x ~ Hemispheric Lighting
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- Georgia Brooks
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1 Irradiance and Incoming Radiance Imagine a sensor which is a small, flat plane centered at a point ~ x in space and oriented so that its normal points in the direction n. This sensor can compute the total light energy received per unit time in watts. n x ~ After dividing by the area of the sensor, we can say that the energy flux that the sensor receives per unit area is the irradiance level E n (x ~ ). This irradiance level can be explained by making reference to the incoming radiance L(w, x ~ ), which is a measure of the light energy arriving at point x ~ from direction w. By integrating this incoming radiance over the entire hemisphere H n centered in direction n at x ~ we can compute the irradiance level E n (x ~ ): E n (x ~ ) = L(w, ~ x ) w n d w Hn The w n term models the usual "cosine of the angle between the normal and the light direction vector" term that we are used to seeing in diffuse lighting calculations. In computer graphics, we are mostly interested in the light reflected from the surface, since that light is what we see when we look at the surface. To a first approximation, we can assume that the incoming energy E n (x ~ ) gets multiplied by a reflectivity term ρ(x ~ ) and then gets reflected equally in all directions throughout the upper hemisphere H n to make an outgoing radiance term L(x ~, v ), where v is the view direction from the surface toward the viewer. Hemispheric Lighting One of the simplest incoming radiance functions models an outdoor scene as a sky color for all directions ω in the upper half-sphere of directions and a ground color for all directions ω in the lower half-sphere of directions. This L(w, ~ x ) is independent of the position ~ x.
2 θ I will not present a derivation here, but the result of this hemispherical illumination is an irradiance term where E n (x ~ ) = Hn L(w, x ~ ) w n d w = a Sky_Color + ( - a) Ground_Color - sin θ θ / a = sin θ θ > / Here θ is the angle between the normal n and the vector that points up vertically. This simple irradiance model produces reasonable results when rendering outdoor scenes, and can be combined with a traditional diffuse lighting calculation that models lighting from a point source "sun" at some specific location in the sky. Environment Mapping A technique that is occasionally used in computer graphics for various applications is environment mapping. In this technique we imagine that some object is contained inside an imaginary cube. Imagine further that we could stand in the center of that cube and make six snapshots of the scene by looking out at the scene through the center of each of the six faces in turn. In one version of this technique we make textures from each of those snapshots and essemble these six face textures into a single texture that could be wrapped around the interior of the cube. Another more sophisticated version of this idea uses a sphere instead of a cube to enclose the object. We then make a single texture that shows everything we could possibly see from the center of the sphere looking outward and then paste that texture onto the interior of the sphere. This texture is known as a spherical environment map.
3 Irradiance Mapping Imagine a polygon positioned in a complex scene with many areas of light and shadow. Suppose also that we could reduce everything that we could see from some point on the surface of the polygon to a spherical environment map positioned over that point. Just as above, we could model the illumination at that point on the surface by E n (x ~ ) = L(w, ~ x ) w n d w Hn The only difference this time around is that L(w, x ~ ) will be a complicated function that captures all the various light and dark colors we would see if we were to look out into the surrounding scene in the direction w from the point x ~. One simplifying assumption we can make is that the object we are trying to illuminate is small relative to the objects that surround it in the scene. In that case, to a first approximation L(w, x ~ ) is independent of x ~ : E n = L(w ) w n d w Hn An important optimization we can apply at this point is to precompute values for E n and place them in a special kind of environment map, called an irradiance map. By using this irradiance map to lookup diffuse illumination values and using the environment map to compute specular terms, we can produce a high quality lighting model. 3
4 Technical details - integrating over an environment map To compute an irradiance value from an environment map we have to compute an integral. E n = Hn L(w ) w n d w = S L(ω) M(n ω) d ω For simplicity here I have replaced the integral over the hemisphere H n with the integral over the entire sphere S, and at the same time replaced the dot product w n with a term M(w n), where M(x) is a function that evaluates to x when x is positive and to when x is negative. To emphasize the fact that we are integrating over a sphere, I have also replaced the direction vectors w with angles ω. The term L(ω) is a color value sampled from an environment map. In practice, most environment maps are actually cube maps made up of discrete texture pixels (texels). In that case, the integral over all directions in an environment map can be replaced by a sum over all texels in the six faces of a cube map. E n = L i M(n ω ) i dω i i Here L is the color of texel i, ω is the direction to that texel in the cube map, and dω is the solid angle subtended by that texel. 4
5 If we treat one of the faces of the cube map as a texture, we would access the texels via texture coordinates s and t. Since we are treating the texture as the face of a cube centered at the origin, we can replace texture coordinates with a more convenient coordinate system in x and y: (x = -, y = ) (s =, t = ) (x =, y = ) (s =, t = ) (x =, y = -) To transform coordinate systems, we would do x y = - - s t In the x, y coordinate system it is easy to compute the direction vector. All we have to do is to normalize the vector ( x i, y i, ) and it becomes a direction vector ω. i Computing the solid angle dω i subtended by a particular texel is a little more involved. Texels near the center of the texture subtend a somewhat larger angle, while texels near the corner take up a smaller solid angle when we map the cube texture onto a sphere. Here is a reference that explains how to compute the solid angle correctly from the positions x i and y. i Using a shader to compute E n Computing E n as described above is a very compute intensive operation, because we have to compute the integral over the cube map for each new direction vector n that we want to work with. At the same time, the details for different values of n are highly repetitive. This suggests that we should enlist the aid of shaders to compute this mapping for us. Here is the outline of a strategy that makes it possible to do this.. Use OpenGL to render a two dimensional square centered at the origin with sides of length. This square represents one of the sides of our cubical irradiance map. Given the mapping above we can map any point ( x, y ) on the square to a direction vector n. 5
