Faceting Artifact Analysis for Computer Graphics

Transcription

1 Faceting Artifact Analysis for Computer Graphics Lijun Qu Gary Meyer Department of Computer Science and Engineering and Digital Technology Center University of Minnesota Abstract The faceting signal, defined in this paper as the difference signal between a rendering of the original geometric model and a simplified version of the geometric model, is responsible for the faceting artifacts commonly observed in the renderings of coarse geometric models. We analyze the source of the faceting artifacts and develop a perceptual metric for the visibility of the faceting signal. To illustrate the utility of the faceting signal, we describe two applications. First, we present a perceptually guided level of detail system that accounts for lighting, texturing, distance and other factors that affect the performance of the human visual system. Second, we calculate the lower bound contrast of a texture that can be used to hide the faceting artifacts by using the masking properties of the human visual system. 1 Introduction Faceting artifacts due to tessellation are commonly present in computer generated pictures. To combat this problem, the computer graphics community has taken two approaches. First, geometric models of higher resolution can be used. The faceting artifacts will disappear as the screen projection of each element of the geometric model reaches the scale of each pixel in the image. Second, shading models superior to flat shading can be used. For example, Phong shading decreases the faceting appearance because it includes a better normal approximation as part of the shading calculation. However, both approaches require more computation and thus penalize the frame rate. Visual perception provides another avenue for solving this problem without compromising the rendering speed. This solution can be accomplished by carefully selecting the surface texture so as to obscure the faceting artifacts. Visual masking has been used to mask certain artifacts in computer graphics for a long time. Bolin and Meyer [3] show that fewer samples are required in a ray tracing program when the image contains many frequencies, because the image content can hide the sampling noise. Ferwerda et al. [9] present a model of visual masking for computer graphics and show that surface texture can be used to mask faceting artifacts. However, most approaches use the principles of visual masking without identifying the artifact signal that is to be hidden. Knowing the artifact signal is critical for the application of visual masking. In order to apply the principle of visual masking, we need to understand the frequency composition of both the artifact signal and the signal that serves as the masker. In this paper, we develop the concept of the faceting signal, defined as the difference signal between two images, one produced with the polygonal representation and another made with the ideal representation of a geometric model. The faceted appearance in a computer generated picture can be characterized by the faceting signal. It is important to understand the nature of the faceting signal, for example, its contrast, amplitude and frequency distribution. This information can reveal solutions to the faceting problem. Even though it is very important to understand the nature of the faceting signal, it remains an open research topic. In this paper, we develop the concept of the faceting signal, and create a perceptual metric for determining the visibility of the faceting signal. We present two applications that can benefit from this analysis. First, we present a level of detail system in which a level of detail model is selected for display based on the visibility of its faceting signal. Second, we predict the minimum contrast necessary for a texture to successfully hide the faceting signal for a geometric model. 1.1 Contributions The contributions of the paper are the following: Develop the concept of the faceting signal. Analyze the source that affects the faceting signal. Develop a perceptual metric for determining the visibility of the faceting signal.

