A quick trip through geometrical shape comparison
|
|
|
- Rosaline Stafford
- 9 years ago
- Views:
Transcription
1 A quick trip through geometrical shape comparison Patrizio Frosini 1,2 1 Department of Mathematics, University of Bologna, Italy 2 ARCES - Vision Mathematics Group, University of Bologna, Italy [email protected] Topics in Mathematics Bologna, 11 April 2013 Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
2 Plan of the talk Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
3 Outline 1 The "trivial" problem of comparing shapes 2 Trying to put some geometrical order in the chaos 3 Some theoretical results and applications Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
4 The "trivial" problem of comparing shapes 1 The "trivial" problem of comparing shapes 2 Trying to put some geometrical order in the chaos 3 Some theoretical results and applications Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
5 The "trivial" problem of comparing shapes What is shape? A "trivial" question: What is shape? SIMPLE ANSWER: Shape is what is left after removing scale and rotation. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
6 The "trivial" problem of comparing shapes What is shape? "There is always an easy solution to every human problem neat, plausible, and wrong." Henry Louis Mencken ( ) Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
7 The "trivial" problem of comparing shapes What is shape? Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
8 The "trivial" problem of comparing shapes What is shape? What is a pipe? Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
9 The "trivial" problem of comparing shapes What is shape? Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
10 The "trivial" problem of comparing shapes What is shape? Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
11 The "trivial" problem of comparing shapes What is shape? Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
12 The "trivial" problem of comparing shapes What is shape? Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
13 The "trivial" problem of comparing shapes What is shape? Pipe and passport of René and Georgette Magritte-Berger Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
14 The "trivial" problem of comparing shapes What is shape? Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
15 The "trivial" problem of comparing shapes What is shape? Buttons Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
16 The "trivial" problem of comparing shapes What is shape? Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
17 The "trivial" problem of comparing shapes What is shape? Cups Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
18 The "trivial" problem of comparing shapes The key role of the observer Every comparison of properties involves the presence of an observer perceiving the properties a methodology to compare the properties It follows that shape comparison is affected by subjectivity. Let us give some examples illustrating this fact. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
19 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer: Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
20 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer: Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
21 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer: Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
22 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer: Julian Beever Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
23 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer: Julian Beever Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
24 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer: Julian Beever Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
25 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer: Magnifying glass or cup of coffee? Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
26 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer: A surfacing submarine? Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
27 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer: Bump Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
28 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer: A yellow frisbee is floating in the air Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
29 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer: What is going on here? Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
30 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer: Coffee or owl? Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
31 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer: Duck or rabbit? Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
32 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer: The black shapes are NOT the camels, the narrow stripes below the shapes are. The black shapes are the shadows of the camels, as this photo was taken from overhead. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
33 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer: How many rabbits? Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
34 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer: Michael Jantzen, "Deconstructing the Houses". Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
35 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer: Crooked House (Krzywy Domek) by Szotyńscy and Zaleski, Sopot, Poland. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
36 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer: Crooked House (Krzywy Domek) by Szotyńscy and Zaleski, Sopot, Poland. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
37 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer: Crooked House (Krzywy Domek) by Szotyńscy and Zaleski, Sopot, Poland. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
38 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer: Crooked House (Krzywy Domek) by Szotyńscy and Zaleski, Sopot, Poland. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
39 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer: Crooked House (Krzywy Domek) by Szotyńscy and Zaleski, Sopot, Poland. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
40 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer: Crooked House (Krzywy Domek) by Szotyńscy and Zaleski, Sopot, Poland. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
41 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer:. "Glued in Florence" by Christiaan Triebert Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
42 The "trivial" problem of comparing shapes The key role of the observer Truth often depends on the observer: Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
43 The "trivial" problem of comparing shapes The key role of the observer The concept of shape is subjective and relative. It is based on the act of perceiving, depending on the chosen observer. Persistent perceptions are fundamental in order to approach this concept. Science is nothing but perception. Plato Reality is merely an illusion, albeit a very persistent one. Albert Einstein Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
