Godfried T. Toussaint

New York University Abu Dhabi, Dubayy, Dubai, United Arab Emirates

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Publications (216)168 Total impact

  • Juan F. Beltran · Xiaohua Liu · Nishant Mohanchandra · Godfried T. Toussaint
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    ABSTRACT: Two approaches to measuring the similarity between symbolically notated musical rhythms are compared with each other and with human judgments of perceived similarity. The first is the edit-distance, a popular transformation method, applied to the symbolic rhythm sequences. The second approach employs the histograms of the inter-onset-intervals (IOIs) calculated from the rhythms. Furthermore, two methods for dealing with the histograms are also compared. The first utilizes the Mallows distance, a transformation method akin to the Earth-Movers distance popular in computer vision, and the second extracts a group of standard statistical features, used in music information retrieval, from the IOI-histograms. The measures are compared using four contrastive musical rhythm data sets by means of statistical Mantel tests that compute correlation coefficients between the various dissimilarity matrices. The results provide evidence from the aural domain, that transformation methods such as the edit distance are superior to feature-based methods for predicting human judgments of similarity. The evidence also supports the hypothesis that IOI-histogram-based methods are better than music-theoretical structural features computed from the rhythms themselves, provided that the rhythms do not share identical IOI histograms.
    No preview · Article · Mar 2015 · International Journal of Pattern Recognition and Artificial Intelligence
  • S.M. Oh · G.T. Toussaint · E.D. Demaine · M.L. Demaine
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    ABSTRACT: A measure of dissimilarity (distance) is proposed for comparing origami crease patterns represented as geometric graphs. The distance measure is determined by minimum-weight matchings calculated between the edges as well as the vertices of the graphs being compared. The distances between pairs of edges and pairs of vertices of the graph are weighted linear combinations of six parameters that constitute geometric features of the edges and vertices. The results of a preliminary study performed with a collection of 45 crease patterns obtained from Mitani's ORIPA web page, revealed which of these features appear to be more salient for obtaining a clustering of the crease patterns that appears to agree with human intuition.
    No preview · Article · Jan 2015
  • William A. Sethares · Godfried T. Toussaint
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    ABSTRACT: Because of its minimalist construction, Steve Reich’s Clapping Music provides an ideal crucible for an examination of the role of expressive timing and expressive timbre in musical performance. Seven recordings of Clapping Music are studied. The analysis begins by dissecting the seven performances into hundreds of segments that contain either single handclaps or double handclaps. These are then studied in many ways: the consistency with which performers are able to maintain the timbre of their claps, the height percept of a handclap and its meaning for such a percussive sound, the accuracy in time with which performers can maintain a steady pulse. When clapping together, the timbre of the clap pair is dependent not only on the individual timbres but also on the timing of the claps. It is possible to measure the accuracy in time with which performers synchronize the double claps. The section-by-section timing deviations correlate significantly with a simple measure of complexity: the more complex the pattern, the greater the timing deviations.
    No preview · Article · Aug 2014 · Journal of New Music Research
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    ABSTRACT: Several methods for reducing the running time of support vector machines (SVMs) are compared in terms of speed-up factor and classification accuracy using seven large real world datasets obtained from the UCI Machine Learning Repository. All the methods tested are based on reducing the size of the training data that is then fed to the SVM. Two probabilistic methods are investigated that run in linear time with respect to the size of the training data: blind random sampling and a new method for guided random sampling (Gaussian Condensing). These methods are compared with k-Nearest Neighbour methods for reducing the size of the training set and for smoothing the decision boundary. For all the datasets tested blind random sampling gave the best results for speeding up SVMs without significantly sacrificing classification accuracy.
    No preview · Article · Jan 2014
  • Godfried T Toussaint · Juan F Beltran
