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Geometric Deformation-Displacement Maps

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Geometric Deformation-Displacement Maps Geometric Deformation-Displacement Maps Gershon Elb er Computer Science Department Technion Haifa 32000, Israel Email: [email protected] Abstract Texture mapping, bump mapping, and displacement maps are central instruments in computer graph- ics aiming to achieve photo-realistic renderings. In all these techniques, the mapping is typically one- to-one and a single surface lo cation is assigned a single texture color, normal, or displacement. Other sp ecialized techniques have also b een develop ed for the rendering of supplementary surface details such as fur, hair, or scales. This work presents an extended view of these pro cedures and allows one to precisely assign a single surface lo cation with few continuously deformed displacements, each with p ossibly di erent texture color or normal, employing trivariate functions in a similar way to FFDs. As a consequence, arbitrary regular geometry could b e employed as part of the presented scheme as supplementary surface texture details. This work also augments recent results on texturing and parameterization of surfaces of arbitrary top ologies by providing more exible control over the phase of texture mo deling. By completely and continuously parameterizing the space ab ove the surface of the ob ject as a trivari- ate vector function, we are able, in this work, to not only control the mapping of the texture on the surface but also to control this mapping in the volume surrounding the surface. Additional Key Words and Phrases: Curves & Surfaces, Texture Mapping, Bump Mapping, Trivariates, Deformations, Warping, FFD. 1 Intro duction Texture mapping [11] is an essential apparatus in computer graphics that increases the photo-realism of any synthetic rendering scheme. A mapping is established from any p oint on the surface of the rendered ob ject into the texture space. The surface p oint is then assigned the color of the resp ective lo cation found in the texture space. In this pap er we will b e concentratating on the texture mapping techniques that relate to shap e alteration. The bump map [1, 11] is a rst attempt to mo dulate the normals of the vertices of the rendered ob jects instead of their assigned colors. Here also, at each surface lo cation, we assign a small p erturbation to the normal, based on some mapping from that surface lo cation to some normal p erturbation texture function. The bump mapping technique is highly successful in conveying a bumpy shap e while the geometry remains smo oth. The illusion created by the bump mapping is quite convincing. The researchwas supp orted in part by the Fund for Promotion of Research at the Technion, I IT, Haifa, Israel. 1 Geometric Deformation-Displacement Maps Elb er 2 Yet, in silhouetted areas the surface remains smo oth and no casting of self shadows due to this bumpiness can o ccur, since the geometry itself is not mo di ed. Displacement maps [3,11] take the next natural step and allow actual mo di cation of the surface. Typically, this map is represented as a height eld that mo dulates the amount each surface lo cation is elevated in one direction, typically the normal direction. Let S u; v b e the surface to displace and let nu; v b e its unit normal eld. Then, given a displacement as a scalar height eld, du; v , the geometry of the new, mo dulated, surface equals, S u; v = Su; v + nu; v du; v : 1 d The computation or the approximation of the normal eld of S u; v can b e quite involved and is, in d general, considered a computationally complex task that hinders the use of such techniques in real time. @S @S n can serve as a basis Since the partials of S u; v span the tangent plane of S , if regular, ; ; @u @v 3 for IR and hence can b e used to prescrib e a displacement in an arbitrary direction. Nonetheless, typically only one displacement direction can b e prescrib ed p er u; v surface lo cation. This displacement is, in most cases, in the normal direction. In [15], an approach that intro duces texels is prop osed. Quoting from [15], \a texel is a three dimensional array that holds the visual prop erties of a collection of micro-surfaces". The geometry is replaced by its visual prop erties, gathering the scattering and re ectance functions. It is this concept that attempts to mo del the volume ab ove the surface that triggered this work. Trilinear texels are also used in [19]. Due to the linearity of the texel, one surface lo cation is assigned one displacement direction. The texels are continuous but their normals are not. Similarly, b ecause the base of the texel is a bilinear form, it is imp ossible to precisely follow non linear p olynomial surfaces. One advantage of the approachof[15,19] stems from its abilitytoray trace the scene without the need to duplicate the geometry b ehind the texture function, remapping the general ray, as it is traced, into the canonical texel. This approach can also handle texture mapping that is one-to-many. That is, a single surface lo cation can b e mapp ed to several p oints ab ove the surface. In this work, we further extend this approachinto non linear trivariate functions, as an alternativescheme to b etter manage the space ab ove the surface. Three-dimensional supplementary details were also added to the surface of the ob ject using other means. Typically, this texturing pro cess could b e divided into two. In the rst, the texture placement phase, the lo cations at which the texture elements are to b e placed are determined. A typical approach to texture placement employs some kind of a parameterization. Then, in the second, the texturemodeling phase, the texture elements are lo cally molded to t their nal shap e. In [10], an attempt was made at creating three-dimensional details, such as scales or thorns, over the surface. Natural cellular developmentwas simulated in [10]toward texture generators, for the placement Geometric Deformation-Displacement Maps Elb er 3 phase of the individual elements. The texture mo deling phase in [10] exploited a geometric mo deler that is parametric. Cellular texture generators as a to ol for texture placementhave b een receiving signi cant attention recently, and for example, in [17] brick textures are investigated for architectural mo dels. The question of texture placement is of ma jor concern in irregular p olygonal meshes of arbitrary top ology, as is the question of seamless tiling of a rep eatable texture over these domains. These questions have, of late, captured the attention of researchers. For example, [24] employs a vector eld over the geometry to achieve this prop er placement and [25] derives a parameterization and lo cal tangents for each vertex of the mesh, attempting to reach the same goal. Simulation of reaction-di usion [23, 26] is another scheme used to create textures for surfaces with no parameterization. In this work, we extend the notion of displacement maps and relax several of its constraints. By using non linear p olynomial trivariate functions to derive the texture mapping function, continuous normal elds may b e created. The presented approach employs the same to ol as used in freeform deformations FFD [21, 4] and hence the resulting texture undergo es a displacement transformation as well as a deformation that 3 co erces it to follow the precise shap e of the underlying surface. As a consequence, any geometry in IR could b e employed as supplementary detail texture, making the texture mo deling phase as general as it can b e. Moreover, any ob ject can serve as the geometric deformation-displacement map DDM of itself, forming a complete closure. [10, 17, 23,24, 25, 26] are samples from the state of art in this area of texture placement, which seeks a prop er distribution of the texture elements on the surface, whether a square texture tile, a thorn, a brick or hair. Herein, we augment these techniques by providing extended texture mo deling capabilities. Having a prop er texture placement, we allow the texture function to fully and smo othly encompass the third dimension ab ove the surface. That is, given a surface lo cation to place the texture along, the DDM maps the surface lo cation to more than one textured or displaced p oint. Moreover, byhaving a complete and continuous parameterization of the surface domain and ab ove it, we are able to fully adapt the displaced texture to the shap e of the surface as well as o er an ecient estimation scheme for the normals of the deformed and displaced geometry. This pap er is organized as follows. In Section 2, we p ortray the prop osed displacement approach, and in Section 3, we present our ecient estimation scheme for the normals of the displaced surface. In Section 4, some examples are presented and nally,we conclude in Section 5. 2 Geometric Displacement Algorithm Consider O , a given ob ject to b e textured and let S u; v , u; v 2 [0; 1] b e a regular parametric surface n u; v b e the unit normal eld of on the b oundary of O . Denote S u; v as the base-surface.Further, let Geometric Deformation-Displacement Maps Elb er 4 w nu; v w T u; v ; w B v u S u; v Figure 1: The trivariate T u; v ; w function is de ned above surface S u; v .
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