44 Trying to put some geometrical order in the chaos 1 The "trivial" problem of comparing shapes 2 Trying to put some geometrical order in the chaos 3 Some theoretical results and applications Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
45 Trying to put some geometrical order in the chaos Our formal setting In shape comparison objects are not accessible directly, but only via measurements made by an observer. The comparison of two shapes is usually based on a family F of "measuring functions", which are defined on a set M (set of measurements) and take values in a set V (set of measurement values). Each function in F represents a measurement obtained via a measuring instrument. In most cases, the family F of measuring functions is invariant with respect to a given group G of transformations, that depends on the type of measurement we are considering. A G-invariant pseudo-metric d F is available for the set F, so that we can quantify the difference between the measuring functions in F. (pseudo-metric = metric without the property d(x, y) = 0 = x = y) Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
46 Trying to put some geometrical order in the chaos Example 1 M = S 2 (the globe s surface), V = R (the set of temperatures) Every f F is a function associating each point of S 2 with its temperature. G is the set of rigid motions of S 2 (we observe that F G = F) We can set d F (f 1, f 2 ) = inf g G sup x M f 1 (x) f 2 (g(x)) Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
47 Trying to put some geometrical order in the chaos Example 2 M is the vase surface, V = R 3 (the set of colors) Every f F is a function associating each point of M with its color. G is the set of rotations of M around the z-axis (we observe that F G = F) We can set d F (f 1, f 2 ) = inf g G sup x M f 1 (x) f 2 (g(x)) Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
48 Trying to put some geometrical order in the chaos Example 3 A different way of measuring the vases... M = S 1, V = {the set of pictures} Every f F is a function associating each point of S 1 with a picture. G is the set of rotations around the z-axis, and F G = F We can set d F (f 1, f 2 ) = inf g G sup x M f 1 (x) f 2 (g(x)) Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
49 Trying to put some geometrical order in the chaos Our shape pseudo-distance d F (formal definition) Assume that the following objects are given: A set M. Each point x M represents a measurement. A set V. Each point v V represents the value taken by a measurement. A set F of functions from M to V. Each function f F describes a possible set of results for all measurements in M. A group G acting on M, such that F is invariant with respect to G (i.e., for every f F and every g G we have that f g F). A pseudo-metric d F defined on the set F, that is invariant under the action of the group G (in other words, if f 1, f 2 F and g G then f 2 g F and d F (f 1, f 2 ) = d F (f 1, f 2 g)). We call each pair (F, d F ) a (pseudo-)metric shape space. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
50 Trying to put some geometrical order in the chaos An interesting case It often happens that M is a topological space and V is a metric space, endowed with a metric d V. In this case the functions in F are assumed to be continuous, and the group G is assumed to be a subgroup of the group of all self-homeomorphisms of M. As an example, let us think of a CT scanning. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
51 Trying to put some geometrical order in the chaos An interesting case In this example M = S 1 represents the topological space of all directions that are orthogonal to a given axis; V = R represents the metric space of all possible quantities of matter encountered by the X-ray beam in the considered direction. Every f F is a function taking each direction in S 1 to the quantity of matter encountered by the X-ray beam along that direction. G is the group of the rotations of S 1 (F G = F). We can set for f 1, f 2 F. d F (f 1, f 2 ) = inf sup g G x M f 1 (x) f 2 (g(x)) Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
52 Trying to put some geometrical order in the chaos Other interesting cases If V = R k we can use the shape pseudo-distance d F (f 1, f 2 ) = inf sup g G x M f 1 (x) f 2 (g(x)) for f 1, f 2 F. The functional sup x M f 1 (x) f 2 (g(x)) quantifies the change in the measurement induced by the transformation g. The pseudo-metric d F is produced by the attempt of minimizing this functional, varying the transformation g in the group G, and is called natural pseudo-distance. More generally, if V is a metric space endowed with the metric d V, we can set d F (f 1, f 2 ) = inf d V (f 1 (x), f 2 (g(x))) for f 1, f 2 F. sup g G x M Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
53 Trying to put some geometrical order in the chaos Other interesting cases If V = R and M is a compact subset of R m, we can set after fixing p 1. ( ) 1 d F (f 1, f 2 ) = inf f 1 (x) f 2 (g(x)) p p dx g G x M The functional ( x M f 1(x) f 2 (g(x)) p dx ) 1 p quantifies the change in the measurement induced by the transformation g. If G is the group of all isometries of M, d F is a pseudo-metric that is invariant under the action of G. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
54 Some theoretical results and applications 1 The "trivial" problem of comparing shapes 2 Trying to put some geometrical order in the chaos 3 Some theoretical results and applications Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
55 Some theoretical results and applications Usual assumptions In the rest of this talk we will assume that M is a topological space and V is a metric space. These assumptions allow us to require that if two measurements are close to each other (in some reasonable sense), then the values obtained by these measurements are close to each other, too. The functions in F will be assumed to be continuous. The group G will be assumed to be a subgroup of the group Homeo(M) of all self-homeomorphisms of M. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
56 Some theoretical results and applications Usual assumptions Why do we just consider self-homeomorphisms of M? Why couldn t we use, e.g., relations on M? Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