2 Develop a level of detail management system guided by the faceting signal. Develop a surface texturing algorithm directed by the faceting signal. 2 Prior Work Flat Shading Gouraud Shading Phong Shading Error analysis in computer graphics: The computer graphics community often has to make a compromise between rendering speed and image quality. Not only does error analysis provide an estimate to the error in a computer generated image, it also often sheds light on how to control the error. Crow [7] analyzed the aliasing artifacts in computer generated images and presented several solutions to the problem of anti-aliasing. Error analysis has also been proposed for global illumination algorithms. Lischinski et al. [15] described a method for determining a posteriori bounds and estimates for local and total errors in radiosity solutions. Arvo et al. [1] proposed a framework for estimating errors in global illumination algorithms. Visual difference metrics (VDMs): Visual difference metrics [8, 16] were originally developed in the imaging science industry to evaluate the effects of changing parameters in an imaging system. Several metrics incorporating most aspects of VDMs were developed for realistic image synthesis [9, 5]. These metrics have proven to be very useful for computer graphics applications. Lately, Ramanarayanan et al. [22] propose the concept of visual equivalence and develop a metric for visual equivalence. Perceptually guided rendering: The computer graphics community has a long history of exploring the properties of the human visual system to accelerate expensive global illumination algorithms [4, 28, 27]. Not only does visual perception provide a way to allocate computational resources, it also serves as stopping condition for expensive algorithms. Other research includes using perception to get away with less computation by using visual masking of the visual system. Walter et al. [29] describes a system that accelerates global illumination by using texture masking. Perceptually Guided Level of Detail: More recently, researchers have begun to apply visual perception to polygonal techniques. Watson et al. [31] studied visual fidelity of polygonal models. Reddy [24, 25] explored extensively the properties of the visual system and their application in level of detail systems. Luebke et al. [18] developed a mesh simplification algorithm guided by the contrast sensitivity function of the human visual system. Williams et al. [32] extended the result of Luebke et al. [18] to lit and textured models. Other properties of the visual system have also been explored in level of detail systems [6, 11, 12, 21]. Please refer to the LOD book [17] for more details Figure 1. The shading profile for a cylinder using flat shading, Gouraud shading, and Phong shading. 3 Faceting Artifact Analysis We define the faceting signal as the difference signal between a rendering of a highly detailed geometric model, serving as the gold standard, and a rendering of a simplified version of the geometric model under the same image synthesis conditions. The faceting signal of an object is created by placing a sphere of cameras around the ideal polygonal representation of an object and around a simplified version of the object, taking images of each object under the same image synthesis conditions, and subtracting each pair of images. The difference signal between these two images is the faceting signal at that certain viewing circumstance. Similar to [24, 14], we place a sphere of cameras at the vertices of a rhombicuboctahedron to capture the faceting signal of a geometric model. Figure 1 shows the shading profile for a scanline of a regularly tessellated cylinder using different shading models. We assume a directional light source shines perpendicular to the cylinder surface. Figure 2(a), (c), (e) illustrate the faceting signals for different shading models. In this section, we explain how the faceting signal changes as a function of surface curvature, lighting, shading model, textures, and some perceptual aspects that affect the visibility of the faceting signal. A cylinder is used to simplify the illustration. 3.1 Surface Curvature It is well known that silhouette edges of an object are most problematic for low resolution models. The faceting signal can be used to explain why silhouette area can be an issue for low resolution models. Figure 1 shows a scanline of the faceting signal of a cylinder using flat shading, Gouraud shading, and Phong shading. For flat shading, it is obvious that the steps become narrower and higher along the silhouettes. Bump mapping [2] allows for simulating 2

3 Flat Shading (a) Smooth Shading (c) (e) Phong Shading Flat Shading (b) Smooth Shading (d) Phong Shading Figure 2. The faceting signals corresponding to different shading models and their frequency distributions. the details of an object using a lower resolution model but has problem with silhouettes. Several techniques [26, 33] have been developed to address this problem. We can illustrate why the silhouettes are problematic by analyzing the faceting signal of a cylinder. We derive two equations that describe how the faceting signal changes as a function of surface location and surface curvature. We assume a directional light source shines perpendicular to the cylinder surface. Let s further assume that the cylinder sits on top of the xz plane and the center of the cylinder is located at the origin. We focus our analysis on the diffuse component of the shading I for the cylinder. I(x, r) = n l = r2 x 2 where x runs from the center of the cylinder to the right. Take the derivative against x, we have: x di(x) = r r 2 x dx = 1 2 r dx (1) (r/x) 2 1 di(x) represents the faceting signal as a function of x. Its absolute value increases as x increases. This explains why the amplitude of the faceting signal increases along the silhouettes, as shown in Figure 2(a). r (f) Take its derivative against radius r, we have x 2 di(r) = r 2 dr (2) r 2 x2 di(r) represents the faceting signal as a function of r. It is clear that Equation 2 is a decreasing function of r, which explains that the amplitude of the faceting signal becomes smaller as the radius of the cylinder becomes bigger (smaller curvature). 