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行政院國家科學委員會補助專題研究計畫成果報告
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計畫類別：■個別型計畫 □整合型計畫
計畫編號：NSC 89－2213－E－006－059－
執行期間： 88 年 08 月 01 日至 89 年 07 月 31 日
計畫主持人：謝 中 奇
共同主持人：
本成果報告包括以下應繳交之附件：
□赴國外出差或研習心得報告一份
□赴大陸地區出差或研習心得報告一份
□出席國際學術會議心得報告及發表之論文各一份
□國際合作研究計畫國外研究報告書一份
執行單位：國立成功大學工業管理科學系
中 華 民 國 89 年 10 月 31 日
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行政院國家科學委員會專題研究計畫成果報告
國科會專題研究計畫成果報告撰寫格式說明
Pr epar ation of NSC Pr oject Repor ts
計畫編號：NSC 89－2213－E－006－059－
執行期限：88 年 08 月 01 日至 89 年 07 月 31 日
主持人：謝中奇 國立成功大學工業管理科學系
計畫參與人員：丘國昌 國立成功大學工管所
鄭勝銘 國立成功大學工管所
一、中文摘要 虛擬立體成像是一種在虛擬環境中利 用物體之二維投影來呈現物體本身的技 術，其中二維投影之取得通常是依據一固 定參考點從各個角度對物體投射而產生。 由於二維投影成像快速及仿真效果極佳， 在數位化之物體表現已占有重要地位。直 覺上，投影數目的增加會提高物體呈現的 解析度，但此舉無法保証物體的能見度 (visibility) 是最大的。因此，本研究之重點 在求得物體在虛擬立體成像上最大之能見 度，以提高物體之表現力；首將物體的能 見度映成在單位球體上以產生能見度圖 (visibility map) ，從而建立優化模式，並發 展演算法以求得最佳投影集合。 關鍵詞：虛擬立體成像、物體呈現、二維 投影、能見度 AbstractVirtual holography is a methodology which utilizes a set of two-dimensional projections to represent an object in a virtual environment where the 2D projection set is usually taken from all viewpoints with respect to a fixed reference point. Though more projections tend to reveal more details of an object, the full or maximum visibility of the object is not guaranteed. In this study, the visibility of an object is mapped onto a unit sphere and a minimal number of uniform grids, representing viewpoints, which span over the decomposed visibility map is sought. This study will enable better object
representation in virtual holography, hence having significant contributions in many practical applications such as in digital museums, mobile presentations and virtual manufacturing.
Keywords: object representation,
two-dimensional projections, visibility, virtual holography
二、緣由與目的
Realistic representation and fast rendering of a virtual object are two of the most important aspects in the context of virtual environment. A new methodology, called virtual holography, is thus developed
for such purposes that utilizes the two-dimensional project set of an object. As the projection set is taken uniformly from all viewpoints along circular paths with respect to a fixed reference point, the representation is simple and applicable to either a computer-aided design (CAD) model or a physical object.
Though more projections from different viewpoints tend to reveal more details of an object, keeping a large number of projections is not feasible in considerations of storage space and network bandwidth. One intuitive way is to compress the projection data so that a few more projections are permitted without increasing the storage size [1][2][3][4]. In [3], for example, data compression is done by deploying vector quantization and entropy coding. Yet, the resulting projection set might not be able to reveal the full or maximum visibility of an object that is
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important from users’ perspective. It is thus motivated in this study that the viewpoints of the object need to be constructed such that maximum visibility can be achieved [5][6][7][8][9]. Clearly, data compression techniques can then be deployed to reduce the storage size of the representation.
It is assumed in this study that a CAD model has two degrees of freedom, i.e., a CAD model can only be rotated either vertically or horizontally. The rest of the paper is organized as follows. In Section 3, the approach of attaining the visibility map of a CAD model is introduced, followed by the representation of the viewpoints. An optimization formulation for obtaining a minimal number of viewpoints is proposed in Section 4. Section 5 discusses the proposed algorithm and concludes with a brief summary.
