網格參數化技術及其於三維著色系統之應用

行政院國家科學委員會專題研究計畫 成果報告

網格參數化技術及其於三維著色系統之應用

研究成果報告(精簡版)

計 畫 類 別 ： 個別型

計 畫 編 號 ： NSC 94-2213-E-009-088-

執 行 期 間 ： 94 年 08 月 01 日至 95 年 07 月 31 日

執 行 單 位 ： 國立交通大學資訊工程學系(所)

計 畫 主 持 人 ： 莊榮宏

計畫參與人員： 博士班研究生-兼任助理：陳治君、李汪曄、何丹期、簡民昇

碩士班研究生-兼任助理：林奕均

處 理 方 式 ： 本計畫可公開查詢

中 華 民 國 95 年 12 月 26 日

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ABSTRACT

Surface painting is a procedure that allows the users to paint onto a surface directly. The painting strokes are stored in a texture via surface parameterization techniques. In cur-rent surface painting systems, the underlying surface para-meterization is fixed during the painting process. Such a parameterization is not sensitive to the frequency spectrum of the color signal introduced by painting strokes. To asso-ciate the regions of higher color signal variation with more texture samples, we need to do the re-parameterization ac-cording to user’s strokes at interactive rates. We propose a re-parameterization scheme that is based on an iterative-optimization aiming to allocate more texture samples for regions of high signal variation and to perform at an inter-active rate as well.

Keywords

surface parameterization, surface painting, texture mapping

1.

INTRODUCTION

Surface painting(also called 3D painting) is a technique that allows the users to paint directly onto a 3D surface. If the discretization of the surface is fine enough, user can directly paints on the vertices of the surface. However, in general, the desired precision for the color is greater than the geometric detail of the model. Assuming that a surface is provided with a parameterization, it is convenient to store colors in the parameterized texture space. In current surface painting systems, the underlying mesh parameterization is predefined and fixed during the painting process. Such a parameterization is not sensitive to the frequency spectrum of the color signal as the result of painting strokes, and in consequence, may introduce distortion at arbitrary locations and waste texture space in areas of no stroke. Moreover, current surface painting systems parameterize the surface only based on the geometric aspects. Even though these systems provide tools allowing users to adjust the underlying parameterization, but it is not intuitive for normal users.

Most surface parameterization schemes assume no prior know-ledge of the signal, and take only surface’s geometry information into account. For surface painting, we want to allocate more texture samples in regions of greater signal detail by doing the re-parameterization on the fly according to the painting strokes. Moreover, the re-parameterization should be fast enough to achieve an interactive rate.

Our proposed method first derives an initial parameteriza-tion, and then, during the painting process, analyze the color signal frequency introduced by painting strokes, and utilizes an iterative optimization to do the re-parameterization to

interactively allocate more texture samples for regions with high color signal variation. The proposed method is simple to implement and works well for models with either low or large polygon count.

We describes a novel and simple framework of the re-param- eterization necessary for future surface painting sys-tems. Along the way to achieve this goal, we present the following contributions:

• Propose a modified signal metric L2

sthat measures the

geometry and signal stretch of a parameterization.

• Propose an interactive approach for the

re-parameter-ization aiming to increase the sampling ratio in regions with high signal variation.

2.

PREVIOUS WORK

Parameterization is a mapping from two dimension to higher dimension. Several schemes have been proposed that flatten a surface region and establish a parameterization over the last decade in computer graphics [3, 4, 7, 12, 13]. Sander et al. proposed a non-linear stretch that integrate the sum of squared singular values over the map [10]. We refer to this metric as geometry stretch. The parameterization is derived by a coarse-to-fine optimization scheme that min-imizes the geometry stretch over the map. Note that the resulting parameterization may encounter parametric crack problem.

Sander et al. developed a signal-stretch metric that com-bines both surface area and surface signal bandwidth [9]. It is shown that the stretch metric is related to SAE (Signal-Approximation Error) - the difference between a signal de-fined on the surface and its reconstruction. Sander’s sig-nal stretch can be seen as the extension of the geometry stretch. It is, however, not suitable for surface painting since it utilizes an expensive global optimization, and cannot sup-port the interactive re-parameterization required after each stroke is painted.

