtree in terms of the number of border pixels of the picture it represents, and a proof that the bound is sharp to within a factor of two. The bound is then used to analyze the above-mentioned algorithm and could also be used to allocate static memory for storing linear quadtrees. This work was originally motivated by the purpose of determining the age of a fish scale by counting its growth zones, which have the structure of (roughly) concentric annular regions: in a digitized, segmented picture these regions appear most often as connected subsets of black pixels on a white background. A
Data Reduction Algorithm for Phar Curces. JAMES ROBERG& Department of Electrical and Computer Engineering, Northwestern University, Evanston, Illinois. Received June 20, 1983; revised February 2, 1984 and May 8, 1984.
A planar curve may be represented by a sequence of connected line segments. Existent algorithms for reducing the number of line segments used to represent a curve are examined. An efficient (linear time), computationally simple algorithm is developed. This algorithm achieves a high degree of data reduction while producing a representation that is accurate for even the most complex curves. of Qnadtreesjiom Quadtree MedkalAxis Tramforms. Department, University of Maryland, College Park, Maryland revised May 24,1984.
Reconstruction
HANAN SAMET. Computer Science 20742. Received January 21, 1984;
An algorithm is presented for reconstructing a quadtree from its quadtree medial axis transform (QMAT). It is useful when performing operations for which the QMAT is well suited (e.g., thinning of an image). The algorithm is a postorder tree traversal which propagates the subsumption of each BLACK QMAT node in the eight possible directions. Analysis of the algorithm shows that its average execution time is proportional to the number of leaf nodes in the quadtree. The algorithm also serves to reinforce the appropriateness of the definition of the quadtree skeleton which does not permit a BLACK quadtree node to require more than one element of the quadtree skeleton for its subsumption.
SURVEY Segmer&ation and Coding Techniques. MORTON NADLER. Department of Electrical Engineering, Virginia Tech, Blacksburg, Virginia 24061. Received March 5, 1984; revised May 5, 1984.
Document
A critical survey it. given of literature on document scan&g, text/diagram/half-tone-image segmentation, coding, comprt.. sion, and raster/vector conversion. Short abstracts are given of each of the papers covered. Each item i, referenced by keyword with complete cross-referencing.
NOTES on Linear Qwsdbws. MICHAEL A. BAUER. The Department of Computer Science, The University of Western Ontario, London, Ontario N6A .5BO, Canada. Received November 11, 1983; revised June 11,1984.
Set Operations
ABSTRACTS OF PAPERS ACCEPTED FOR PUBLICAT:ION
137
Linear quadtrees provide the potential of very efficient algorithms for imag,e processing. This paper demonstrates the correctness of and analyzes the complexity of several fundamental operations on linear quadtrees. In particular, an algorithm for computing the intersection, union, and pairwise difference of two linear quadtrees is presented and analyzed. These operations provide the basis for many more complex image processing techniques and, hence, their efficiency and correctness are extremely important. It is shown that these algorithms are linear in the number of nodes in the linear quadtrees involved in the operations. The paper also provides a brief introduction to linear quadtrees and some of their properties. On Dgerent Metho& Based on the Karhunen-Loeve Expanskm Md Used in Image AnaiysQ. M. H. SAVOJI AND R. E. BURGE. Department of Physics, Queen Elizabeth College, University of London, Campden Hill Road, London W8 7AH, United Kingdom. Received August 6, 1984; revised June 13, 1984. The aim here is to show that algorithms based on the Karhunen-Loeve expansion (KLE) and used for compression and feature extraction, come under the same generalized eigen data problem and can be looked upon from a unifying point of view. The KL expansion of 2-class data of signal and noise from the multidetector STEM images of ferritin molecules is studied as an example. Compression in KLE space is used for noise suppression (filtering) and dimensionality reduction (feature extraction) of these images. Note on the Determination of a Digital Straight Line from Chain Co&s. R. SHOUCRI,R. BENESCH,AND S. THOMAS. Department of Mathematics and Computer Science, Royal Military College, Kingston, Ontario K7L 2W3, Canada. Received August 1, 1983; accepted May 15, 1984. The K-curvature method is used to derive the average period of a chain cod.e for a digitized straight line. It is shown that this method is equivalent to calculating the second-order difference of the digitized line which exhibits in a simple way the periodicity of the chain code.
ABSTRACTS OF PAPERS ACCEPTED FOR PUBLICATION
Regions, Hoh, and neir Nesting Level in Time Prqwrtiomd to the Border. H. H. ATKINSON, I. GARGANTINI, AND T. R. S. WALSH. Department of Computer Science, The University of Western Ontario, London, Ontario N6A 5B7, Canada. Received January 15, 1984; revised May 31, 1984.
