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 .
S u; v , p ointing outside of O . Then, let see Figure 1,
T u; v ; w = Su; v + nu; v w; 2
b e a trivariate function. The three-dimensional parametric space of T is the in nite b ox B :u; v ; w ,
u; v 2 [0; 1], w>0. T u; v ; w parameterizes the volumetric neighb orho o d of S u; v and exactly equates
with S for w =0.
De nition 1 Apoint P is considered ab ove base-surface S u; v if there exist u ;v ;w ,
0 0 0
w > 0, such that P = T u ;v ;w . Su ;v is denoted the supp ort-p osition of P .
0 0 0 0 0 0
Similarly,
De nition 2 Apoint P is considered b elow base-surface S u; v if there exist u ;v ;w ,
0 0 0
w < 0, such that P = T u ;v ;w . Su ;v is denoted the supp ort-p osition of P .
0 0 0 0 0 0
Point P maybeabove one supp ort-p osition of the base-surface and b elow another. Alternatively,a
p oint can b e ab ove or b elow two di erent supp ort-p ositions. Metho ds to detect and p ossibly eliminate
3 3
such singularities in FFD mappings and related applications are now known. Consider T : IR IR . T is
one-to-one or regular if and only if the determinant of the Jacobian of T never vanishes. This constraint
on the Jacobian was computed to detect singular conditions in FFDs in [12], and was symb olically derived
to extract the b oundary of a sweep surface op eration in [7].
In this work, we are mainly concerned with the volume near the surface, for small values of w , and
therefore such a singularity or dual supp ort-p osition is exp ected to b e uncommon. Furthermore, we seek
an algorithm that needs no inverse function evaluation, which is more ecient, and completely circumvents
the computation diculties that such a singularity might p ose.
Given a bivariate parametric displacement texture function D r;t= ur;t;vr;t;wr;t 2B, the
emb edding of D in T yields See Figure 2
T D = T ur;t;vr;t;wr;t
= S ur;t;vr;t + nur;t;vr;tw r;t: 3
Geometric Deformation-Displacement Maps Elb er 5
D r;t
T D
Figure 2: The mapping of the geometric displacement D r;t using the trivariate function T u; v ; w .
For the sp ecial case where u = r and v = t, D r;t is reduced to the traditional displacement mapping
technique, b ecoming an explicit height eld ab ove the r;t= u; v plane. Equation 3 now assumes the
form of see also Equation 1
T D = T r;t;wr;t
nr;twr;t; 4 = S r;t+
while w r;t serves as the height eld displacement along the direction of the normal. Hence, the traditional
displacement mapping technique is a sp ecial, explicit case of the parametric form presented in Equation 3.
What can b e gained by using the more general parametric displacement representation is the key question
we are ab out to examine as part of this work.
Trivariate functions were intro duced to the graphics community in the much cited work of [21]on
3 3
warping applications. Trivariate functions were employed as mappings T : IR ! IR . Bending, stretching,
and warping op erators were represented using trivariate functions that b ent, stretched or warp ed a subspace
3
of IR . The work of [21], that is also known as freeform deformations FFD, has many derivations such
as the Extended FFD [4], and non tensor pro duct FFD representations [18]. All these derivations share
the abilitytoemb ed a given ob ject in the domain of the trivariate, and that ob ject undergo es the same
non-linear transformation along with the entire subspace around the ob ject.
While an exceptionally general and p owerful ob ject mo di cation op erator, the dicult question has
always b een how can one intuitively derive the prop er trivariate function to p erform a certain warping
op eration. In this work, we b orrow these p owerful capabilities of the trivariate functions as a warping to ol
while the trivariate function is de ned to follow the three-dimensional domain ab ove the surface b oundary
of ob ject O , following Equation 2.
Twotyp es of parametric displacement maps, D r;t, could now b e employed, typ es we denote covering
displacement maps and casual displacement maps:
De nition 3 D r;t= ur;t;vr;t;wr;t, r;t 2 [0; 1] is considereda covering displace-
ment map if 8u ;v 2 [0; 1], 9r ;t 2 [0; 1] such that ur ;t = u , vr ;t = v .