57 Some theoretical results and applications A result that suggests not to use relations in our setting The following result highlights a problem about doing that: Non-existence Theorem Let M be a Riemannian manifold. Let us endow Homeo(M) with the uniform convergence metric d UC : d UC (h, h ) = max x M d M (h(x), h (x)) for every h, h Homeo(M), where d M is the geodesic distance on M. Then (Homeo(M), d UC ) cannot be embedded in any compact metric space (K, d K ) endowed with an internal binary operation that extends the usual composition between homeomorphisms in Homeo(M) and commutes with the passage to the limit in K. In particular, Homeo(M) cannot be embedded in such a way into the set of binary relations on M. P. Frosini, C. Landi, No embedding of the automorphisms of a topological space into a compact metric space endows them with a composition that passes to the limit, Applied Mathematics Letters, 24 (2011), n. 10, Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
58 Some theoretical results and applications A result that suggests not to use relations in our setting In plain words, the previous theorem shows that, if our space of measurements M is a Riemannian manifold, no reasonable embedding of the set of all self-homeomorphisms of M into another compact metric space (K, d K ) exists. In particular, there does not exist any reasonable embedding into the space of binary relations on M. This is due to the fact that any such embedding couldn t preserve the usual composition between homeomorphisms in Homeo(M) and commute with the passage to the limit in K. Remark The previous theorem can be extended to topological spaces that are far more general than manifolds. It is sufficient that they contain a subset U that is homeomorphic to an n-dimensional open ball for some n 1. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
59 Some theoretical results and applications Some theoretical results about the natural pseudo-distance Until now, most of the results about the natural pseudo-distance have been proven for the case when M is a closed manifold, G = Homeo(M), and the measuring functions take their values in V = R. Here we recall three of these results: Theorem (for curves) Assume that M is a closed curve of class C 1 and that f 1, f 2 : M R are two functions of class C 1. Then, if d = inf g Homeo(M) max x M f 1 (x) f 2 (g(x)), at least one of the following properties holds: d equals the distance between a critical value of f 1 and a critical value of f 2 ; d equals half the distance between two critical values of f 1 ; d equals half the distance between two critical values of f 2. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
60 Some theoretical results and applications Some theoretical results about the natural pseudo-distance For example, in this case the natural pseudo-distance d equals half the distance between two critical values of f 1 : Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
61 Some theoretical results and applications Some theoretical results about the natural pseudo-distance Theorem (for surfaces) Assume that M is a closed surface of class C 1 and that f 1, f 2 : M R are two functions of class C 1. Then, if d = inf g Homeo(M) max x M f 1 (x) f 2 (g(x)), at least one of the following properties holds: d equals the distance between a critical value of f 1 and a critical value of f 2 ; d equals half the distance between two critical values of f 1 ; d equals half the distance between two critical values of f 2 ; d equals one third of the distance between a critical value of f 1 and a critical value of f 2. (OPEN PROBLEM: Is the last case possible?) Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
62 Some theoretical results and applications Some theoretical results about the natural pseudo-distance Theorem (for manifolds) Assume that M is a closed manifold of class C 1 and that f 1, f 2 : M R are two functions of class C 1. Then, if d = inf g Homeo(M) max x M f 1 (x) f 2 (g(x)), at least one of the following properties holds: a positive odd integer m exists, such that m d equals the distance between a critical value of f 1 and a critical value of f 2 ; a positive even integer m exists, such that m d equals the distance between two critical values either of f 1 or of f 2. (OPEN PROBLEM: Is the case m 3 possible?) Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
63 Some theoretical results and applications Some references about the natural pseudo-distance for topological spaces P. Frosini, M. Mulazzani, Size homotopy groups for computation of natural size distances, Bulletin of the Belgian Mathematical Society - Simon Stevin, 6 (1999), P. Donatini, P. Frosini, Natural pseudodistances between closed topological spaces, Forum Mathematicum, 16 (2004), n. 5, P. Donatini, P. Frosini, Natural pseudodistances between closed surfaces, Journal of the European Mathematical Society, 9 (2007), P. Donatini, P. Frosini, Natural pseudodistances between closed curves, Forum Mathematicum, 21 (2009), n. 6, Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
64 Some theoretical results and applications Natural pseudo-distance and size functions The natural pseudo-distance is usually difficult to compute. Lower bounds for the natural pseudo-distance can be obtained by computing the size functions. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
65 Some theoretical results and applications Definition of size function Given a topological space M and a continuous function f = (f 1,..., f k ) : M R k, Lower level sets For every u R k, M f u = {x M : f (x) u}. (u = (u 1,..., u k ) v = (v 1,..., v k ) means u j v j for every index j.) Definition (Frosini 1991) The Size Function of (M, f ) is the function l that takes each pair (u, v) with u v to the number l(u, v) of connected components of the set M f v that contain at least one point of the set M f u. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
66 Some theoretical results and applications Example of a size function, in the case that the measuring function has only one component Here the measuring function equals the distance from C. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
67 Some theoretical results and applications Example of a size function, in the case that the measuring function has only one component Here the measuring function equals the distance from C. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
68 Some theoretical results and applications Example of a size function, in the case that the measuring function has only one component Here the measuring function equals the distance from C. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
69 Some theoretical results and applications Example of a size function, in the case that the measuring function has only one component Here the measuring function equals the distance from C. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
70 Some theoretical results and applications Example of a size function, in the case that the measuring function has only one component Here the measuring function equals the distance from C. sizeshow.jar+cerchio.avi Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
71 Some theoretical results and applications We observe that each size function can be described by giving a set of points (vertices of triangles in figure) Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