Hanrahan and Haeberli firstly proposed the concept of three dimensional surface painting, in which the color sig-nal is stored directly in mesh vertices [6]. Based on this method, the shading result is interpolated between mesh vertices, though we could not reveal rich texture detail. Igarashi and Cosgrove stored the paint strokes image that occurred for each pose as separate charts packed into a tex-ture atlas [8]. Mesh triangles affected by painting strokes are found and projected onto a two dimensional domain to form an atlas. Similarly, each subsequent stroke is stored in a new atlas. When the painting process complete, all the atlas are packed together to form the final texture atlas. The major disadvantage of the method is that a stroke that overlapping other strokes may appear in more than one at-las. In such cases, texture space may be wasted. Carr and Hart proposed a method aiming to derive a parameterization that is sensitive to signal distribution [2] base on their prior work, multi-resolution meshed atals (MMA) [1]. In their method, the mesh is first divided into several charts based on the method proposed by Sander et al. [11] to form the MMA tree hierarchy. During the surface painting process, all painting strokes are rendered into texture and then the stroke frequency distributed on the texture is analyzed using graphics hardware. After the analysis, an importance value computed from frequency analysis for previous strokes

Figure 1: The overall process of our method. is attached to each chart and the MMA hierarchy

(quater-nary tree) is re-balanced to generate a new parameterization. Each chart should consist of a quite large number of faces in order to reduce distortion introduced by the parameteri-zation. Moreover, the re-balancing is more significant with more number of charts. Therefore this method is suitable for meshes with large number of triangles.

3.

A RE-PARAMETERIZATION

FRAME-WORK FOR SURFACE PAINTING

To optimize the sampling resolution in parametric space, our basic idea is to increase the resolution of regions with high signal variation while decreasing the resolution of oth-ers. Our parameterization optimization framework for sur-face painting comprises the following steps as shown in Fig-ure 1:

1. Transform the closed-surface ΩT into an open-surface

Ω0

T using topological surgery, construct a global initial

parameterization for the surface mesh, and generate a base texture based on the parameterization.

2. Resample painting strokes into the base texture. Ana-lyze the signal frequency on base texture, generate im-portance map and geometry stretch map using graph-ics hardware.

3. Apply a uniform grid G underlying the parameteri-zation domain, in which each point of G is assigned a L2

s stretch value derived from importance map and

geometry stretch map, and then perform a two-stage

optimization to get an optimized uniform grid Gopt.

4. Re-parameterize Ω0

T according to the optimized

uni-form grid Gopt, and resample the painting strokes

ac-cording to the new parameterization.

3.1

Initial parameterization

To parameterize surface ΩT onto a planar domain, ΩT

should be topologically equivalent to a disk. If ΩT is a

closed-surface, we perform the topological surgery proposed in [5] to transform ΩT to an open-surface Ω0T that is

equiv-alent to a topological disk.

Although many parameterization techniques are adequate to derive a global initial parameterization, the one aiming to guarantee uniform sampling and preserve conformality structure of the input mesh is most preferable. Here, we use the method proposed by Yoshizawa et al. [13] due to its preferable properties and requires solving a simple, sparse linear system, which is usually handled in a matter of sec-onds using Conjugate Gradient solver with good precondi-tioning.

3.2

Stroke sampling

During painting process, we use the method proposed by Carr et al. [2] to sample painting stroke into texture. Each paint stroke applied in the same object pose (i.e. modelview coordinates of the model) is rendered directly into base tex-ture map using graphics hardware. For this task, we need a stroke buffer for storing the painting data and a depth buffer for the depth of current object pose.

The resampling is done by a vertex shader and a frag-ment shader. The vertex shader transforms the world space position into model view coordinates and then swaps each vertex’s model view coordinates with its texture coordinates.

(a) (b) Figure 2: Problems of four-tap filter.

