Visual data mining based on differential topology: a survey <s> Analyzing samples of a function R n → R m <s> I: Preliminaries on Manifolds.- 1. Manifolds.- 2. Differentiable Mappings and Submanifolds.- 3. Tangent Spaces.- 4. Partitions of Unity.- 5. Vector Bundles.- 6. Integration of Vector Fields.- II: Transversality.- 1. Sard's Theorem.- 2. Jet Bundles.- 3. The Whitney C? Topology.- 4. Transversality.- 5. The Whitney Embedding Theorem.- 6. Morse Theory.- 7. The Tubular Neighborhood Theorem.- III: Stable Mappings.- 1. Stable and Infinitesimally Stable Mappings.- 2. Examples.- 3. Immersions with Normal Crossings.- 4. Submersions with Folds.- IV: The Malgrange Preparation Theorem.- 1. The Weierstrass Preparation Theorem.- 2. The Malgrange Preparation Theorem.- 3. The Generalized Malgrange Preparation Theorem.- V: Various Equivalent Notions of Stability.- 1. Another Formulation of Infinitesimal Stability.- 2. Stability Under Deformations.- 3. A Characterization of Trivial Deformations.- 4. Infinitesimal Stability => Stability.- 5. Local Transverse Stability.- 6. Transverse Stability.- 7. Summary.- VI: Classification of Singularities, Part I: The Thom-Boardman Invariants.- 1. The Sr Classification.- 2. The Whitney Theory for Generic Mappings between 2-Manifolds.- 3. The Intrinsic Derivative.- 4. The Sr,s Singularities.- 5. The Thom-Boardman Stratification.- 6. Stable Maps Are Not Dense.- VII: Classification of Singularities, Part II: The Local Ring of a Singularity.- 1. Introduction.- 2. Finite Mappings.- 3. Contact Classes and Morin Singularities.- 4. Canonical Forms for Morin Singularities.- 5. Umbilics.- 6. Stable Mappings in Low Dimensions.- A. Lie Groups.- Symbol Index. <s> BIB001 </s> Visual data mining based on differential topology: a survey <s> Analyzing samples of a function R n → R m <s> There is a growing need to be able to accurately and efficiently search visual data sets, and in particular, 3D shape data sets. This paper proposes a novel technique, called Topology Matching, in which similarity between polyhedral models is quickly, accurately, and automatically calculated by comparing Multiresolutional Reeb Graphs (MRGs). The MRG thus operates well as a search key for 3D shape data sets. In particular, the MRG represents the skeletal and topological structure of a 3D shape at various levels of resolution. The MRG is constructed using a continuous function on the 3D shape, which may preferably be a function of geodesic distance because this function is invariant to translation and rotation and is also robust against changes in connectivities caused by a mesh simplification or subdivision. The similarity calculation between 3D shapes is processed using a coarse-to-fine strategy while preserving the consistency of the graph structures, which results in establishing a correspondence between the parts of objects. The similarity calculation is fast and efficient because it is not necessary to determine the particular pose of a 3D shape, such as a rotation, in advance. Topology Matching is particularly useful for interactively searching for a 3D object because the results of the search fit human intuition well. <s> BIB002 </s> Visual data mining based on differential topology: a survey <s> Analyzing samples of a function R n → R m <s> Part I. Classification of Singular Fibers: Preliminaries Singular Fibers of Morse Functions on Surfaces Classification of Singular Fibers Co-existence of Singular Fibers Euler Characteristic of the Source 4-Manifold Examples of Stable Maps of 4-Manifolds.- Part II. Universal Complex of Singular Fibers: Generalities Universal Complex of Singular Fibers Stable Maps of 4-Manifolds into 3-Manifolds Co-orientable Singular Fibers Homomorphism Induced by a Thom Map Cobordism Invariance Cobordism of Maps with Prescribed Local Singularities Examples of Cobordism Invariants.- Part III. Epilogue: Applications Further Developments References List of Symbols Index. <s> BIB003 </s> Visual data mining based on differential topology: a survey <s> Analyzing samples of a function R n → R m <s> Generalizing the concept of a Reeb graph, the Reeb space of a multivariate continuous mapping identifies points of the domain that belong to a common component of the preimage of a point in the range. We study the local and global structure of this space for generic, piecewise linear mappings on a combinatorial manifold. <s> BIB004 </s> Visual data mining based on differential topology: a survey <s> Analyzing samples of a function R n → R m <s> Contour trees and Reeb graphs are firmly embedded in scientific visualization for analysing univariate (scalar) fields. We generalize this analysis to multivariate fields with a data structure called the Joint Contour Net that quantizes the variation of multiple variables simultaneously. We report the first algorithm for