72 Some theoretical results and applications A matching distance can be used to compare size functions Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
73 Some theoretical results and applications A matching distance can be used to compare size functions Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
74 Some theoretical results and applications Example of a size function, in the case that the measuring function has two components Here the measuring function f = (f 1, f 2 ) is given by the coordinates of each point of the curve M. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
75 Some theoretical results and applications Some references about size functions A. Verri, C. Uras, P. Frosini, M. Ferri, On the use of size functions for shape analysis, Biological Cybernetics, 70 (1993), A. Verri, C. Uras, Metric-topological approach to shape representation and recognition, Image and Vision Computing, 14, n. 3 (1996), C. Uras, A. Verri, Computing size functions from edge maps, International Journal of Computer Vision, 23, n. 2 (1997), S. Biasotti, L. De Floriani, B. Falcidieno, P. Frosini, D. Giorgi, C. Landi, L. Papaleo, M. Spagnuolo, Describing shapes by geometrical-topological properties of real functions, ACM Computing Surveys, vol. 40 (2008), n. 4, 12:1 12:87. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
76 Some theoretical results and applications Persistent homology groups and size homotopy groups Size functions have been generalized by Edelsbrunner and al. to homology in higher degree (i.e., counting the number of holes instead of the number of connected components). This theory is called Persistent Homology: H. Edelsbrunner, D. Letscher, A. Zomorodian, Topological persistence and simplification, Discrete & Computational Geometry, vol. 28, no. 4, (2002). Size functions have been also generalized to size homotopy groups: P. Frosini, M. Mulazzani, Size homotopy groups for computation of natural size distances, Bulletin of the Belgian Mathematical Society, vol. 6, no. 3, (1999). Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
77 Some theoretical results and applications Stability of the matching distance between size functions An important property of the matching distance is that size functions (and the ranks of persistent homology groups) are stable with respect to it. This stability can be expressed by means of the inequality D match d F where d F denotes the natural pseudo-distance. Therefore, the matching distance between size functions gives a lower bound for the natural pseudo-distance. In plain words, the stability of D match means that a small change of the measuring function induces a small change of the size function. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
78 Some theoretical results and applications Comparing shapes Stability allows us to use the matching distance to compare shapes: Upper lines refer to the 2-dimensional matching distance associated with a suitable function f = (f 1, f 2 ) taking values in R 2. Lower lines refer to the maximum between the 1-dimensional matching distances associated with f 1 and f 2, respectively. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
79 Some theoretical results and applications The idea of G-invariant size function Classical size functions and persistent homology are not tailored on the group G. In some sense, they are tailored on the group Homeo(M) of all self-homeomorphisms of M. In order to obtain better lower bounds for the natural pseudo-distance we need to adapt persistent homology, and to consider G-invariant persistent homology. Roughly speaking, the main idea consists in defining size functions and persistent homology by means of a set of chains that is invariant under the action of G. We skip the details of this procedure, that produces better lower bounds for the natural pseudo-distance. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
80 Some theoretical results and applications Uniqueness of models with respect to size functions: the case of curves We have seen that size functions allow to obtain lower bounds for the natural pseudo-distance. What about upper bounds? This problem is far more difficult. However, the following statement holds: Theorem Let f = (f 1, f 2 ), f = (f 1, f 2 ) : S1 R 2 be generic functions from S 1 to R 2. If the size functions of the four pairs of measuring functions (±f 1, ±f 2 ), (±f 1, ±f 2 ) (with corresponding signs) coincide, then there exists a C 1 -diffeomorphism h : S 1 S 1 such that f h = f. Moreover, it is unique. P. Frosini, C. Landi, Uniqueness of models in persistent homology: the case of curves, Inverse Problems, 27 (2011), Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
81 Some theoretical results and applications Conclusions In this talk we have illustrated a general metric scheme for geometrical shape comparison. The main idea of this model is that in shape comparison objects are not accessible directly, but only via measurements made by an observer. It follows that the comparison of two shapes is usually based on a family F of functions, which are defined on a topological space M and take values in a metric space (V, d). Each function in F represents a measurement obtained via a measuring instrument and, for this reason, it is called a "measuring function". In most cases, the set F of measuring functions is invariant with respect to a given group G of transformations, that depends on the type of measurement we are considering. After endowing F with a natural pseudo-metric, we have presented some examples and results, motivating the use of this theoretical framework. Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
82 Some theoretical results and applications Closing salutation Il catalogo delle forme è sterminato: finché ogni forma non avrà trovato la sua città, nuove città continueranno a nascere. Dove le forme esauriscono le loro variazioni e si disfano, comincia la fine delle città. The catalogue of forms is endless: until every shape has found its city, new cities will continue to be born. When the forms exhaust their variety and come apart, the end of cities begins. Italo Calvino, Le città invisibili Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
83 Some theoretical results and applications Patrizio Frosini (University of Bologna) Geometrical shape comparison 11 April / 83
Topological Data Analysis Applications to Computer Vision
Topological Data Analysis Applications to Computer Vision Vitaliy Kurlin, http://kurlin.org Microsoft Research Cambridge and Durham University, UK Topological Data Analysis quantifies topological structures
Let H and J be as in the above lemma. The result of the lemma shows that the integral
Let and be as in the above lemma. The result of the lemma shows that the integral ( f(x, y)dy) dx is well defined; we denote it by f(x, y)dydx. By symmetry, also the integral ( f(x, y)dx) dy is well defined;
Introduction to Topology
Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................