3.2 General Lighting Level The magnitude of the faceting signal becomes stronger as the light source becomes brighter. This is due to the linearity of the shading intensity with regard to the lighting intensity in the lighting equation. By using the linearity of Fourier analysis, we can conclude that in the frequency domain the magnitude of the frequencies increases as the intensity of the general lighting level increases. 3.3 Shading Model Different shading models can have a dramatic impact on the faceting signal. Figure 2 shows pictures of the faceting signals and their frequency distributions for three shading models: flat shading, smooth shading, and phong shading. Flat shading causes clear faceting artifacts; smooth shading shows less faceting artifacts; Phong shading is almost equivalent to the ideal signal and is much better than the previous two shading methods in terms faceting artifacts. This is due to the fact that different shading models use different surface normal approximations to calculate the pixel intensities. 3.4 Texture Mapping Textures on the surface can also have an impact on the visibility of the faceting signal. Texture mapping can be considered as a way to alter the faceting signal. There are different ways that a texture can be mapped onto a surface and thus change the faceting signal. For example, texels are multiplied with the shading of a fragment for diffuse color texturing. When a diffuse color texture is applied to a surface, due to multiplication, the magnitude of the faceting signal is reduced, thus decreasing the visibility of the faceting signal. 3.5 Perceptual Aspects The viewing distance of an observer also impacts the visibility of the faceting signal by affecting the perceived frequency of the faceting signal (in unit of cycles per visual 3

4 degree). As the viewing distance increases, the visual angle subtended by an object becomes smaller. Therefore, its size on the image plane becomes smaller. This means that the frequencies of the object becomes higher, and the frequencies of its faceting signal also become higher. Since the human visual system is less sensitive to stimuli of high frequencies, the faceting artifacts are less visible as the viewing distance increases. This effect can explain why a coarse geometric object looks better when the viewing distance increases. Other parameters also affect the visibility of the faceting signal such as the adaptation state of the observer, eccentricity, etc. 4 Characterization of the Faceting Signal 4.1 Root Mean Square Metric for the Faceting Signal The root mean square image metric has traditionally been used for image processing tasks because it is relatively easy to deal with mathematically. Lindstrom and Turk [14] observe that the goal of mesh simplification is to produce a simplified model that is visually similar to the original model and they therefore propose to use root mean square image metric to prioritize edge collapse operations. In our faceting signal analysis framework, the facetness of a coarse geometric model can be described using the root mean square image difference as: d RMS (M 1, M 2 ) = 1 m n (yij mn y1 ij )2 i=1 j=1 where M 1 and M 2 represent two versions of a geometric model. yij represents the pixel value of a rendering of the first model at pixel location (i, j), yij 1 represents the pixel value of a rendering of the second model at pixel location (i, j). 4.2 A Perceptual Metric for the Faceting Signal Despite its advantages, the root mean square error metric does not correlate well with the visibility of the faceting signal. In this section, we propose a perceptual metric for determining the visibility of the faceting signal, taking into account the luminance non-linearity and contrast sensitivity function of the human visual system. A perceptual metric can further take into account other viewing parameters such as the lighting adaptation level, observer distance, etc, which all affect the visibility the faceting signal. The structure of our perceptual metric is closely related visual difference metrics [8, 16]. It consists of three stages: amplitude nonlinearity, contrast sensitivity function, and psychometric function. The result from contrast sensitivity function filtering is passed through the psychometric function and the result is turned into probability of visibility. These three stages are discussed in the following Luminance Nonlinearity Processing It is well known that the visual sensitivity and perception of lightness are nonlinear function of luminance. The amplitude nonlinearity describes the sensitivity variations as a function of background luminance. Various models have been proposed in the