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    ABSTRACT: A mathematical measure of pattern complexity based on subsymmetries possessed by the pattern, previously shown to correlate highly with empirically derived measures of cognitive complexity in the visual domain, is found to also correlate significantly with empirically derived complexity measures of perception and production of auditory temporal and musical rhythmic patterns. Not only does the subsymmetry measure correlate highly with the difficulty of reproducing the rhythms by tapping after listening to them, but also the empirical measures exhibit similar behavior, for both the visual and auditory patterns, as a function of the relative number of subsymmetries present in the patterns.
    No preview · Article · Jan 2013 · Perception
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    Godfried T. Toussaint · Constantin Berzan
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    ABSTRACT: Previous experiments with low dimensional data sets have shown that Gabriel graph methods for instance-based learning are among the best machine learning algorithms for pattern classification applications. However, as the dimensionality of the data grows large, all data points in the training set tend to become Gabriel neighbors of each other, bringing the efficacy of this method into question. Indeed, it has been conjectured that for high-dimensional data, proximity graph methods that use sparser graphs, such as relative neighbor graphs (RNG) and minimum spanning trees (MST) would have to be employed in order to maintain their privileged status. Here the performance of proximity graph methods, in instance-based learning, that employ Gabriel graphs, relative neighborhood graphs, and minimum spanning trees, are compared experimentally on high-dimensional data sets. These methods are also compared empirically against the traditional k-NN rule and support vector machines (SVMs), the leading competitors of proximity graph methods.
    Preview · Conference Paper · Jul 2012
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    Yang Liu · Godfried T. Toussaint
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    ABSTRACT: Several methods for the mathematical notation, representation, and visualization of musical rhythm at the symbolic level are illustrated and compared in terms of their advantages and drawbacks, as well as their suitability for particular applications.
    Preview · Article · Jan 2012
  • Binay K.bhattacharya · Asishmukhopadhyay · Godfried T.toussaint
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    ABSTRACT: A simple polygon P is said to be weakly extrenally visible from a line segment L if it lies outside P and for every point p on the boundary of P there is a point q on L such that no point in the interior of lies inside P. In this paper, a linear time algorithm is proposed for computing a shortest line segment from which P is weakly externally visible. This is done by a suitable generalization of a linear time algorithm which solves the same problem for a convex polygon.
    No preview · Article · Nov 2011 · International Journal of Computational Geometry & Applications
  • Yang Liu · Godfried T. Toussaint
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    ABSTRACT: The marble pavement of the Cathedral in the Tuscan city of Siena in Italy has been described as one of the marvels of the world. Over the centuries much has been written about its biblical and political characters, the stories depicted in its figurative mosaics, the artists responsible for creating the mosaics, the types of marble used and the history of their construction. The many frieze patterns framing the figurative mosaics are noteworthy examples of geometric design, and yet, they have been conspicuously overlooked in the literature concerning this pavement. Here, the geometric frieze patterns found on the pavement, walls and ceiling of the Siena Cathedral are analysed in terms of their underlying geometric structure, the optical effects, such as multi-stable perception, that they engender in the viewer and a typology of patterns of repetition.
    No preview · Article · Sep 2011 · Journal of Mathematics and the Arts
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    Csaba D. Tóth · Godfried T. Toussaint · Andrew Winslow
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    ABSTRACT: An open edge of a simple polygon is the set of points in the relative interior of an edge. We revisit several art gallery problems, previously considered for closed edge guards, using open edge guards. A guard edge of a poly- gon is an edge that sees every point inside the polygon. We show that every simple non-starshaped polygon ad- mits at most one open guard edge, and give a simple new proof that it admits at most three closed guard edges. We characterize open guard edges, and derive an algorithm that finds all open guard edges of a simple n-gon in O(n) time in the RAM model of computation. Finally, we present lower bound constructions for simple polygons with n vertices that require [n/3] open edge guards, and conjecture that this bound is tight.