Surface bundles over S 1, the Thurston norm, and the Whitehead link
Surface bundles over S 1, the Thurston norm, and the Whitehead link Michael Landry August 16, 2014 The Thurston norm is a powerful tool for studying the ways a 3-manifold can fiber over the circle. In
Comments on Quotient Spaces and Quotient Maps
22M:132 Fall 07 J. Simon Comments on Quotient Spaces and Quotient Maps There are many situations in topology where we build a topological space by starting with some (often simpler) space[s] and doing
Metrics on SO(3) and Inverse Kinematics
Mathematical Foundations of Computer Graphics and Vision Metrics on SO(3) and Inverse Kinematics Luca Ballan Institute of Visual Computing Optimization on Manifolds Descent approach d is a ascent direction
Metric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
1 Local Brouwer degree
1 Local Brouwer degree Let D R n be an open set and f : S R n be continuous, D S and c R n. Suppose that the set f 1 (c) D is compact. (1) Then the local Brouwer degree of f at c in the set D is defined.
4. Expanding dynamical systems
4.1. Metric definition. 4. Expanding dynamical systems Definition 4.1. Let X be a compact metric space. A map f : X X is said to be expanding if there exist ɛ > 0 and L > 1 such that d(f(x), f(y)) Ld(x,
Geometry and Topology from Point Cloud Data
Geometry and Topology from Point Cloud Data Tamal K. Dey Department of Computer Science and Engineering The Ohio State University Dey (2011) Geometry and Topology from Point Cloud Data WALCOM 11 1 / 51
Metric Spaces Joseph Muscat 2003 (Last revised May 2009)
1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought of
Notes on metric spaces
Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.
Big Ideas in Mathematics
Big Ideas in Mathematics which are important to all mathematics learning. (Adapted from the NCTM Curriculum Focal Points, 2006) The Mathematics Big Ideas are organized using the PA Mathematics Standards
COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction
COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact
How To Understand General Relativity
Chapter S3 Spacetime and Gravity What are the major ideas of special relativity? Spacetime Special relativity showed that space and time are not absolute Instead they are inextricably linked in a four-dimensional
1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
Random graphs with a given degree sequence
Sourav Chatterjee (NYU) Persi Diaconis (Stanford) Allan Sly (Microsoft) Let G be an undirected simple graph on n vertices. Let d 1,..., d n be the degrees of the vertices of G arranged in descending order.