6 . We render the square to a framebuffer with dimensions size by size pixels, where size is the desired size of our irradiance cube map texture. 3. In the fragment shader, we translate the interpolated fragment positions we are given into direction vectors n and construct loops that sum over the six faces of the environment map: E n = L i M(n ω ) i dω i i 4. When rendering is done, we convert the image in the frame buffer to a texture for use by our irradiance cube map. Here is a reference that shows how to render an image to a texture. Environment maps versus irradiance maps Here are some pictures that illustrate how enviroment maps relate to irradiance maps. The first image below is a typical cubical environment map. 6
7 7
8 Here is that same environment map mapped onto a sphere. Next we have the irradiance map derived from this environment map: Irradiance Mapping with Spherical Harmonics In, Ramamoorthi and Hanrahan published a paper describing a much more efficient technique for computing irradiance maps. In this paper they used spherical harmonics to decompose the terms L(ω) and M(n ω) in the irradiance integral E n = L(ω) M(n ω) d ω and subsequently greatly reduced the amount of time needed to compute an irradiance map. Their technique is based on the use of spherical harmonics. Since we have not encounted this concept before, some basic background is in order. The spherical harmonic functions Y l,m (θ,φ) are a set of orthogonal functions defined on the unit sphere. These functions are defined in terms of the complex valued spherical harmonics 8
9 Y l m (θ,φ) = N e i m φ P l m (cos θ) where P l m (x) is the m th Legendre polynomial of order l and N is a normalization factor that depends on l and m. The functions Y l m (θ,φ) arise in the solution of the polar form of the Laplace equation. In terms of the complex-valued spherical harmonics. the real-valued spherical harmonics are given by m m -m ( Y l + (-) Yl ) m > Y l,m = Y l m = m m -m (Y l - (-) Yl ) m < i Because the spherical harmonic functions are orthonormal on the unit sphere, any function defined on the unit sphere can be described as a linear combination of spherical harmonics: f(θ,φ) = l l = m = -l f l,m Y l,m (θ,φ) the coefficients f l,m are computed by integrating the target function against the spherical harmonics: f l,m = f(θ,φ) Y l,m (θ,φ) sin θ d θ d φ In the Ramamoorthi and Hanrahan paper the key insight was that that the integral we need to compute is in the form of a convolution E n = L(ω) M(n ω) d ω E n = L(ω) f(n, ω) d ω This is a useful observation, because convolutions map to products when we transform to the space of spherical harmonic coefficients. Most importantly, if the function product is particularly simple and straightforward. f(n, ω) is rotationally symmetric the mapping to a Ramamoorthi and Hanrahan determined that I(n) can be computed more easily by computing the coefficients of L and M with respect to the spherical harmonics and then forming E l,m = 4 M l, L l,m l + where M l, = L l,m = / cos θ Y l, (θ,φ) sin θ d θ d φ L(θ,φ) Y l,m (θ,φ) sin θ d θ d φ 9
10 This result is useful because the coefficients M l, and the term L, and can be precomputed: 4 / l + are independent of the illumination A l = 3 4 M l, = l + (-) l/ - (l + )(l - ) l! l l! ( ) l = l odd and > l even More simply, the first few terms are A = A = 3 A = 4 Once we have computed the coefficients E l,m we can recover the function E n via an inverse transform: E n = l l = m = -l E l,m Y l,m (n) Ramamoorthi and Hanrahan pointed out that although the sum above is an infinite sum, we can compute E n accurate to about % by using only the terms up through and including l =. In practice this means that we only have to compute 9 E l,m terms: E, = A L, = L, E,m = A L,m = L,m for m = -,, 3 E,m = A L,m = 4 L,m for m = -, -,,, Practical summary of the method To compute our estimate E n for any normal vector n we compute E n = l l = m = -l E l,m Y l,m (n) E l,m = A l L l,m Since the A l are precomputed and fixed, we only have to compute the L l,m terms:
11 L l,m = L(θ,φ) Y l,m (θ,φ) sin θ d θ d φ Since the L l,m terms are independent of the normal direction n we can precompute these terms from the environment map ahead of time. This will allow us to precompute the E l,m terms ahead of time, pass them down to a fragment shader, and compute E n very quickly and efficiently when we need it in the fragment shader. (In fact, since E n values typically do not vary all that much across a typical polygon, we can compute E n values in a vertex shader and use interpolation for the fragment shader.) Since the values L(θ,φ) are in practice computed by doing lookups in an environment map, once again it makes the most sense to reduce the L l,m integral to a sum over texels in the environment map: L l,m = L i Y l,m (ω ) i dω i i To compute these sums in OpenGL we will need to fetch the pixels in the 6 textures that make up the environment map. Recall that in an earlier step we computed these textures by rendering views of the scene into a framebuffer. As we are doing this step, we can fetch the data in the frame buffer and dump that data into an array array of 3*width*height floats in OpenGL via the functionglreadpixels: glreadpixels(,,width,height,gl_rgb,gl_float,array); A convenient way to handle the spherical harmonic terms is to represent them in terms of rectangular coordinates x, y, and z. If ( x, y, z ) is a point on the unit sphere, the spherical harmonics of interest are Y, (x,y,z) = Y,- (x,y,z) = Y, (x,y,z) = Y, (x,y,z) = Y,- (x,y,z) = Y,- (x,y,z) = 3 y 4 3 z 4 3 x 4 5 x y 5 y z Y, (x,y,z) = 4 Y, (x,y,z) = 5 (3 z - ) 5 x z
12 Y, (x,y,z) = 4 5 ( x - y ) It is important to note that as we read pixels from the environment map, those pixels will correspond to locations ( x, y, z ) that are not on the unit sphere. Those pixel coordinates will need to be normalized to points on the unit sphere before using the formulas above.
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