UNC Chapel Hill Computer Science Technical Report 97-041
(also available in Adobe Acrobat format)
|University of North Carolina|
Normal distribution mapping is an extension of traditional texture mapping methods to normal and shading information. The surface normal direction and shading for a patch of surface are represented by a statistical distribution of normal directions. The normal distribution map is a map of these distributions as they vary over the surface. Normal distribution mapping admits considerable freedom in the choice of probability density function. We propose using a distribution based on a 3D Gaussian for this purpose. This distribution can be easily filtered for map antialiasing and cheaply evaluated for surface shading.
In the same way that texture mapping provides separation between the sampling of the surface color and the surface geometry, normal distribution mapping separates the sampling of the surface normals and surface geometry. With a normal distribution MIP-map, the normal and shading information can be appropriately filtered and resampled based on pixel coverage. These two properties mean that normal distribution mapping can improve the sampling and filtering of shading information on polygonal approximations surfaces, either polygonalization of smooth surfaces or coarse level of detail surfaces.
Since the normal distribution map uses a single representation that combines properties of bump maps with surface shading information, it also provides continuous transitions between these domains. In fact, calling it a "transition" is not accurate since normal distribution mapping uses a single unified representation for both domains.
CR Categories and Keywords: G.3 Probability and statistics; I.3.3 Picture/Image Generation; I.3.7 Three-Dimensional Graphics and Realism; Surface Shading; BRDF; texture mapping; bump mapping; multiresolution analysis
One way to think about polygonal representations of smooth surfaces is in terms of sampling and reconstruction. Each vertex is a sample of the smooth surface. We reconstruct an approximation to that surface from the vertex sample locations. The quality of the polygonal approximation depends on the sampling rate and distribution - how many vertices were used and where they were placed.
Sometimes we will also use a color and normal at each vertex to reconstruct the color and shading of the smooth surface. However, the sampling that is most appropriate for the surface geometry is not necessarily the best choice for reconstructing the surface color, and the best rate for the color is not best for the geometry. Texture mapping addresses this problem by divorcing the sampling of the color from the sampling of the geometry.
Figure 1: Color and normal distribution map textures on a simple object, with filtering on both done using a MIP-map. a) color and normal distribution map textures on a simple object. b) same textures at a different MIP-map scale (under-sampled for this image size). Note how the specular highlight blurs and elongates in the direction of the ripples.
While texturing solves the problems of color sampling, we still sample surface normals at the vertices (see Figure 1). Gouraud shading and Phong shading provide different schemes for reconstructing the smooth surface shading from the sampled normals. We have observed in recent work in level of detail and multiresolution surface approximation, that appropriate sampling for the surface geometry does not necessarily provide appropriate sampling for the normals either. For example, simplification envelopes [Cohen96] can guarantee that the polygonal approximation to a surface is correct to within one pixel. Yet the surface highlights may change drastically between one level of detail and the next (see Figure 2). We are aware of work in progress to incorporate surface normal and curvature information in the simplification process, but this will result in excess polygons, not necessary to represent the geometry faithfully. If we can separate the normal sampling from the geometry sampling, it will be possible to do model simplification based only on geometric constraints, without introducing extra polygons.
Figure 2: a) and b) Two images of the same smooth object using different numbers of polygons. Notice the changes in the specular highlights. c) and d) The same polygonalized objects with normal distribution mapping.
Williams also noticed problems sampling the surface highlights in image space, which he called highlight aliasing [Williams83]. One form of highlight aliasing occurs when geometry normals change rapidly with respect to the image sampling frequency. This is particularly apparent for regular bumps, where the bump highlights beat against the image samples. The second form of highlight aliasing occurs on very shiny surfaces when the highlight size is close to the spacing between image samples. His solution used an illumination MIP-map for the specular component of the surface illumination, though this solution is only applicable for non-local lighting.