ON FIBER DIAMETERS OF CONTINUOUS MAPS
ON FIBER DIAMETERS OF CONTINUOUS MAPS PETER S. LANDWEBER, EMANUEL A. LAZAR, AND NEEL PATEL Abstract. We present a surprisingly short proof that for any continuous map f : R n R m, if n > m, then there
Fuzzy Probability Distributions in Bayesian Analysis
Fuzzy Probability Distributions in Bayesian Analysis Reinhard Viertl and Owat Sunanta Department of Statistics and Probability Theory Vienna University of Technology, Vienna, Austria Corresponding author:
1 Error in Euler s Method
1 Error in Euler s Method Experience with Euler s 1 method raises some interesting questions about numerical approximations for the solutions of differential equations. 1. What determines the amount of
Visualization of General Defined Space Data
International Journal of Computer Graphics & Animation (IJCGA) Vol.3, No.4, October 013 Visualization of General Defined Space Data John R Rankin La Trobe University, Australia Abstract A new algorithm
Geometric Transformations
Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
2. Length and distance in hyperbolic geometry
2. Length and distance in hyperbolic geometry 2.1 The upper half-plane There are several different ways of constructing hyperbolic geometry. These different constructions are called models. In this lecture
Separation Properties for Locally Convex Cones
Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam
Managerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay. Lecture - 13 Consumer Behaviour (Contd )
(Refer Slide Time: 00:28) Managerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay Lecture - 13 Consumer Behaviour (Contd ) We will continue our discussion
m i: is the mass of each particle
Center of Mass (CM): The center of mass is a point which locates the resultant mass of a system of particles or body. It can be within the object (like a human standing straight) or outside the object
Mathematical Induction
Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,
Bernice E. Rogowitz and Holly E. Rushmeier IBM TJ Watson Research Center, P.O. Box 704, Yorktown Heights, NY USA
Are Image Quality Metrics Adequate to Evaluate the Quality of Geometric Objects? Bernice E. Rogowitz and Holly E. Rushmeier IBM TJ Watson Research Center, P.O. Box 704, Yorktown Heights, NY USA ABSTRACT
The Heat Equation. Lectures INF2320 p. 1/88
The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)
Cyber Graphics. Abstract. 1. What is cyber graphics? 2. An incrementally modular abstraction hierarchy of shape invariants
Preprint of the Keynote Paper: Tosiyasu L. Kunii, Cyber Graphics, Proceedings of the First International Symposium on Cyber Worlds (CW2002), November 6-8 2002 Tokyo, Japan, in press, IEEE Computer Society
Logo Symmetry Learning Task. Unit 5
Logo Symmetry Learning Task Unit 5 Course Mathematics I: Algebra, Geometry, Statistics Overview The Logo Symmetry Learning Task explores graph symmetry and odd and even functions. Students are asked to
Mathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
Linear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
Current Standard: Mathematical Concepts and Applications Shape, Space, and Measurement- Primary
Shape, Space, and Measurement- Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two- and three-dimensional shapes by demonstrating an understanding of:
SOLIDWORKS: SKETCH RELATIONS
Sketch Feature: The shape or topology of the initial sketch or model is important, but exact geometry and dimensions of the initial sketched shapes are NOT. It recommended to work in the following order:
Math 1B, lecture 5: area and volume
Math B, lecture 5: area and volume Nathan Pflueger 6 September 2 Introduction This lecture and the next will be concerned with the computation of areas of regions in the plane, and volumes of regions in
TOPOLOGY OF SINGULAR FIBERS OF GENERIC MAPS
TOPOLOGY OF SINGULAR FIBERS OF GENERIC MAPS OSAMU SAEKI Dedicated to Professor Yukio Matsumoto on the occasion of his 60th birthday Abstract. We classify singular fibers of C stable maps of orientable
Creating Repeating Patterns with Color Symmetry
Creating Repeating Patterns with Color Symmetry Douglas Dunham Department of Computer Science University of Minnesota, Duluth Duluth, MN 55812-3036, USA E-mail: [email protected] Web Site: http://www.d.umn.edu/
MATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform
MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish
Classification and Structure of Periodic Fatou Components
Classification and Structure of Periodic Fatou Components Senior Honors Thesis in Mathematics, Harvard College By Benjamin Dozier Adviser: Sarah Koch 3/19/2012 Abstract For a given rational map f : Ĉ Ĉ,
Metric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence
CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs
CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce
Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights
How To Know The Limit Set Of A Fuchsian Group On A Boundary Of A Hyperbolic Plane
Reading material on the limit set of a Fuchsian group Recommended texts Many books on hyperbolic geometry and Kleinian and Fuchsian groups contain material about limit sets. The presentation given here
A Non-Linear Schema Theorem for Genetic Algorithms
A Non-Linear Schema Theorem for Genetic Algorithms William A Greene Computer Science Department University of New Orleans New Orleans, LA 70148 bill@csunoedu 504-280-6755 Abstract We generalize Holland
Fixed Point Theorems in Topology and Geometry
Fixed Point Theorems in Topology and Geometry A Senior Thesis Submitted to the Department of Mathematics In Partial Fulfillment of the Requirements for the Departmental Honors Baccalaureate By Morgan Schreffler
MA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 3 Binary Operations We are used to addition and multiplication of real numbers. These operations combine two real numbers
Chapter 7. Permutation Groups
Chapter 7 Permutation Groups () We started the study of groups by considering planar isometries In the previous chapter, we learnt that finite groups of planar isometries can only be cyclic or dihedral
Mathematical Research Letters 1, 249 255 (1994) MAPPING CLASS GROUPS ARE AUTOMATIC. Lee Mosher
Mathematical Research Letters 1, 249 255 (1994) MAPPING CLASS GROUPS ARE AUTOMATIC Lee Mosher Let S be a compact surface, possibly with the extra structure of an orientation or a finite set of distinguished
Tennessee Mathematics Standards 2009-2010 Implementation. Grade Six Mathematics. Standard 1 Mathematical Processes
Tennessee Mathematics Standards 2009-2010 Implementation Grade Six Mathematics Standard 1 Mathematical Processes GLE 0606.1.1 Use mathematical language, symbols, and definitions while developing mathematical
Representing Geography
3 Representing Geography OVERVIEW This chapter introduces the concept of representation, or the construction of a digital model of some aspect of the Earth s surface. The geographic world is extremely
North Carolina Math 2
Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4.