Counting
Given a 2-dimensional black-and-white digital picture it is sometimes of interest to determine all the black components (together with the number of white holes in each one) and evaluate their level of nesting. Algorithms for solving this problem appear in the literature, but without time-complexity analyses. The algorithm proposed in this paper separates all maximally connected subsets and counts holes, regions, and nesting levels in time proportional to the number of components multiplied by the number of border pixels. The efficiency of this algorithm is due to the use of linear quadtrees to represent pictures. This paper also contains an upper bound for the size of a linear quadtree in terms of the number of border pixels of the picture it represents, and a proof that the bound is sharp to within a factor of two. The bound is then used to analyze the above-mentioned algorithm and could also be used to allocate static memory for storing linear quadtrees. This work was originally motivated by the purpose of determining the age of a fish scale by counting its growth zones, which have the structure of (roughly) concentric annular regions: in a digitized, segmented picture these regions appear most often as connected subsets of black pixels on a white background. A
Data Reduction Algorithm for Phar Curces. JAMES ROBERG& Department of Electrical and Computer Engineering, Northwestern University, Evanston, Illinois. Received June 20, 1983; revised February 2, 1984 and May 8, 1984.
A planar curve may be represented by a sequence of connected line segments. Existent algorithms for reducing the number of line segments used to represent a curve are examined. An efficient (linear time), computationally simple algorithm is developed. This algorithm achieves a high degree of data reduction while producing a representation that is accurate for even the most complex curves. of Qnadtreesjiom Quadtree MedkalAxis Tramforms. Department, University of Maryland, College Park, Maryland revised May 24,1984.
Reconstruction
HANAN SAMET. Computer Science 20742. Received January 21, 1984;
An algorithm is presented for reconstructing a quadtree from its quadtree medial axis transform (QMAT). It is useful when performing operations for which the QMAT is well suited (e.g., thinning of an image). The algorithm is a postorder tree traversal which propagates the subsumption of each BLACK QMAT node in the eight possible directions. Analysis of the algorithm shows that its average execution time is proportional to the number of leaf nodes in the quadtree. The algorithm also serves to reinforce the appropriateness of the definition of the quadtree skeleton which does not permit a BLACK quadtree node to require more than one element of the quadtree skeleton for its subsumption.
SURVEY Segmer&ation and Coding Techniques. MORTON NADLER. Department of Electrical Engineering, Virginia Tech, Blacksburg, Virginia 24061. Received March 5, 1984; revised May 5, 1984.
Document
A critical survey it. given of literature on document scan&g, text/diagram/half-tone-image segmentation, coding, comprt.. sion, and raster/vector conversion. Short abstracts are given of each of the papers covered. Each item i, referenced by keyword with complete cross-referencing.
NOTES on Linear Qwsdbws. MICHAEL A. BAUER. The Department of Computer Science, The University of Western Ontario, London, Ontario N6A .5BO, Canada. Received November 11, 1983; revised June 11,1984.
Set Operations
ABSTRACTS OF PAPERS ACCEPTED FOR PUBLICAT:ION
137
Linear quadtrees provide the potential of very efficient algorithms for imag,e processing. This paper demonstrates the correctness of and analyzes the complexity of several fundamental operations on linear quadtrees. In particular, an algorithm for computing the intersection, union, and pairwise difference of two linear quadtrees is presented and analyzed. These operations provide the basis for many more complex image processing techniques and, hence, their efficiency and correctness are extremely important. It is shown that these algorithms are linear in the number of nodes in the linear quadtrees involved in the operations. The paper also provides a brief introduction to linear quadtrees and some of their properties. On Dgerent Metho& Based on the Karhunen-Loeve Expanskm Md Used in Image AnaiysQ. M. H. SAVOJI AND R. E. BURGE. Department of Physics, Queen Elizabeth College, University of London, Campden Hill Road, London W8 7AH, United Kingdom. Received August 6, 1984; revised June 13, 1984. The aim here is to show that algorithms based on the Karhunen-Loeve expansion (KLE) and used for compression and feature extraction, come under the same generalized eigen data problem and can be looked upon from a unifying point of view. The KL expansion of 2-class data of signal and noise from the multidetector STEM images of ferritin molecules is studied as an example. Compression in KLE space is used for noise suppression (filtering) and dimensionality reduction (feature extraction) of these images. Note on the Determination of a Digital Straight Line from Chain Co&s. R. SHOUCRI,R. BENESCH,AND S. THOMAS. Department of Mathematics and Computer Science, Royal Military College, Kingston, Ontario K7L 2W3, Canada. Received August 1, 1983; accepted May 15, 1984. The K-curvature method is used to derive the average period of a chain cod.e for a digitized straight line. It is shown that this method is equivalent to calculating the second-order difference of the digitized line which exhibits in a simple way the periodicity of the chain code.