Some of these sampling problems could be avoided by using a normal map. We will be referring to a number of different maps on the surface. These are functions defined over the surface, usually through some texture coordinate system. For example, a texture map gives a color at every point on the surface; a normal map gives a normal at every point on the surface; a bump map gives a perturbation of the normal at every point on the surface. In general, unless there is another already common term for a map, we will name the map after the thing being mapped.
While the normal map decouples geometry sampling density from normal sampling density, it is still necessary to properly filter it to avoid aliasing. For color textures, pre-filtering techniques are commonly used speed the filtering process during rendering. For example, MIP-mapping computes a pyramid of pre-filtered versions [Williams83]. We can consider creating a hierarchy of maps for any map, and can apply standard MIP-mapping and signal processing techniques on any map which behaves linearly.
While normals may be linearly combined and averaged, directly filtering the normal map does not yield the expected results. Consider the example of sheet of aluminum foil that has been crumpled then flattened again. When examined closely, it appears to be a shiny surface with surface normals pointing in many directions. From a sufficient distance, the wrinkles are no longer apparent, and it appears to be a single flat surface with a common normal. However, at this scale the sheet no longer produces a clean reflection - it appears less shiny. A filtered version of the normal map correctly shows the new common normal direction, but does not reflect the changes in surface shading.
The solution is to incorporate information about the distribution of normals in the closer view into the shading computations in the more distant view. We need an approximation to the bidirectional reflectance distribution function (BRDF) of the surface at the distant scale that is derived from a combination of the BRDF and normals at the close scale.
Cabral et al. produced a BRDF for a distant view of a bump mapped surface using Monte Carlo ray tracing techniques [Cabral87]. The close view of the bump map was modeled as a height field, and random rays (modulated by the close scale BRDF) were traced to produce a histogram representation of the distant scale BRDF for the entire bump map. This distant BRDF could be fit by an appropriate approximation function for computation. Westin et al. used similar methods to produce a milliscale BRDF from surface geometry and microscale BRDF [Westin92].
Both of these produced BRDFs at discrete scales. In a follow-on to the Cabral et al. work, Becker and Max addressed the transition between representations at different scales [Becker93]. To blend between bump mapping and BRDF representations, they blended between the color produced by each. They also allowed a hierarchy of scales, with Perlin's technique of band-limiting the frequency of the bumps [Perlin84]. This technique uses a frequency cut-off at each scale to decompose the bumps into a part to be rendered using the BRDF and a part to be rendered using bump-mapping.
Fournier eliminated the problem of blending between bump mapping and BRDF by combining both into an approximation to the BRDF, which varied over the surface [Fournier92]. He could also handle a hierarchy of scales using a pyramid of these approximations, similar to a MIP-map. This is the approach we use for our normal distribution maps.
A common approach to approximating a BRDF is to split it into separate components for diffuse and specular reflection. The diffuse component can be represented as simple Lambertian shading. The specular component can be represented as a probability distribution of normals. This is usually derived by imagining the surface as composed of millions of perfectly reflective microfacets, randomly distributed about the average normal direction (see Figure 3). These facets are too small to see, but can be modeled with geometric optics. For any particular light and view direction, there is some probability of having facets with the normal direction that reflects the light to the viewer. This probability can be found by looking directly at the value of the probability density function, and can be used directly to determine the percentage of light that reflects from the light to viewer.
Figure 3: Surface microfacets. a) The surface is assumed to be made of millions of tiny facets. The facets are used to find a probability distribution of facet normal directions. b) The surface is rendered as a geometrically flat surface with the normal distribution used to reproduce the shading effects of the facets.