16.3 Fredholm Operators
Lectures 16 and 17 16.3 Fredholm Operators A nice way to think about compact operators is to show that set of compact operators is the closure of the set of finite rank operator in operator norm. In this
Logic, Algebra and Truth Degrees 2008. Siena. A characterization of rst order rational Pavelka's logic
Logic, Algebra and Truth Degrees 2008 September 8-11, 2008 Siena A characterization of rst order rational Pavelka's logic Xavier Caicedo Universidad de los Andes, Bogota Under appropriate formulations,
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
A Short Proof that Compact 2-Manifolds Can Be Triangulated
Inventiones math. 5, 160--162 (1968) A Short Proof that Compact 2-Manifolds Can Be Triangulated P. H. DOYLE and D. A. MORAN* (East Lansing, Michigan) The result mentioned in the title of this paper was
Problem of the Month: Cutting a Cube
Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:
WORKSHOP ON TOPOLOGY AND ABSTRACT ALGEBRA FOR BIOMEDICINE
WORKSHOP ON TOPOLOGY AND ABSTRACT ALGEBRA FOR BIOMEDICINE ERIC K. NEUMANN Foundation Medicine, Cambridge, MA 02139, USA Email: [email protected] SVETLANA LOCKWOOD School of Electrical Engineering
Math 231b Lecture 35. G. Quick
Math 231b Lecture 35 G. Quick 35. Lecture 35: Sphere bundles and the Adams conjecture 35.1. Sphere bundles. Let X be a connected finite cell complex. We saw that the J-homomorphism could be defined by
Graphical degree sequences and realizations
swap Graphical and realizations Péter L. Erdös Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences MAPCON 12 MPIPKS - Dresden, May 15, 2012 swap Graphical and realizations Péter L. Erdös
UNIFORMLY DISCONTINUOUS GROUPS OF ISOMETRIES OF THE PLANE
UNIFORMLY DISCONTINUOUS GROUPS OF ISOMETRIES OF THE PLANE NINA LEUNG Abstract. This paper discusses 2-dimensional locally Euclidean geometries and how these geometries can describe musical chords. Contents
Shortcut sets for plane Euclidean networks (Extended abstract) 1
Shortcut sets for plane Euclidean networks (Extended abstract) 1 J. Cáceres a D. Garijo b A. González b A. Márquez b M. L. Puertas a P. Ribeiro c a Departamento de Matemáticas, Universidad de Almería,
Non Parametric Inference
Maura Department of Economics and Finance Università Tor Vergata Outline 1 2 3 Inverse distribution function Theorem: Let U be a uniform random variable on (0, 1). Let X be a continuous random variable
THE BANACH CONTRACTION PRINCIPLE. Contents
THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,
Offline sorting buffers on Line
Offline sorting buffers on Line Rohit Khandekar 1 and Vinayaka Pandit 2 1 University of Waterloo, ON, Canada. email: [email protected] 2 IBM India Research Lab, New Delhi. email: [email protected]
Arrangements And Duality
Arrangements And Duality 3.1 Introduction 3 Point configurations are tbe most basic structure we study in computational geometry. But what about configurations of more complicated shapes? For example,
SHINPEI BABA AND SUBHOJOY GUPTA
HOLONOMY MAP FIBERS OF CP 1 -STRUCTURES IN MODULI SPACE SHINPEI BABA AND SUBHOJOY GUPTA Abstract. Let S be a closed oriented surface of genus g 2. Fix an arbitrary non-elementary representation ρ: π 1
The Tangent Bundle. Jimmie Lawson Department of Mathematics Louisiana State University. Spring, 2006
The Tangent Bundle Jimmie Lawson Department of Mathematics Louisiana State University Spring, 2006 1 The Tangent Bundle on R n The tangent bundle gives a manifold structure to the set of tangent vectors
C H A P T E R Regular Expressions regular expression
7 CHAPTER Regular Expressions Most programmers and other power-users of computer systems have used tools that match text patterns. You may have used a Web search engine with a pattern like travel cancun
3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
Section 7.2 Area. The Area of Rectangles and Triangles
Section 7. Area The Area of Rectangles and Triangles We encounter two dimensional objects all the time. We see objects that take on the shapes similar to squares, rectangle, trapezoids, triangles, and
Representation of functions as power series
Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions
A fast and robust algorithm to count topologically persistent holes in noisy clouds
A fast and robust algorithm to count topologically persistent holes in noisy clouds Vitaliy Kurlin Durham University Department of Mathematical Sciences, Durham, DH1 3LE, United Kingdom [email protected],