constructing the Joint Contour Net and demonstrate that Contour Trees for individual variables can be extracted from the Joint Contour Net. <s> BIB005 </s> Visual data mining based on differential topology: a survey <s> Analyzing samples of a function R n → R m <s> This is a survey article on recent developments in visualization of large data, especially that of multivariate volume data. We present two essential ingredients. The first one is the mathematical background, especially the singularity theory of differentiable mappings, which enables us to capture topological features of given multivariate data in a mathematically rigorous way. The second one is a new development in computer science, called the joint contour net, which can encode topological structures of a given set of multivariate data in an efficient and robust way. Some applications to real data analysis are also presented. <s> BIB006 </s> Visual data mining based on differential topology: a survey <s> Analyzing samples of a function R n → R m <s> In this paper, we first classify singular fibers of proper $C^\infty$ stable maps of 3-dimensional manifolds with boundary into surfaces. Then, we compute the cohomology groups of the associated universal complex of singular fibers, and obtain certain cobordism invariants for Morse functions on compact surfaces with boundary. <s> BIB007
Data analysis based on differential topology has long been primarily focused on data samples on univariate (scalarvalued) functions f : R n → R. Nonetheless, with the recent advent of high-performance computers and sensors of high resolution, we are more likely to tackle data samples of multivariate functions f : R n → R m , m > 1. For example, illuminating the global structure of 3D fields of multiple scalar values such as temperature, pressure, humidity, etc. poses a very important technical problem for weather forecast, and the visual data mining based on differential topology is again expected to help us to tackle the challenging problems. A natural extension of the previous approach for multivariate data is to extract the intersection among inverse images of multiple function values first, and then track the topological changes inherent in that intersection with respect to the multivariate function value changes. For a multivariate function f : R n → R m and a point c ∈ R m , the inverse image is called a fiber BIB003 . A more rigorous mathematical definition goes as follows. For two multivariate functions f 0 and f 1 : R n → R m and points y 0 and y 1 in the range R m , we say that the fibers of f 0 and f 1 over the points y 0 and y 1 , respectively, are equivalent if for some open neighborhoods and ϕ : U 0 → U 1 with ϕ(y 0 ) = y 1 which make the following diagram commutative: Therefore, a fiber of a differentiable multivariate function over a point in the range refers to such an equivalence class. Note that this information encodes the semi-local behavior of the function around the whole inverse image of a point, and not just the inverse image as a set. A point p ∈ R n is a singular point if the rank of the Jacobian matrix is strictly less than min{n, m}, where and (x 1 , x 2 , . . . , x n ) are the coordinates of R n . In general, we can characterize singular points as the points in the domain where a topological change occurs in the fibers. A fiber is a singular fiber if it contains a singular point. Furthermore, the set of all singular points is called the Jacobi set of f and is often denoted by J(f ). As the reader can easily imagine, recognizing singular fibers is very important in visualizing a given set of large data. In practice, the analysis of multivariate data samples begins with extracting the Jacobi set, which has been tackled by Edelsbrunner et al.  BIB004 . They successfully developed an algorithm for extracting such Jacobi sets from functions to R 2 by identifying sample points where the gradients of the two corresponding scalar functions are parallel to each other. However, the algorithm extracts singular points only individually from the given data samples, and cannot identify the topological changes in fibers; in particular, we cannot get any information on the types of the singular points. Thus, identifying only the Jacobi set does not help us very much to identify global structures inherent in the entire data. Therefore, it remained as an important problem to identify the global structures of a given large data by seeking the underlying connectivity among the singular points or singular fibers. http://www.pacific-mathforindustry.com/content/6/1/4 In order to represent a given set of data in a hierarchical fashion, an extension of the Reeb graph is useful. For a function f : R n → R m , by contracting each connected component of its inverse images to a point, we get a space R f , which is called the Reeb space BIB004 of f . This is expected to play an important role similar to that of a contour tree. Quite recently, the above long lasting problem has been solved again by Carr et al.  