三、能見度 (Visibility)
As a visibility map (V-Map) is closely related to a Gaussian map (G-Map), we will introduce the notion of G-Map first.
A G-Map of the object is the region on the surface of the unit sphere that represents each normal to the surface of an object. Since the normals give the directions in which the point is visible from infinity, a V-Map of the object can be attained by computing the set of points, each of which differs from every normal on the surface by at most 90 degrees. An efficient computation of V-Map is as follows [5]:
1. Compute the G-Map of an object 2. Compute the spherical convex hull,
SCH(G-Map), of the G-Map
3. Take the dual of SCH(G-Map), which is the desired V-Map
In Step 2, a set of points on the sphere is convex if for any two points in the set, the (shorter) great arc connecting them lies entirely in the set. To further speed up the computation of the V-Map, some numerical approximation techniques can be deployed.
四、最佳化模式 (Optimization)
Given the V-Map of an object computed in the previous section, we formulate an optimization problem which aims to minimize the number of uniform grids, representing the viewpoints, on the two-dimensional spherical coordinate systems such that the disjoint regions of the V-Map can be covered, hence achieving maximal visibility. The algorithm is outlined as follows:
1. For each allowable size of uniform grids in one dimension (starting from the smallest integral value to the largest one)
1.1 Find the optimal orientation such that the V-Map is covered maximally.
1.2 If every region of the V-Map is covered, stop the iteration and return the best orientation and the grid size; otherwise, increase the grid size by one and repeat Step 1.
In Step 1.1, some heuristic techniques can be deployed to find the optimal orientation since the underlying solution space is highly nonlinear. In this study, we utilize genetic algorithms to find the optimal orientation for each grid size.
五、討論 (Discussion)
We have proposed an algorithm for obtaining the representation of an object that gives the maximal visibility. Our study has significant contribution on image-based object representation and thus finds many applications such as in digital museums, E-commerce and mobile presentation. Nevertheless, some improvements are plausible and will be considered in the future study. For instance, a hybrid algorithm that combines genetic algorithms and simulated annealing might be a more efficient means of achieving global optimal solutions.
六、參考文獻 (Refer ences)
[1] S.E. Chen, “QuickTime VR – An image-based approach to virtual environment,” Proc. SIGGRAPH ’95, pp. 29-38.
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[2] S.J. Gortler, R. Grzeszczuk, R. Szeliski and M.F. Cohen, “The lumigraph,” Proc. SIGGRAPH ’96, pp. 11-20.
[3] M. Lovy and P. Hanrahan, “Light field rendering,” Proc. SIGGRAPH ’96, pp. 31-42.
[4] S.M. Seitz and C.R. Dyer, “Viewing
morphing,” Proc. SIGGRAPH ’96, pp. 21-30.
[5] T.C. Woo and B.F. von Turkovich, “Visibility map and its applications to numerical control,” Annals of CIRP, Vol. 39, No. 1, pp. 451-454.
[6] L.L. Chen and T.C. Woo, “Computational geometry on the spher with application to automated machining,” ASME Trans. J. Mechanical Design, Vol. 114, June 1992, pp. 288-295.
[7] L.L. Chen, S.Y. Chou and T.C. Woo,
“Partial visibility for selecting a parting direction in mold and die design,” J. of Manufacturing Systems, Vol. 14, No. 5, 1995, pp. 319-330.
[8] L.L. Chen, S.Y. Chou and T.C. Woo,
“Parting directions for mold and die design,” Computer-Aided Design, Vol. 25, No. 3, 1993, pp. 233-239.
[9] K. Tang and T.C. Woo, “Maximum
intersection of spherical polygons and workpiece orientation for 4- and 5-axis machining,” ASME Trans. J. Mechanical Design, Vol. 114, September 1992, pp. 477-485.
[10] G. Elber and E. Zussman, “Cone visibility decomposition of freeform surfaces,” Computer-Aided Design, Vol. 30, No. 4, 1998, pp. 315-320.
[11] C.C. Hsieh, “Smoothing representation in virtual holography,” NSC project #87-2218-E-324-005.