Quasi-static evolution and congested transport
Quasi-static evolution and congested transport Inwon Kim Joint with Damon Alexander, Katy Craig and Yao Yao UCLA, UW Madison Hard congestion in crowd motion The following crowd motion model is proposed
Copyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass
Centre of Mass A central theme in mathematical modelling is that of reducing complex problems to simpler, and hopefully, equivalent problems for which mathematical analysis is possible. The concept of
Section 16: Neutral Axis and Parallel Axis Theorem 16-1
Section 16: Neutral Axis and Parallel Axis Theorem 16-1 Geometry of deformation We will consider the deformation of an ideal, isotropic prismatic beam the cross section is symmetric about y-axis All parts
14.11. Geodesic Lines, Local Gauss-Bonnet Theorem
14.11. Geodesic Lines, Local Gauss-Bonnet Theorem Geodesics play a very important role in surface theory and in dynamics. One of the main reasons why geodesics are so important is that they generalize
Nonzero degree tangential maps between dual symmetric spaces
ISSN 1472-2739 (on-line) 1472-2747 (printed) 709 Algebraic & Geometric Topology Volume 1 (2001) 709 718 Published: 30 November 2001 ATG Nonzero degree tangential maps between dual symmetric spaces Boris
FIBRATION SEQUENCES AND PULLBACK SQUARES. Contents. 2. Connectivity and fiber sequences. 3
FIRTION SEQUENES ND PULLK SQURES RY MLKIEWIH bstract. We lay out some foundational facts about fibration sequences and pullback squares of topological spaces. We pay careful attention to connectivity ranges
tr g φ hdvol M. 2 The Euler-Lagrange equation for the energy functional is called the harmonic map equation:
Notes prepared by Andy Huang (Rice University) In this note, we will discuss some motivating examples to guide us to seek holomorphic objects when dealing with harmonic maps. This will lead us to a brief
Intersection of a Line and a Convex. Hull of Points Cloud
Applied Mathematical Sciences, Vol. 7, 213, no. 13, 5139-5149 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.213.37372 Intersection of a Line and a Convex Hull of Points Cloud R. P. Koptelov
About the inverse football pool problem for 9 games 1
Seventh International Workshop on Optimal Codes and Related Topics September 6-1, 013, Albena, Bulgaria pp. 15-133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute
11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
Minnesota Academic Standards
A Correlation of to the Minnesota Academic Standards Grades K-6 G/M-204 Introduction This document demonstrates the high degree of success students will achieve when using Scott Foresman Addison Wesley
TOOLS FOR 3D-OBJECT RETRIEVAL: KARHUNEN-LOEVE TRANSFORM AND SPHERICAL HARMONICS
TOOLS FOR 3D-OBJECT RETRIEVAL: KARHUNEN-LOEVE TRANSFORM AND SPHERICAL HARMONICS D.V. Vranić, D. Saupe, and J. Richter Department of Computer Science, University of Leipzig, Leipzig, Germany phone +49 (341)
Lessons on Teaching Undergraduate General Relativity and Differential Geometry Courses
Lessons on Teaching Undergraduate General Relativity and Differential Geometry Courses Russell L. Herman and Gabriel Lugo University of North Carolina Wilmington, Wilmington, NC Abstract We describe the
Explorations. Math Moves! Experiencing ratio and proportion
Explorations Math Moves! Experiencing ratio and proportion Inquiry and prolonged engagement are essential to math Science centers are known for promoting active, hands-on inquiry in science. Mathematics
7 Gaussian Elimination and LU Factorization
7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
6.3 Conditional Probability and Independence
222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted
AP Calculus AB Syllabus
Course Overview and Philosophy AP Calculus AB Syllabus The biggest idea in AP Calculus is the connections among the representations of the major concepts graphically, numerically, analytically, and verbally.
Mathematics Scope and Sequence, K-8
Standard 1: Number and Operation Goal 1.1: Understands and uses numbers (number sense) Mathematics Scope and Sequence, K-8 Grade Counting Read, Write, Order, Compare Place Value Money Number Theory K Count
Computational Geometry. Lecture 1: Introduction and Convex Hulls
Lecture 1: Introduction and convex hulls 1 Geometry: points, lines,... Plane (two-dimensional), R 2 Space (three-dimensional), R 3 Space (higher-dimensional), R d A point in the plane, 3-dimensional space,