The fragment shader is applied to render the new base tex-ture by taking the stroke buffer, depth buffer, and the origi-nal base texture as input. The alpha channel in stroke buffer represents the existence of paint strokes to ensure that only the strokes can overwrite the existing base texture. The depth buffer is used to prevent paint being applied to invisi-ble portions of the model. This process is performed for the stroke painted at each pose.

3.3

Importance map and geometry stretch map

To analyze the base texture for finding regions that re-quires additional samples, a four-tap gradient magnitude filter is used in [2] to find undersampled regions. The four-tap filter fetches four samples from the input texture, and outputs the result in half resolution. For the four-tap gradi-ent magnitude filter, some gradigradi-ent features will be missed. For example, as shown in Figure 2, each red rectangle repre-sents 4 pixels on the texture, and we detected the gradient in s-direction of paint (a), but not in paint (b).

Here we modify previous four-tap filter. For each pixel on the base texture, we calculated its magnitude of the gradi-ent using fragmgradi-ent shader arithmetic by cgradi-entral difference. Actually, this is the Sobel Filter in the filed of image process-ing. The filter is applied for each pixel of the base texture, therefore the output image is the same resolution as the base texture. Figure 3 demonstrates the result of two filters which shows that our modified filter is more accurate than four-tap filter.

Besides the importance map, another map called geom-etry stretch map is also computed. This geometric stretch

map stores L2_{stretch value for each face on parametric}

do-main as shown in Figure 1(g). We normalize the value of geometric stretch of each face to lie between 0 and 1. Next, we render the mesh on parametric domain using the nor-malized geometric stretch value as the color of the face.

3.4

The

L2

s

stretch

After the generation of importance map and geometry

stretch map, the L2

s stretch is derived from these two maps.

As mentioned in [9], the signal stretch can have zero gradient since the signal may be locally constant on a region of the surface. Therefore, a tiny fraction of geometry stretch is

added into the energy function to be minimized. The L2

s

stretch is defined as follows:

L2 s(s, t) =  1 − L2_{(s, t) , if E} h(s, t) = 0 1 − (α · L2_{(s, t) + β · E} h(s, t)) , otherwise.

where Eh(s, t) is the signal stretch proposed by Sander et al.

[9], and L2_{(s, t) is the geometry stretch [10]. The two values}

Figure 3: Four-tap filter and our filter. are obtained from importance map and geometry stretch map, respectively. In the region with signal variation, we use the weighted geometric stretch and signal stretch as in [9]. Otherwise, in the region without signal variation, we purely take the geometry stretch into account to prevent

undersampling in regions with no signal variation. The L2

s

stretch could be considered as the extension of signal stretch.

3.5

Iterative optimization based on uniform

grid

For interactive applications, the parameterization proposed by Sander et al. [9] has two major problems when it is ap-plied to surface painting systems. First, since the signal introduced by painting strokes is not constant over the tri-angle, numerical integration is used to compute the signal stretch on each triangle. All the mesh triangles are subdi-vided into 64 sub-triangles and the signal stretch are eval-uated at all the vertices. The second drawback is that the optimization process proposed by Sander et al. is a non-linear, global optimization. As a result, the parameteriza-tion is expensive and therefore not suitable for interactive surface painting applications.

To reduce the cost of computing signal stretch, instead of subdividing each triangle, we derive the signal stretch on parametric domain. We apply an N × N uniform grid G to the parametric domain in which the initial parameterization lies as shown in Figure 4. These grid points, rather than the

mapping of mesh vertices, are used to sample L2

s stretch on

parametric domain, that is, we compute L2

s stretch for each

grid point. Such an approach allows us to control the sam-pling resolution. Moreover, the grid is used to be the target for stretch optimization. By doing this, the computational complexity of performing optimization will be dependent on the resolution of the grid, rather than the mesh.

We then optimize G by the following steps:

1. For each point N ∈ G, derive L2

s(N ) from importance

map and geometry stretch map using graphics hard-ware.

2. For each interior point Ni∈ G in turn, compute Nfi= P N0∈1-ring of NiL 2 s(N0) · N0 P N0∈1-ring of N_iL2s(N0) , set Ni= fNi.

3. Repeat 1 and 2 until kfNi− Nik < ² for every i.

(a) The face model with a painting stroke.