The Blinn [Blinn77] model adds a term for geometric attenuation (from shadowing and masking occlusion effects). Shadowing, of course, is when one part of the object blocks the path between the light and surface. Masking is when part of the object blocks the path between viewer and surface. Dealing with these effects can be troublesome and expensive. Many rendering systems do not deal with the shadowing effects (they do deal with masking since this is the visibility problem). Bump maps, generally do not account for either shadowing or masking. The best known technique for dealing with occlusion effects in the context of bump mapping and shading is the work of Max and Becker [Becker93]. Their technique could be adapted to our normal distribution maps. We have elected not to do this in an attempt to keep the computational cost of the method low. Note also that Ward, while he was addressing a different problem, achieved good fit to measured BRDFs without any geometric attenuation [Ward92].
So far, we have a BRDF map over the surface. By allowing the BRDF to vary over the surface, we combine bump mapping and surface shading effects into a single representation, avoiding problems that occur when transitioning between these effects when they are modeled separately. We approximate this BRDF map a normal distribution map, basically approximating each BRDF by a statistical distribution of surface normal directions. The main choice that remains to be made is the probability distribution to use. In searching for a distribution, we are interested in three key properties. First, it should be able to reproduce a Gaussian-like distribution of normals. Second, it must be cheap to evaluate during shading. Third, it should be easy to filter for producing a MIP-map. If possible, we would like the parameters of the distribution to combine linearly, so standard filtering and MIP-map lookup techniques can be applied directly.
Microfacet shading models are of interest since they also model the statistical distribution of surface normals, albiet locally. These models generally assume that the only variation is due to the microfacet orientations. Otherwise, the surface is flat. This is reasonable for the context where these models are used. To understand what it means to represent the normals for a larger section of surface, we use a tool from differential geometry, the Gauss map [Koenderink90]. Every point on the surface has a unit normal associated with it. That unit normal can be interepreted as a point on a unit sphere. Thus, the normals of the surface define a mapping between the surface and the unit sphere. This mapping is called the Gauss map. When the mapping is applied to a particular surface, the result is called the Gauss map of the surface (see Figure 4).
Figure 4. 2D example of a Gauss map for a section of a curve. This is the mapping of the curve onto the unit circle based on curve normal direction. It is analogous to the 3D Gauss map, which maps a surface onto the unit sphere. Compare this figure to Figure 3.
For our purposes, we are not interested in the shape or connectivity of the Gauss map, but its density. We can model the density of the Gauss map with a probability density function on the sphere. Each texel in the normal distribution map models the distribution of normals on the Gauss map for a section of surface. The texels in a normal distribution MIP-map cover progressively larger pieces of surface at each MIP-map level.
Microfacet models generally use probability distributions over a hemisphere of the Gauss map, pointing outward from the local surface direction. Even if we start with these hemispherical distributions, as we combine them at coarser MIP-map levels, the probability of normals below the "horizon" increases. As a result, we are interested in probability distributions over the entire Gauss map. That is to say, probability distributions on a sphere.
Any distribution on the hemisphere can be used on the sphere by using a zero probability for any normals in the other hemisphere. This is the approach taken by Fournier, who used distributions based on mixtures of Gaussians, which he approximated as "Phong Peaks" for cheaper shading [Fournier92]. If we want our density function to represent the normals on the other side of the sphere we need a different type of distribution. Also, where Fournier placed more emphasis on the accuracy of the approximation to the BRDF, we place more emphasis on speed. We want a way to achieve most of the visual improvement without having to compute mixtures of up to fifty Gaussians. If possible, we would also like to avoid the hours non-linear optimization to fit the approximations in the pre-processing stages.
Microfacet models are created by applying geometric optics to simple shapes. Blinn used `V' shaped grooves [Blinn77] for one shading model and spheres for another [Blinn82]. Poulin used parallel cylindrical grooves [Poulin90]. Neyret used this geometric technique to derive a distribution for use in volume shading that covered the full Gauss map [Neyret95]. He used a single randomly sized and oriented ellipsoid to produce his distribution of normals. Neyret's work is interesting because, when producing an octtree representation of the volume, parent cells were represented by a single ellipsoid that approximated all of the ellipsoids in the children cells. This allows basic filtering operations to always result in an approximation of the same form. We will come back to this idea later.