0 0 0 0 0 0 0 0 0 0
Geometric Deformation-Displacement Maps Elb er 6
In other words, any displacement map that is onto, and hence spans the entire surface is a covering
displacement map. Clear examples of covering maps are the traditional bump and displacement maps. A
covering displacement map can b e discontinuous in w r;t and hence still present gaps in its range.
De nition 4 D r;t, r;t 2 [0; 1] is considereda casual displacement map if it is not a covering
displacement map.
If D r;tisnotacovering displacement map, the base-surface will not b e replaced by the displacement
map, but will merely b e augmented by it. That is, the displacement map is serving supplementary purp oses
only.Anobvious example of a casual texture is hair. We will present examples for b oth in Section 4.
0
De nition 5 Given a C continuous D r;t, r;t 2 [0; 1], and assume P and Q areboundary
points of D r;t.If
8P =x; 0;z 2 D; 9Q =x; 1;z 2 D and
8P =x; 1;z 2 D; 9Q =x; 0;z 2 D and
8P =0;y;z 2 D; 9Q =1;y;z 2 D and
8P =1;y;z 2 D; 9Q =0;y;z 2 D;
0
we say that D r;t is a periodic C continuous p erio dic displacement map.
A p erio dic displacement map can seamlessly tile the base-surface. Higher continuity constraints, such
k
as tangent plane continuity, could b e imp osed as well, providing a G continuous p erio dic displacement
map.
The advantage of using p erio dic displacement maps may b e found in the smaller size of the represen-
tation. Since the result is exp ected to contain large amounts of geometry, the b ene ts are evident. We
can express details in the surface only once and duplicate and tile these details as necessary. Clearly, b oth
covering and casual displacement maps could b e made p erio dic. Here, one can nd another advantage
k k 1
in using trivariate functions as displacement to ols. If S u; v is C and D r;tis C continuous, T D
k 1
is C continuous. In other words, the continuity of the DDM, even when tiles are used, is completely
k
governed by the base-surface, S , and its tiled texture, D . Clearly, for a tiled texture, D must b e a G
continuous p erio dic displacement map.
In Section 2.1 we present the comp osition computation of T D in the piecewise p olynomial and/or
rational domains. Then, in Section 2.2. we present the necessary computation of this comp osition, when
D is p olygonal.
Geometric Deformation-Displacement Maps Elb er 7
2.1 Polynomial/Rational Comp osition Algorithm
Assume the base-surface S u; v is a rational freeform surface. It is unfortunate that,
T u; v ; w = S u; v + nu; v w;
@Su;v @Su;v
@u @v
= S u; v + w; 5
@Su;v @Su;v
@u @v
n. One simple way of approximating a unit is not rational due to the normalization that is imp osed in
size normal eld would b e to simply normalize all the control p oints in the rational eld of nu; v =
@Su;v @Su;v
. Moreover, by applying re nement[2]tonu; v b efore this normalization of the control
@u @v
p oints takes place one can improve this approximation and, in fact, converge as closely as needed to a unit
size normal eld.
Another, more rigorous approachwas presented in [16]. A rational scalar eld mu; v is approximated
p
as mu; v = hnu; v ;nu; v i only to approximate nu; v as
nu; v
nu; v = : 6
mu; v
Both schemes converge to the exact unit normal, under re nement.
Let D r;t= ur;t;vr;t;wr;t T b e a p olynomial parametric displacement. Having a p olyno-
mial representation to T , T D r;t could b e precisely evaluated into a deformation-displacement surface,
as a comp osition. Here, we present the comp osition pro cess for the p olynomial domain in detail whereas
the extension to rationals is simple. Without loss of generality, assume S , D , and T are all in the B ezier
representation. Then, the DDM equals
n n n
u v w
X X X
T D = P B ur;tB v r;tB wr;t;
ij k i;n j;n k;n
u v w
i=0 j =0
k =0