Proc. IEEE Int. Workshop on Multimedia Signal Processing, Siena, Italy, Sept. 29-Oct. 1, 2004
Visual Salience-Guided Mesh Decomposition Hsueh-Yi Sean Lin∗† , Hong-Yuan Mark Liao† , and Ja-Chen Lin∗ ∗ Department of Computer and Information Science National Chiao-Tung University, Hsinchu, Taiwan 300 Email: {hylin,jclin}@cis.nctu.edu.tw † Institute of Information Science Academia Sinica, Taipei, Taiwan 115 Email:
[email protected]
Abstract— In this paper, we propose a novel mesh decomposition scheme called “visual salience-guided mesh decomposition”. The concept of part salience is originated from cognitive psychology and it asserts the salience of a part can be determined by (at least) three factors: the protrusion, the boundary strength, and the relative size of a part. We try to convert these “conceptual” rules into “real” computational processes and then use them to “guide” a 3D mesh decomposition process. The experimental results have shown that the proposed scheme is indeed effective and powerful in decomposing a 3D mesh into significant components.
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Fig. 1. Illustration of (a) part boundaries and (c)-(d) part cuts on a 2D silhouette (Re-sketched from [1]).
3D shape databases, the organization from parts to the whole allows us to conduct a “part-in-whole” searching process. In addition, extracting significant components based on different salient features allows us to execute an active construction of visual parts from a 3D model.
I. I NTRODUCTION ENERALLY, decomposition is a leverage to obtain the componential representation from a whole. After the decomposition step is executed, the decomposed components can be individually selected, grouped, and analyzed based on the properties of interest. The underlying assumption of the existing 3D object decomposition methods [3], [4] is based on psychologists’ definition of a part, which is motivated by “a uniformity of a nature” [1]. Furthermore, the principle of transversality1 is regarded as a guide to find the part shapes. Among the existing methods, the methods that aim at tracing concave regions [3] adopted the minima rule2 to generate the part boundaries. As to the clustering-based methods [3] and the merging processes [4], Gestalt laws of similarity and proximity are frequently used to capture the homogeneous characteristics of parts. Obviously, some existing 3D object decomposition methods did attempt to extract part shapes and boundaries by mimicking human visual perception on 3D shapes. However, none of the existing 3D object decomposition methods takes salience of parts into account. In this paper, we propose a novel mesh decomposition scheme called “visual salience-guided mesh decomposition”, which bases the decomposition on the theory of part salience borrowed from cognitive psychology [1]. The potential applications of the new approach are as follows. For example, in
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II. R EVIEW OF H OFFMAN AND S INGH ’ S OF V ISUAL S ALIENCE According to Hoffman and Singh’s theory [1], there are at least three factors that determine the salience of a part: the protrusion, the boundary strength, and the relative size. Based on different visual salience perceived by human visual mechanism, the 2D silhouette shown in Fig. 1(a) may have different interpretations and decomposition results. For example, the 2D silhouette might be interpreted as an alien’s head with a pair of protrusive ears when the salience of a part is determined primarily by its protrusion (i.e., the part cuts in Fig. 1(b) is adopted). On the other hand, the 2D silhouette might be interpreted as an unidentified flying object when the part salience is determined primarily by its relative size (i.e., the part cuts in Fig. 1(c) is adopted). As a result, the salience of a part plays an important role in human visual processes. However, the quantitative definitions for part salience proposed by Hoffman and Singh [1] were made under the assumption that a part and its boundary are found in advance. In the sense of perceptual organization [5], this drawback to some extent limits the power of Hoffman and Singh’s theory of visual salience. In this paper, we shall propose a new 3D mesh decomposition scheme that incorporates the psychological theory of visual salience, in such a way that the mesh decomposition process is as close as possible to the human visual perception mechanism.
1 Transversality [1]: When two arbitrarily shaped surfaces are made to interpenetrate, they always meet in a contour of concave discontinuity of their tangent planes. 2 Minima rule [1]: All negative minima of the principal curvatures (along their associated lines of curvature) form boundaries between parts.
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III. V ISUAL S ALIENCE -G UIDED M ESH D ECOMPOSITION A. Modeling the Protrusion as the Degree of Center In our investigation, we found that the integral function proposed by Hilaga et al. [2] is more suitable for the purpose of protrusion characterization. Therefore, we adopt the integral function described in [2] to characterize the protrusion of a part. Here, in contrast to [2], the integral function is constructed on the dual graph of a given mesh, G = (V, E), where V and E represent the set of dual vertices and the set of dual edges, respectively. A dual vertex v ∈ V is referred to the center of mass of a face in the original mesh while a dual edge (u, v) ∈ E links the center-of-mass of two adjacent faces and intersects at the midpoint of the edge shared by the two faces. Let area(v) denote the area occupied by a dual vertex v and area(V ) denote the total area of the object surface. The protrusion degree at a dual vertex v can be defined as [2]: X µ(v) = g(v, bi ) · area(bi ), (1) i
where {b0 , b1 , . . . } are the base dual-vertices, which are used to approximate the above integral P function. In addition, area(bi ) is the area of the base and i area(bi ) = area(V ). Furthermore, g(v, bi ) returns the shortest geodesic distance between the dual vertex v and the base vertex bi . Since the function µ(v) defined in Eq. (1) is not invariant to scaling transformation, a normalized version of µ(v) is defined as [2]: Protrusion(v) =
µ(v) − minu∈V µ(u) . maxu∈V µ(u)
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Using the normalized protrusion degree defined in Eq. (2), we are able to calculate a numeric value (ranging from 0 to 1) for each dual vertex located on a 3D mesh. The farther a dual vertex is apart from the center of a 3D mesh, the larger the protrusion degree will be. Fig. 2(a) shows the protrusion characterization of the dinopet model. B. Choosing the Salient Representatives of Parts The criterion used for selecting salient representatives is to find the local maxima of protrusion degrees. Given a dual vertex r∈V , the dual vertex is chosen as a salient representative of a part if the following condition is satisfied: Protrusion(r) = max {Protrusion(v)|∀v ∈ V, g(r, v) < thrp } , (3) where thrp represents the size of an observation window for finding a local maximum, with which we can control the range of influence of a protrusive stimulus. Note that since the observation windows of local maxima are subject to overlap, only one of them will be chosen as a salient representative. Fig. 2(b) shows the six salient representatives of parts chosen from the dinopet model.
C. Modeling the Boundary Strength as the Border Area Change Inspired by the concept of locale turning (as described in [1]), we propose to use the Dijkstra’s algorithm to explore how the surface evolves in the locale3 of a boundary. For the purpose of clarity, we propose to split the computational process for modeling the boundary strength into two steps: Step 1. Establishing the Candidate Locales: Given a salient x representative 0 1 of a part r, a set of candidate locales {Lr } = Lr , Lr , · · · is established using a modified version of the single-source Dijkstra’s algorithm. For the purpose of simplicity and later use, we drop the subscript r in the subsequent descriptions and denote the xth candidate locale as: Lx = {v|∀v ∈ V, x · e ≤ D(v) < (x + 1) · e} for x ∈ {0, . . . , l − 1}, (4) where D(v) returns the shortest distance from the source r to a dual vertex v, in terms of geodesic distance and protrusive difference. e represents the extent of a candidate locale, in which the boundary evolution is explored. l=bmaxv∈V D(v)/ec is the number of candidate locales established. Since each salient representative produces a set of candidate locales, each set of candidate locales will overlap one another. We, therefore, propose to establish a constrained set of candidate locales that always ends whenever a termination base is touched. The constrained set of candidate locales, L, is a set of consecutive locales satisfying the following constraint: m