The real result of any geometric microfacet model is a distribution of normals. Other shading models have used distributions without any reference to geometry, based only on the distribution's ability to approximate the BRDF [Ward92]. So we may consider other distributions if they have the right Gaussian character and are easier to compute or filter.
One way to create new distributions is to a way to generalize existing ones. For example, the `V' shaped microfacet grooves are usually assumed to have a Gaussian distribution of slopes. The resulting distribution can also be seen as a projection of a Gaussian onto the hemisphere as follows: define a Gaussian distribution on a plane and place this plane tangent to the top of the hemisphere. To find the probability density at a point on the hemisphere, project from the center of the sphere, through the point, onto the plane (see Figure 5 a). The value of the Gaussian density function on the plane defines a density for the point on the sphere.
Gaussian projected to hemisphere:
One of the first probability distributions we tried was an extension of this to the full sphere. Instead of projecting from the center of the sphere, we tried projecting from the opposite pole (a stereographic projection).
Gaussian projected to full sphere:
The result is a Gaussian-like distribution, defined over the whole Gauss map (see Figure 5 b). Unfortunately, this distribution is quite difficult to combine and filter, and the approximation we get when we combine a mixture of two distributions into a single one is not a good one.
Figure 5. Creating a distribution on the sphere by projection. a) The density function for a Gaussian distribution of microfacet slopes can also be created by projection from the center of the sphere. b) A similar density function over the full sphere.
The final class of distributions we consider are not based on microfacet models at all. Johnson gives a number of probability distributions on the circle and sphere [Johnson87]. These distributions are formed by starting with a distribution in Euclidean 3-space, and using the conditional probability of the distribution when restricted to the surface of the sphere. This opens up the possibility of working in Euclidean space, where distributions tend to be better behaved, instead of on the sphere.
Figure 6:2D example of a distribution on a circle created by restricting a 2D Gaussian to the circle. This can be extended to create a distribution on a sphere from a 3D Gaussian.
A requirement for any distribution is that it be able to reproduce the tight 2D Gaussian-like cluster of normal directions on shiny surfaces. If we consider possible 3D distributions to produce Gaussian behavior, the first distribution that comes to mind is the 3D Gaussian. A low-variance Gaussian centered on the surface of the Gauss map can certainly create a Gaussian-like cluster of normals on the surface of the Gauss map. The mean of the 3D Gaussian is the average normal direction for that point on the shiny surface. The distribution defines a small Gaussian blob in space around that average normal direction (see Figure 6).
Further, simple 3D Gaussians are easy to combine linearly. The best Gaussian fit to a mixture of two 3D Gaussians has a mean that is an average of the means of the two original Gaussians. The variances do not combine linearly, but the second moments about the origin do [Johnson87].
with mean vector and covariance matrix
Second central moment:
Second central moment:
To clarify the mixture of these 3D Gaussians, consider a shiny bump on a surface. Two nearby locations will both be characterized by low-variance distributions with slightly different means. Low-variance because the surface is shiny, differing means because the normal directions on different points on the bump are different. When we combine an equal mixture of these two distributions, the new mean is the average of the two original means. This means that the total average normal direction is the average of the original two normal directions. The variance increases along the axis between the two original means, while remaining the same in the other directions. The variation in original normal directions turns into additional uncertainty in the distribution of normals.
Figure 7:Mixture of 2D Gaussians: a) A shiny bump, sampled at two points, each with a different normal. b) The concentric circles are contours of the 2D Gaussians on the Gauss map for each sample point. The thickness of the circles indicates indicates the actual probability density induced on each Gauss map. c) The two 2D Gaussians from (b) are combined into a single 2D Gaussian. Note the increase in variance.