literature such as log model and various power laws. In this paper, we have chosen to use the cube root model of the nonlinearity of the visual system, which can be described as: r o = r 1/3 i where output value r o is obtained by returning the cube root of the input value r i Contrast Sensitivity Function The contrast sensitivity function of the human visual system describes the variations in visual sensitivity as of function of spatial frequencies. In our implementation, we have chosen to use the contrast sensitivity function described by Daly [8]. This function models the sensitivity of the visual system as a function of radial spatial frequency ρ in c/deg, orientation θ in degrees, light adaptation level l in cd/m 2, image size i 2 in visual degrees, lens accommodation due to viewing distance d in meters, and eccentricity e in degrees. S(ρ, θ, i 2 ρ, d, e) = P min[s 1 (, l, i 2 ), S 1 (ρ, l, i 2 )]; r a r e r θ r a =.856 d.14, where d is the viewing distance in meters. r e = 1/(1 + ke), where e is the eccentricity in visual degrees, k =.24, r θ = (1 ob)/2 cos(4θ) + (1 + ob)/2, where ob =.78. S 1(ρ,l,i 2 )=((3.23(ρ 2 i 2 ).3 ) 5 +1).2 A l ɛρe (B l ɛρ) 1+.6e B l ɛρ where A l =.81(1 +.7/l).2 and B l =.3(1 + 1/l) Psychometric Function The psychometric function describes the probability of detection as a function of the signal contrast [19]. An example of the psychometric function is shown below: P (c) = 1 e (c/α)β (3) where P (c) is the probability of detecting a signal of contrast c, α is the detection threshold, and β describes the 4

5 slope of the psychometric function and is largely invariant across many tasks Discussion Even though our perceptual metric is based on psychophysical data, we still carried out a few test cases that allowed us to verify the validity of our visibility measure experimentally. Figure 3 shows a patch of a sine wave and its visibility maps for viewing distances of 1, 2, and 3 meters. Brighter regions in the visibility map indicate higher probability of visibility. For a sine wave, the peaks and troughs are shown on the visibility map. As the distance increases, the peaks and troughs become narrower, which indicates that its visibility decreases as the viewing distance increases. The average probability of visibility for each viewing distance is.71,.52, and.41, respectively. Besides viewing distance, we also performed experiments to verify that our metric works under different luminance values. Figure 3. A sine wave and its visibility map at viewing distance 1, 2, and 3 meters. As an example of a more complicated faceting signal and its visibility map, Figure 4 shows an example of the faceting signal (left) of the Stanford Bunny and its visibility map (right) at viewing distance of 1 meter. From the visibility map, we can observe that the silhouette edges are more visible. Figure 4. The left image shows an example of the faceting signal and the right image shows its visibility map. Figure 5 (a) shows the magnitude of the faceting signal changes calculated with the root mean square image difference metric as the model size becomes smaller. Figure 5 (b) shows the how the probability of visibility for the faceting signal changes calculated with the perceptual metric developed in Section 4.2 as the model size changes. Both error measures become larger as the model size becomes smaller. Different curves represent the error metric for the faceting signal at different camera viewing distances. The two figures show similar trend as the model size decreases and as the camera moves farther away from the model. One of the differences is that the perceptual metric takes into account observer viewing distance while the root mean square error metric does not. The root mean square error metric does take into account general lighting intensities. 5 Level of Detail Selection The perceptual metric described above has direct application in perceptually guided level of detail management. A level of detail system mostly deals with three major issues: creation, selection, and switching. LOD creation includes generating several representations of a geometric model, which include both static level of detail models and dynamic data structures from which a level of detail model can be extracted at runtime. Selection deals with the problem of selecting the most appropriate model based on a given set of rendering and viewing parameters, or a given error threshold. Switching deals with how we change from one LOD model to another LOD model and minimizes the switching artifacts. In this section, our effort is focused on the selection problem of the LOD system. We describe a LOD system that uses our visibility measure in the selection of level of detail models. 5.1 Faceting Signal Visibility Pre- Computation Similar to the perceptually guided level of detail system described by Reddy [24], our perceptually driven level of detail system has an off-line preprocessing stage and a runtime stage. The preprocessing stage employs our perceptually guided visibility metric and calculates a table of average probability of visibility for use in the LOD runtime. We pre-compute the average probability of visibility at different lighting intensities, observer distances, and camera distances. Please note that we take the average lighting intensity of an environment because the space of lighting is large and it can be difficult to pre-calculate. One of the fundamental differences between our approach and Reddy s is that our system computes the visibility of the faceting signal while Reddy s system computes the visibility of the image content. The off-line pre-computation process can be described as: for each lighting intensity 5