(b) Initial

parame-terization. (c) Initial uniformgrid G.

(d) Importance

map. (e) Stretch on sam-ple points (64 × 64).

(f) Optimized sam-ple points Gopt.

Figure 4: The optimization result base on one iter-ation.

Figure 4(a) illustrates the face model with a red painting stroke and the resulting importance map is shown in Figure 4(d). A 64 × 64 uniform grid G is applied to the parametric

domain, where each sample point is assigned a L2

s stretch

value as shown in Figure 4(e) (we take only signal stretch into account in this case). Figure 4(f) shows the optimized

grid Gopt, where the sample points are more sparse in the

regions with signal variation. The optimization procedure on the grid points is illustrated in Figure 5. Figure 5(a) depicts the grid points and the corresponding parametric

domain with signal distributed. Since the L2

s stretch values

of p2, p7 and p12are smaller than that of p0, p5and p10, p1, p6 and p11 are moved toward p0, p5 and p10. Similarly, p3, p8 and p13 are moved toward p4, p9 and p14; as shown in

Figure 5(b)(c). 4 8 14 2 7 6 5 9 10 11 12 13 0 1 3

(a) Initial grid.

14 8 4 3 7 6 5 9 10 11 12 13 0 1 2 (b) The optimiza-tion process. 14 4 8 13 1 6 11 0 2 7 5 9 10 12 3 (c) The optimized grid.

Figure 5: Chart diagram of the optimization

process.

The optimization procedure is an iterative optimization process, in which the local optimization optimizes a grid

point in one iteration. After the optimization, we will get

an optimized uniform grid Gopt. On Gopt, grid points will

become dense in the regions with high L2

sstretch (lower

sig-nal variation), and sparse otherwise. After the optimiza-tion process, the underlying parameterizaoptimiza-tion will be re-computed by barycentric interpolation according to the op-timized grid points as described in next section.

3.6

Re-parameterization

After optimizing the initial uniform sample points G, we re-parameterize the parameterization by the barycentric in-terpolation based on the optimized uniform sample points

Gopt. For each vertex vj∈ VI, let Nj0, Nj1, Nj2and Nj3be

the sample points of the cell that contains vj. Barycentric

coordinates w0, w1, w2 and w3 are derived such that vj=

3

X

i=0

wi· Nji.

The new position of vj will be

vjopt= 3 X i=0 wi· Noptji , where Nj0 opt, N j1 opt, N j2 optand N j3

optare the homologous points

of Nj0_{, N}j1_{, N}j2 _{and N}j3 _{in Gopt.}

3.7

Stroke resampling over optimized

para-meterization

Finally, we resample the base texture based on the opti-mized parameterization. The sampling process is similar to the method mentioned in section 3.2. The only difference is that now we have two texture coordinates, i.e. parameter values tprevand toptfor each vertex, which are derived from

initial parameterization φ and optimized parameterization

φopt, respectively. As described in section 3.2, we first swap

each vertex’s model view coordinates with its current

tex-ture coordinates toptin vertex shader, and then we resample

painting strokes to form a new base texture. The resampling procedure here consists of two step. The first step resamples current painting strokes stored in stroke buffer; step two re-samples previous painting strokes stored in the previous base

texture. Therefore current model view coordinates and topt

are used to sample current stroke from stroke buffer and

tprev is used to sample previous stroke form previous base

texture.

3.8

Two-stage re-parameterization framework

Compared to Sander’s signal-specialized parameterization [9], the proposed framework tends to be a local optimization process. Figure 6 shows the re-parameterization result us-ing a 256x256 uniform sample points. We can see that the relaxation of sample points is bounded inside the cell it lies. As shown by the red arrow in Figure 6, there should be less sample space in these regions with lower signal gradi-ent. However, the movement of the sample points in these

regions is not much due to the fact that the L2

s stretch of

these points are almost the same. Therefore the relaxation works well in the regions with high gradient, but may not work well in the other regions. To solve this problem, a two stage optimization framework is used instead of the single stage optimization.