BIB005 , where they invented an algorithm for constructing a joint contour net as a variant of a Reeb space for data samples of a multivariate function f : R n → R m , m > 1. The basic idea of this algorithm is to quantize the image of function samples in the mdimensional range space R m into a set of small blocks in terms of the m coordinate axes, and then seek the connectivity of the fibers between every pair of adjacent blocks along each coordinate axis, which finally allows us to compose a joint contour net as a network structure over the range space. Figure 7 shows such an example, where two different function values are defined over a 3D polygonal surface to characterize its shape with the corresponding joint contour net. Here, the two function values are the integral of geodesic distance over the polygonal shape BIB002 and surface curvature, where the integral of geodesic distance represents how much the sample point is far away from the object center. Figure 7(a) shows an ellipsoid and its corresponding joint contour net projected to the 2D range space. Since the surface curvature becomes large as the sample point moves away from the object center, the joint contour net is projected to a narrow region between the left bottom corner and the right top corner. Figure 7 (b) presents 3D spectacles where the associated joint contour net becomes more complicated, since the 3D shape model has variation in surface curvature. On the other hand, singular points of the multivariate function can be easily detected as the branches of the joint contour net. An example is demonstrated in Figure 8 , where a certain explicit function V → R 2 is analyzed with V being a bounded domain in R 3 . For each case, the left window exhibits a singular fiber at some function value that is marked in the right window. The joint contour nets indeed allow us to track the connectivity among the singular fibers, while precisely identifying the topological type of each singular fiber still remains to be tackled. In fact, some classification results for singular fibers have been obtained in singularity theory. For scalar functions on 2D domains, we have seen a classification of local topological changes of contour lines in Figure 2 , under the assumption that all the singular points are non-degenerate (see Section 3). For functions V → R 2 , where V is a bounded domain in 3D space, recently a classification of singular fibers and their topological changes has been obtained in BIB007 , under the assumption that the functions are stable. A function f : V → R 2 is stable if for any of its C ∞ approximation g (in the sense of the Whitney C ∞ topology), there exist diffeomorphisms : V → V and ϕ : R 2 → R 2 such that g = ϕ • f • −1 (for details, see BIB001 ). Note that the space of stable functions is open and dense in the space of all smooth functions V → R 2 . The list of singular fibers for such stable functions is as in Figures 9 and 10 . Note that this classifies the singular fibers up to equivalence, and therefore describes completely the topological change of the fibers around the singular fibers. Such a classification clearly helps us to identify the type of a singular fiber for a given set of data. By combining such classification results with the technique of joint contour net, we can visualize a given set of data in an efficient and robust way (for details, see BIB006 ). Figure 8 shows such a visualization result: each curve in the right windows corresponds to a specific singular fiber type in Figure 9 according to its color. Nevertheless, identifying the topological type of each singular fiber is considerably hard, especially for higher dimensional cases, where theoretical investigation has been still devoted for more detailed classification of singular fibers according to their topological types BIB003 . We are also working on this problem so that we can provide learners of differential topology with an interface for visually inspecting singular fibers of multivariate functions on the screen space. Furthermore, by referring to the topological types of the singular fibers embedded in the data domain, we can effectively extract meaningful features from multivariate data samples with minimal cost. Figure 9 First list of singular fibers for stable functions V → R 2 for a 3D bounded domain V: those in this list appear along curves in the range BIB007 . Both the black dot and the black square represent a point: however, the former one is in the interior of V, while the latter one appears on the boundary of V. Void squares represent tangencies with the boundary ∂V. http://www.pacific-mathforindustry.com/content/6/1/4