6 camera distance 1 camera distance 2 camera distance camera distance 1 camera distance 2 camera distance RMS Error.8.6 Probability of Visibility Simplified Models Simplified Models Figure 5. The rms (left) and the probability of visibility (right) of the faceting signal for the Stanford Bunny of different model sizes at different camera distances. for each observer viewing distance for each camera distance for each level of detail model render ideal and current model find the faceting signal find its visibility end for end for end for end for In order to make the computation tractable, we quantize the lighting intensity, observer viewing distance, and camera distance into a set of bins based on system requirements. For example, if we understand beforehand that the observer will be looking at the object at a certain distance, we will focus the pre-computation at that distance range. Similarly, we quantize the lighting intensity, camera distance to the object, etc. Figure 5 (right) shows the average probability of visibility for the Stanford bunny at different camera distances. 5.2 LOD Runtime During runtime, all the parameters such as observer distance, camera distance, general lighting intensity are found. Then, given a visibility threshold, these parameters are used to index into the table to find the appropriate level of detail. One of the advantages of our system is that the user can choose a perceptually meaningful threshold. Also our system allows for specifying the observer s viewing distance. The visibility map pre-calculation is the most expensive part of our system. For example, for the Stanford bunny model at seven different resolutions (simplified from the original using Qslim [1]), on a machine with 1.8G Xeon CPU and 1G memory, it takes about 1 minutes to calculate the visibility of the faceting signal at three different camera distances, one lighting intensity levels, 24 viewing positions, one observer distance. 6 Surface Texturing In this section, we discuss another application of faceting signal analysis for determining surface textures that minimize the faceting artifacts for bicubic subdivision surfaces. We draw upon research in visual masking to make the faceting artifacts less visible by using a texture as a masker that obscures the faceting signal. We select bicubic subdivision surfaces as our model because it is easy to calculate and control its major faceting signal frequencies. We choose to use Perlin noise [2] as surface textures because it is easy to control its frequencies. According to the theory of visual masking, we will seek textures that are similar in frequency and orientation to the faceting signal. Furthermore, the texture must have a certain contrast to be able to hide the faceting signal. The following section derives such a lower bound contrast for the Perlin texture. 6.1 Visual Masking Visual masking refers to the phenomenon where a foreground signal decreases the visibility of a background signal. In the visual masking experiment, the foreground signal used to mask the background signal is called the masker, and the background signal is called the signal [13]. This is a very strong phenomenon which has been explored both in computer graphics for realistic image synthesis and outside of computer graphics in areas such as auditory perception. Several vision models that incorporate masking have been developed in the imaging science industry [8, 16] and more recently in computer graphics [9]. Visual masking studies show that the foreground signal has stronger masking properties when it is similar in frequency and orientation to the 6

7 background signal. Furthermore, the higher the contrast of the foreground signal, the stronger the visual masking effect it produces. In the case of faceting artifacts, the background signal is the faceting signal, and the foreground signal is the texture. Visual masking has proven to be useful in computer graphics in the area of realistic image synthesis [23, 29]. 6.2 Surface Texture Contrast Lower Bound m = (L max L min )/2L where L max and L min are the maximum and minimum intensity of the signal, and L is the average intensity. Using the above equation, the contrasts of the faceting signal and texture are, respectively m 1 = 2i 2 a/2i 1 i 2 = a/i 1, m 2 = 2i 1 b/2i 1 i 2 = b/i 2 Let us represent the ideal shading profile for a geometric model as s 1 (t) = i 1 (t) + a(t) cos(ωt) where i 1 (t) is the piecewise constant approximation of the ideal signal. a(t) cos(ωt) is the localized frequency component of the faceting signal. In a similar manner, let us represent the texture as s 2 (t) = i 2 (t) + b(t) cos(ωt) where i 2 (t) is the localized average intensity of the texture, and b(t) cos(ωt) is the localized frequency component of the texture. The problem is to find the condition in which b(t) cos(ωt) can mask out the faceting signal a(t) cos(ωt). We assume the signal and the masker have the same frequency ω and phase. For clarity, we will drop t from this point onwards. We will now analyze what happens for diffuse texture mapping, where the texture is modulated by the lighting. This can be represented as a pointwise multiplication of the shading profile s 1 (t) and the texture s 2 (t). In this case, the