In the two-stage optimization, we expect that the first stage diminishes the texture sample space in region with

lower signal gradient and the second stage magnifies the tex-ture space in regions with high signal gradient. To achieve this goal, a lower resolution uniform grid is used in the first stage and a high resolution grid in the second stage. Figure 7 illustrates the optimization result using a high resolution grid. As shown in Figure 7(c), only these sample points near

the regions of low L2

s stretch value (high signal variation)

will be moved after the iterative optimization. Other sample points will remain fixed in other regions where neighboring

points have the same L2

s stretch value. Figure 7(d) shows

the final result of the optimization. The regions with lower signal stretch are expected to obtain less texture space. Ap-parently, optimization using a high resolution grid does not work well for this purpose, see the comparison highlighted by the blue circle in Figure 7(a) and Figure 7(d).

The optimization resulting from using a lower resolution grid will have more convergence effect in the regions of high

L2

s stretch (lower signal variation), and allocate less texture

space in these regions. See the comparison shown in Figure 8(a) and Figure 8(d).

(a) Parasaur model(b) Optimized

uni-form sample

points

(c) Base texture

Figure 6: Parasaur model : single stage optimization using 256x256 uniform sample points

(a) Initial grid. (b) The optimization

process.

(c) The relaxed grid points. (d) The optimization result. Figure 7: High resolution uniform grid points. Figure 9(b) shows that the sample points of 16 × 16 res-olution int the first stage and Figure 9(d) is the result of using the sample points of 256 × 256 resolution in the sec-ond stage. We see that the texture space in regions of lower signal gradient is diminished in stage one; as shown in Figure 9(c), while in stage two, more texture space in the regions of high signal gradient are allocated; see Figure 9(e). Fig-ure 10 shows the result of single stage optimization and two stage optimization for comparison. Obviously, the texture

(a) Initial grid. (b) The optimization

process.

(c) The relaxed grid points. (d) The optimization result. Figure 8: Low resolution uniform grid points. space is used more efficiently using the two stage optimiza-tion method, especially in the regions of lower signal gra-dient, see the comparison highlighted by the red arrows in Figure 10.

(a) Parasaur model.(b) Optimized 16 × 16 grid in the first stage. (c) Base texture (Stage 1). (d) Optimized 256 × 256 grid in the second stage.

(e) Base texture

(Stage 2).

Figure 9: Parasaur model : Two-stage optimization using 16 × 16 and 256 × 256 grids.

4.

RESULTS AND PERFORMANCE

ANALY-SIS

All results are performed with a AMD Athlon64 3000+ PC, 512 MB RAM and an NVIDIA GeForce 6800 graphics card. It is running Windows XP with NVIDIA Cg 1.3 com-piler, vp40 vertex shader profile and fp40 fragment shader profile. We use the pBuffer extension for efficient texture rendering.

We compare our result with that based on static para-meterization in current surface painting systems. Figure 11 and Figure 12 show the painting result of our surface

paint-(a) Single stage

optimiza-tion (b) Two stage optimization

Figure 10: Comparison of single and two stage op-timization

ing system on the venus model. The left columns show the result of current surface painting systems, i.e. with fixed underlying parameterization. The right columns show the result of our two-stage optimization process. Our method depicts better texturing quality than that for current surface painting systems.

Figure 13 and Figure 14 demonstrate the painting results of other models. Aliasing occurs in undersampling regions and our method alleviate this problem efficiently.

The strokes of all the results shown in Figure 11 to Figure 14 are painted manually. Figure 15 shows the result where four images is texture mapped to simulate painting strokes. The artifact, blur, occurs due to the fact that texture is undersampled using fixed parameterization as shown in the left column of Figure 15. The right column shows that the result of two-stage optimization is much more better.

Table 1 lists the computation time for initial parameter-ization, two-stage optimization and re-parameterization oc-curs during surface painting process. Because the optimiza-tion procedure is done on parametric domain, the compu-tation cost of two-stage optimization is independent on the face number of input model. The timing required by the two-stage optimization is reasonable for the interactive ap-plication of surface painting systems. Figure 16 shows the optimization time after each stroke is applied on the tricer-atops model. Since the geometry stretch is minimized in the first optimization process, the timing is higher than succeed-ing optimizations.