    Full-text · Conference Paper · Jun 2011
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    ABSTRACT: We address the question: How many edge guards are needed to guard an orthogonal polyhedron of e edges, r of which are reflex? It was previously established [3] that e/12 are sometimes necessary and e/6 always suffice. In contrast to the closed edge guardsused for these bounds, we introduce a new model, open edge guards (excluding the endpoints of the edge), which we argue are in some sense more natural in this context. After quantifying the relationship between closed and open edge guards, we improve the upper bound to show that, asymptotically, (11/72)e (open or closed) edge guards suffice, or, in terms of r, that (7/12)r suffice. Along the way, we establish tight bounds relating e and r for orthogonal polyhedra of any genus. 1
    Full-text · Conference Paper · Jan 2011
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    David Rappaport · Godfried T. Toussaint · Mustafa Mohamad
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    ABSTRACT: Motivated by a problem in music theory of measuring the distance between chords, scales, and rhythms we consider algorithms for obtaining a minimum-weight many-to-many matching between two sets of points on the real line. Given sets \(A\) and \(B\), we seek to find the best rigid translation of \(B\) and a many-to-many matching that minimizes the sum of the squares of the distances between matched points. We provide discrete algorithms that solve this continuous optimization problem, and discuss other related matters.
    Preview · Conference Paper · Jan 2011
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    ABSTRACT: Given a planar polygon (or chain) with a list of edges fe 1 ; e 2 ; e 3 ; : : : ; e n 1 ; e n g, we examine the eect of several operations that permute this edge list, resulting in the formation of a new polygon. The main operations that we consider are: reversals which involve inverting the order of a sublist, transpositions which involve interchanging subchains (sublists), and edge-swaps which are a special case and involve interchanging two consecutive edges. Using these permuting operations, we explore the complexity of performing certain actions, such as convexifying a given polygon or obtaining its mirror image. When each edge of the given polygon has also been assigned a direction we say that the polygon is signed. In this case any edge involved in a reversal changes direction. The complexity of some problems varies depending on whether a polygon is signed or unsigned. An additional restriction in many cases is that polygons remain simple after every permutation.
    Preview · Article · Jan 2011
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    ABSTRACT: In 1994 Grunbaum showed that, given a point set S in R-3, it is always possible to construct a polyhedron whose vertices are exactly S. Such a polyhedron is called a polyhedronization of S. Agarwal et al. extended this work in 2008 by showing that there always exists a polyhedronization that can be decomposed into a union of tetrahedra (tetrahedralizable). In the same work they introduced the notion of a serpentine polyhedronization for which the dual of its tetrahedralization is a chain. In this work we present a randomized algorithm running in O(n log(6) n) expected time which constructs a serpentine polyhedronization that has vertices with degree at most 7, answering an open question by Agarwal et al.
    Full-text · Conference Paper · Dec 2010
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    Yang Liu · Godfried Toussaint
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    ABSTRACT: A geometrical analysis of the meander decorative patterns on a Roman pavement mosaic found at the Roman villa in Chedworth, England, is presented. The analysis reveals that the intricate swastika meander pattern consisting of four closed curves could have been easily constructed using a very simple hypothesized algorithm. The algorithm also explains the design of Roman swastika meanders found throughout the Roman Empire. Connections are indicated between these patterns and the sona traditional art of Angola as well as the kolam traditional art of Tamil South India. The analysis and algorithm described have applications to the classification of geometric mosaic patterns, the design of new patterns, and the reconstruction of mosaics that have been partially destroyed by the ravages of time.
    Full-text · Article · Mar 2010 · Journal of Mathematics and the Arts
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    Godfried Toussaint