We can characterize the 3D Gaussian density function with nine numbers, all of which combine linearly. Three for the mean and six for the matrix of second moments. We can put these nine numbers into a MIP-map, and filtering will work as expected. In fact, in our implementation, we treat these nine numbers as three separate three-color MIP-maps with 16 bits per component. No changes to either MIP-map creation or access were necessary.
For our previous example of the sheet of aluminum foil, in the close view we have Gaussian distributions with small variance (corresponding to a shiny surface), but differing means. In the distant view, all of the Gaussians have similar means, but a larger variance (corresponding to a duller surface). This is exactly the behavior we want. Similarly, for grooved surfaces, the variance will increase more in one direction than the other as the object becomes smaller, modeling the anisotropic reflection.
The most time critical computation is not the creation of the normal distribution MIP-map, but its use during shading. Since the 3D Gaussian-based normal distribution map behaves linearly, the MIP-map lookup computations are straightforward. The result is a single 3D Gaussian to be used as the normal distribution for an image sample. We must compute the normalized halfway vector between the light and viewer and look this up in the density function. Ignoring normalization of the density function for the moment, this just involves plugging the halfway vector into the 3D Gaussian function.
To this point, we have been able to work with the probability distributions without constructing their density functions. We know the form of the density function, but it must still be normalized so the total probability over the entire sphere is one. To do this, we integrate the 3D distribution over the sphere and divide by the result. Unfortunately, the Gaussian distribution does not integrate well over the sphere.
We could compute these integrals numerically at every image sample, but that is far too expensive. Instead, we can do the numerical integration as a pre-process for each element of the normal distribution MIP-map, producing a MIP-map of normalization factors. The resulting tri-linear interpolation is only an approximation to the true normalization factor, but is much cheaper to compute.
An alternate method is to use the 3D Gaussian for MIP-mapping only, and approximate the resulting 3D distribution for shading. For example, since the restriction of a 3D Gaussian to a plane is a 2D Gaussian [Johnson87]. One possible approximation is to choose a plane tangent to the sphere or one through the mean of the 3D Gaussian, and project the the distribution on this plane onto the sphere. To summarize, we approximate the BRDF with a 3D Gaussian for MIP-mapping, then re-approximate with a projected 2D Gaussian for rendering.
As a final option, we can follow the "traditional" graphics approach and the 3D Gaussian, but use an un-normalized density function. It is still necessary to do a partial normalization based on the square root of the covariance matrix to factor out changes due to the changing variance of the 3D Gaussian. Surprisingly, this works reasonably well. Since it is, by far, computationally the least expensive, it was the method used for all the images shown in this paper.
Even if we ignore the normalization of the density function, the computation of the Gaussian is still rather expensive, since it involves an exponential. To make the computation cheaper, we actually use an approximation to the 3D Gaussian. Johnson discusses the class of elliptically contoured distributions, which include the Gaussian distribution.
Density function (for a given g(t)):
Gaussian distribution: for
Lyon shows that the Phong and Pearson Type II both do a good job of approximating a Gaussian [Lyon93]. We choose, as he did, to use the Pearson Type II distribution for its computational simplicity.
Pearson Type II distribution: for
The parameter m is a constant that controls how closely the Pearson Type II distribution matches the Gaussian (we used eight).
We have presented a new method for uniformly handling sampling of surface normals and shading information, and the interaction between the two at different distances. The method uses a probability distribution of surface normal directions as an approximation to the BRDF as it varies over the surface. We propose using a probability distribution based on a 3D Gaussian for this purpose. The parameters definining this distribution behave linearly, so they can be easily filtered using standard filtering techniques. An approximation to this distribution can be used that is fast to evaluate at shading time, or several progressively more expensive alternatives exist for normalizing the distribution if it becomes necessary.
Though shadowing and masking effects cannot occur on bump maps, they can occur if the surface model is more complex. We would like to be able to model these effects. The likely method for creating these models has been previously proposed by Becker.
Finally, we would like to explore multi-resolution simplification using probability density functions based on wavelets which have "nice" recombination behavior.
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