The signal-specialized parameterization proposed by Sander et al.[9] is thought to be the state-of-art work in mesh pa-rameterization which is sensitive to surface signal. We com-pare the parameterization performance between our two-stage optimization framework and signal-specialized para-meterization.

Figure 17(a)(b) show the result of signal-specialized para-meterization using 2048 × 2048 and 128 × 128 texture maps, respectively. Figure 17(c)(d) shows the result of our two-stage optimization under two different texture map resolu-tions. Our result is pretty good under resolution of 256x256 and still fine under resolution of 128x128.

5.

CONCLUSION

We have proposed a rapidly re-parameterization frame-work for surface painting which redistributes texture sample space according to the surface signal variation. A two stage

Figure 11: Painting results of the venus model. Re-sult of a fixed-parameterization (left column) and the result of our two-stage optimized parameteriza-tion (right column).

uniform grid optimization framework is proposed which di-minished sample space in lower gradient regions in stage one and magnifies sample space for high gradient regions in stage two. In addition, this two stage optimization frame-work is suitable for interactive use required by surface paint-ing. For the optimization process, we derived the modified

L2_{metric denoted as L}2

s. The L2smetric takes signal stretch

into account in the regions of signal variation and combines geometry stretch in the regions without signal variation.

Some potential future work are listed as follows:

• Better stroke sampling method The stroke

sam-pling method[2] based on graphics hardware is simple and fast for interactive use. However the result is bad when the resampling was done under either magnifi-cation or minifimagnifi-cation. Perhaps some filter on image space could alleviate this problem.

• Better parameterization metric Since the same

value of geometry and signal stretch does not imply

the equal significance, our proposed L2

s stretch

ac-tially works in a heuristic manner. A good study on the weighted relationship between geometry and sig-nal stretch may enhance the theoretical background of our method. Furthermore, a better metric, especially the one which is more sensitive for the anisotropical distribution of surface signal on parametric domain is a chance to improve the overall quality.

• Hierarchical optimization Optimization based on

adaptive sample points can be utilized to improve the performance. In our two-stage optimization frame-work, the sample points are uniformly distributed on

parametric domain at each step. To use the

Model face Init-param. Two-stage Optimization Re-param. range avg. venus 1396 0.625 0.718 - 3.843 1.784 0.016 triceratops 5660 3.234 0.625 - 3.156 1.739 0.063 face 1162 0.5 1.531 - 3.828 1.690 0.016 horse 7500 4.906 1.031 - 3.125 1.375 0.078

Table 1: Statistics of initial parameterization, two-stage optimization and re-parameterization time (sec.) for four different models.

Figure 12: Back-view of the painting results of the

venus model. Result of a fixed-parameterization

(left column) and the result of our two-stage op-timized parameterization (right column).

points on the regions of high signal gradient to accu-rately grab the signal variation. Less sample points are distributed on the regions of lower signal gradient, thus these regions will be converged more quickly. To achieve the goal, a hierarchy architecture of uniform grid is required to maintain the different resolution of grid points. For sampling, there are two major prob-lems of the hierarchical method. The first one is the determination of high gradient region and lower gra-dient region. A two-pass method will be practical to accomplish this. The second problem is that a

theoret-ical and efficient method to propagate the L2

s stretch

from high resolution grid pints to lower resolution grid points is required. In addition to the problems of sam-pling, an efficient optimization algorithm for the hier-archical gird architecture is also required.

6.

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Figure 13: Painting results of the triceratops model. Result of a fixed-parameterization (left column) and the result of our two-stage optimized parameteriza-tion (right column).

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Figure 15: Four images are texture mapped to

simulate painting strokes. Result of a

fixed-parameterization (left column) and the result of our two-stage optimized parameterization (right col-umn).

Figure 16: The optimization process of triceratops model.

(a) Signal-specialized (b) Signal-specialized

(c) Two-stage

optimiza-tion (d) Two-stagetion

optimiza-Figure 17: Comparison of our result with signal specialized parameterization under different texture map resolutions.