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    ABSTRACT: Many problems concerning the theory and technology of rhythm, melody, and voice-leading are fundamentally geometric in nature. It is therefore not surprising that the field of computational geometry can contribute greatly to these problems. The interaction between computational geometry and music yields new insights into the theories of rhythm, melody, and voice-leading, as well as new problems for research in several areas, ranging from mathematics and computer science to music theory, music perception, and musicology. Recent results on the geometric and computational aspects of rhythm, melody, and voice-leading are reviewed, connections to established areas of computer science, mathematics, statistics, computational biology, and crystallography are pointed out, and new open problems are proposed.
    Preview · Article · Jan 2010 · Computational Geometry
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    ABSTRACT: The Hexachordal Theorem may be interpreted in terms of scales, or rhythms, or as abstract mathematics. In terms of scales it claims that the complement of a chord that uses half the pitches of a scale is homometric to—i.e., has the same interval structure as—the original chord. In terms of onsets it claims that the complement of a rhythm with the same number of beats as rests is homometric to the original rhythm. We generalize the theorem in two directions: from points on a discrete circle (the mathematical model encompassing both scales and rhythms) to a continuous domain, and simultaneously from the discrete presence or absence of a pitch/onset to a continuous strength or weight of that pitch/onset. Athough this is a significant generalization of the Hexachordal Theorem, having all discrete versions as corollaries, our proof is arguably simpler than some that have appeared in the literature. We also establish the natural analog of what is sometimes known as Patterson’s second theorem: if two equal-weight rhythms are homometric, so are their complements.
    Preview · Chapter · Jun 2009
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    ABSTRACT: Musical cyclic rhythms with a cycle length (timespan) of 8 or 16 pulses are called binary; those with 6 or 12 pulses are called ternary. The process of mapping a ternary rhythm of, say 12 pulses, to a rhythm of 16 pulses, such that musicologically salient properties are preserved is termed binarization. By analogy, the converse process of mapping a binary rhythm to a ternary rhythm is referred to as ternarization. New algorithms are proposed and investigated for the binarization and ternarization of musical rhythms with the goal of understanding the historical evolution of traditional rhythms through inter-cultural contacts. The algorithms also have applications to automated rhythmic pattern generation, and may be incorporated in composition software tools.
    Full-text · Article · May 2009
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    Francisco Gómez-Martín · Perouz Taslakian · Godfried Toussaint
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    ABSTRACT: Several operations on Euclidean rhythms based on musical motivations are defined. The operations defined are complementation, alternation, and decomposition. Some mathematical properties are proved for each and the conditions under which a given operation preserves the Euclidean property are examined. Finally, connections are shown to interlocking Euclidean rhythms and tiling canons, and tiling quasi-canons are introduced.
    Full-text · Article · Apr 2009 · Journal of Mathematics and Music
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    Francisco Gómez-Martín · Perouz Taslakian · Godfried Toussaint
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    ABSTRACT: In this paper we investigate the structure of Euclidean rhythms and show that a Euclidean rhythm is formed of a pattern, called the main pattern, repeated a certain number of times, followed possibly by one extra pattern, the tail pattern. We thoroughly study the recursive nature of Euclidean rhythms when generated by Bjorklund's algorithm, one of the many algorithms that generate Euclidean rhythms. We make connections between Euclidean rhythms and Bezout's theorem. We also prove that the decomposition obtained is minimal.
    Full-text · Article · Mar 2009 · Journal of Mathematics and Music

Publication Stats

5k Citations
168.00 Total Impact Points

Institutions

  • 2012-2015
    • New York University Abu Dhabi
      Dubayy, Dubai, United Arab Emirates
  • 2010-2012
    • Harvard University
      • Department of Music
      Boston, MA, United States
  • 1974-2011
    • McGill University
      • • School of Computer Science
      • • Centre for Interdisciplinary Research in Music Media & Technology (CIRMMT)
      Montréal, Quebec, Canada
    • Government of British Columbia, Canada
      Vancouver, British Columbia, Canada
  • 1992
    • University of Kentucky
      • Department of Computer Science
      Lexington, KY, United States
  • 1970-1972
    • University of British Columbia - Vancouver
      • Department of Electrical and Computer Engineering
      Vancouver, British Columbia, Canada