intensity profile for the rendered image is: where ab/2 + ab/2 cos(2ωt) can be omitted because they are higher order terms. (a and b are small compared to i 1 and i 2.) Therefore, we have s(t) = i 1 i 2 + (i 1 b + i 2 a) cos(ωt) where i 1 i 2 represents the localized average intensity after texture mapping, i 2 a cos(ωt) is the difference signal to be masked, and i 1 b cos(ωt) is the masker. Given i 1, i 2, a and b, we can find their relationship using experimental data in psychophysical studies. Since all this experimental data is in the units of contrast, we have to convert i 1, i 2, a, and b into contrasts. The Michaelson contrast is calculated using the equation In our implementation, we have followed the visual masking function proposed by Watson [3]. A similar visual masking function was used in [29]. This formula describes the relationship between the normalized increment threshold c/c and the background normalized contrast c/c as c(c) = C max(1, (c/c) w ) (4) In the contrast masking experiment, an observer is asked to discriminate two images of different contrasts. One image has contrast c, and the second image has contrast c+ c. The increment c is varied to find the c at which the observer can just discriminate from c. When the background contrast is zero, one measures a contrast detection threshold C. w is a constant with value.7. Using equation 4, we can compute the lower bound contrast for the texture to be able to mask the difference signal: m 2 = m 1 + c = m 1 + C max(1, (m 1 /C) w ) (5) 6.3 Results In this section, we report some promising results for bicubic subdivision surfaces. For subdivision surfaces, the major frequency of the faceting signal can be easily calculated by counting the number of subdivisions in object s(t) = s 1 s 2 = i 1 i 2 + (i 1 b + i 2 a) cos(ωt) + ab cos 2 (ωt) space. We therefore generate Perlin texture of the same frequency and use this texture as diffuse color texture. = i 1 i 2 + (i 1 b + i 2 a) cos(ωt) + ab/2 + ab/2 cos(2ωt) Please also note that it is important to match the orientation of the faceting signal and the orientation of the texture in order to achieve maximum masking effect. We can then use Equation 5 to calculate the minimum texture contrast. Figure 6(a) shows a rendering of the Utah teapot with faceting artifacts. We have picked two spots on the teapot, marked by a green dot and a yellow dot, as shown in Figure 6(a). m 1 is the contrast of the faceting signal, and m 2 is the minimum contrast of the texture computed using Equation 5. The following table shows the values of these two parameters at the two spots. Location m 1 m 2 green dot yellow dot

8 We see that the texture in Figure 6(b) can just mask out the faceting artifacts at the green dot. The texture in Figure 6(c) has enough contrast so that it can mask out the faceting artifacts all the way to the yellow dot. Although it was possible to find a texture contrast that would mask the faceting artifacts across the entire top half of the teapot, as can be seen in Figure 6(b) and 6(c) this texture contrast was not sufficient to obscure the artifacts on the bottom portion of the teapot. This is because the direction of minimum curvature for the bottom part of the teapot is orthogonal to the direction of minimum curvature for the top portion. Because the texture being used only has frequency information in one direction, masking of the faceting artifacts in the bottom half of the teapot can only be accomplished by additional subdivision. Figure 6(d) shows the result of further subdividing the lower portion of the teapot to eliminate the faceting artifacts. 7 Conclusions and Future Work In this paper, we introduced the concept of the faceting signal, which describes the artifact signal induced by coarse geometric models. We analyzed how it is affected by surface curvature, lighting, shading model, etc. We then described two metrics that characterize its visibility: the traditional root mean square error metric and a perceptually driven visibility metric. The importance of the faceting signal lies in the fact that it represents the artifact signal for coarse geometric models. The introduction of the faceting signal offers opportunities in designing better perceptual guided algorithms for computer graphics, especially algorithms that take advantage of visual masking, where identifying the artifact signal is critical to come up with a good masker. We also include two examples that illustrate the utility of the faceting signal. First, we developed a perceptually guided level of detail selection system that selects the most appropriate geometric model. Second, we developed a surface texturing algorithm that calculates the lower bound contrast for a texture that can serve as a successful masker. For future work, we would like to explore more applications that can benefit from the analysis of the faceting signal. Also, it would be interesting to extend our surface texturing algorithm to more complicated surfaces and textures. Acknowledgements I would like to thank Stanford computer graphics group for making the Bunny model available, Michael Garland for his QSlim software. References [1] J. Arvo, K. Torrance, and B. Smits. A framework for the analysis of error in global illumination algorithms. In SIG- GRAPH 94: Proceedings of the 21st annual conference on Computer graphics and interactive techniques, pages 75 84, [2] J. F. Blinn. 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