1. CS 325 Introduction to Computer Graphics 03 / 26 / 2010 Instructor: Michael Eckmann

2. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Today’s Topics Questions? Illumination modeling –Ambient –Diffuse –Specular –Refraction

3. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Illumination Illumination models are used to calculate the color of a point on the surface of an object. A surface rendering method uses the calculations from the illumination model to determine the pixel colors of all the projected points in a scene. –Can be done by doing illumination model calculations at each pixel position (like you'll see is done in ray tracing). OR –Can be done by doing a small number of illumination calculations and interpolating between pixels (like is often done in scan line algorithms.)‏

4. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Illumination Diffuse reflection reflects light of equal intensity in all directions, independent of viewing position. Contrast this with specular reflection which reflects light in unequal intensities at different directions, dependent on viewing position. Shiny surfaces result in specular reflection. Dull, matte surfaces result in diffuse reflection. A Lambertian surface is one that reflects light, equally intense in all directions. A Lambertian surface reflects light diffusely. Examples of Lambertian surfaces are matte photographic prints (as opposed to glossy prints), chalk, etc.

6. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Illumination Lambert's cosine law is: –The intensity of the light reflected from a surface is proportional to the cosine of the angle between the surface normal and the direction to the light source. From the diagram on the last slide –If L is a vector from the surface towards the light source and N is a normal vector to the surface and A is the angle between them, then cos A = L ● N / |L||N| –If L and N are of unit length then cos A = L ● N I p is the intensity of a point white light source, k d is the diffuse reflection coefficient of the surface Our Illumination equation for a lambertian surface with a point light source is: I = I p k d L ● N

7. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Illumination To simulate both ambient light and point light sources on diffuse reflecting surfaces, we can simply add in the ambient light term to our previous illumination equation and end up with: I = I a k a + I p k d L ● N This creates a more realistic look. Without the ambient light term, objects will look harshly lit.

8. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Illumination Light is attenuated as it moves through space. The attenuation factor is 1 / d L 2 –where d L 2 is the square of the distance from the light source Light hitting surfaces closer to the light source receive higher energy than further surfaces. Without taking this into account, two equal surfaces in the world at different distances (that overlap eachother in the viewplane), will appear to be one surface in the view plane. Example on the board.

9. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Illumination But, because actual lights are not point light sources, and ambient light isn't exactly accurately modelling light in the world we do not use this attenuation factor exactly as 1 / d L 2 Instead, to get a more realistic look (than if we had used the inverse square law) we use an inverse quadratic function f radatten (d L ) = 1/ (a 0 + a 1 d L + a 2 d L 2 ) where the a coefficients can be adjusted for realism. We use this factor as a multiplier of the point light source term: I = I a k a + f radatten (d L ) I p k d L ● N

10. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Illumination Specular reflection as stated before is a property of a surface where light is reflected in unequal intensities at different directions, dependent on viewing position. A perfect mirror reflects light in only one direction. Less perfectly shiny surfaces reflect light more in one particular direction than other directions. Consider the angle the light and the surface normal makes. The direction that the most intense light is reflected is at an angle which is equal to that angle but on the other side of the surface normal. The next slide has a diagram for specular reflection.

12. Illumination L is a vector in the direction of the light source. N is the surface normal. R is the vector which is in the direction of the maximum specular reflection. V is the vector in the direction toward the viewer. The angle between L and N is the same as the angle between N and R. The angle phi, between R and V affects how much light is reflected in V's direction. If the surface was a perfect mirror, then light is only reflected along R. If the surface is shiny, but not perfect, light is reflected mostly in R's direction but decreases reflection as we go in directions at angles away from R. An illumination model that takes into account non perfect specular reflections is the Phong model.

13. Illumination Notice that if phi is 0, the viewer is in the direction of most reflection. As phi gets larger, less light is reflected in the direction of the viewer. The next slide contains the Phong model which contains a function on phi that acts as above.

14. Illumination The Phong model sets the intensity of the specular reflection proportional to cos ns (phi), where phi is the angle between V and R. ns is the specular reflection exponent and is specified based on the type of surface you are modelling. Very shiny surfaces have ns values of 100 or more and less shiny surfaces can have ns values closer to 1. The Phong equation is: f radatten (d L ) I p W(theta)cos ns (phi)‏ where W(theta) is a property of the surface. It is the specular- reflection coefficient, which is typically set to some constant, otherwise it is a function on theta which is the angle between L and N. See fig. 10-19 in text for W(theta) for a few mat'ls.

15. Illumination

16. Our illumination model with the Phong term added in now becomes I = I a k a + f radatten (d L ) I p k d L ● N + f radatten (d L ) I p W(theta)cos ns (phi)‏ which can be simplified to: I = I a k a + f radatten (d L ) I p (k d L ● N + W(theta)cos ns (phi))‏

17. Illumination I = I a k a + f radatten (d L ) I p (k d L ● N + W(theta)cos ns (phi))‏ Notice the following: –the Phong term is dependent on the point light source and the attenuation factor –our whole illumination model at this point is influenced by ambient light and a point light source. Further, our object has diffuse reflection properties and specular reflection properties. And the phi angle is dependent on where the viewer is.

18. Illumination If we have more than one point light source (which is white light), then we sum up over all the light sources i: I = I a k a + Sum all i [f radatten (d L ) I pi (k d L ● N + W(theta)cos ns (phi))]

19. Illumination If the light sources are not white light sources, then we can seperate out the red, green and blue components. Also, if objects are to be colored, then they reflect different wavelengths and absorb others. Specify ambient light now as 3 components: I aR I aG I aB Specify point light now as 3 components: I pR I pG I pB Specify objects with color: O R O G O B I R = I a R k a O R + Sum all i [f radatten (d L ) I pRi (k d O R L ● N + W(theta)cos ns (phi))] I G and I B can be expressed similarly.

20. Illumination We typically want our Intensities to be in the range 0 to 1. With many light sources that are specified as too intense, computed intensities of objects can easily be too large (> 1). We can normalize the range of calculated intensities into the range 0 to 1. Or we can specify our light source intensities better and if any calculations go over 1, then clip them to intensity 1.

21. Directional light source The light sources that we have discussed so far were point light sources which radiated light in all directions. Real lights in the world do not have this property. Instead, real lights typically are directional. A directional light source can be described as having –a position –a directional vector –an angular limit from that vector –a color The vector will be the axis in a cone shaped volume. This volume is the only part of space that will be illuminated by this light. Anything outside the cone will not be illuminated by this light. See diagram next slide.

24. Directional light source To determine how intense the light is from a directional light source on an object we determine the angle between –the vector V o from the position of the light to the point on the object we're illuminating and –the cone axis vector V l –So, we take the dot product: V o V l = cos alpha, alpha is the angle between them. –If V o is in the same direction as V l, then the light should be the most intense (alpha = 0). The light should get less intense as we go in larger angles away from V l. –We can attenuate the light as function of alpha like so: cos al alpha where al is the attenuation exponent (a property of the light source) --- if al is 0, it's a point light source, the smaller al is, the larger the attenuation is (because the cos is less than 1.)‏

25. Transparent/Translucent surfaces Transparent, opaque, translucent –Transparent refers to the quality of a surface that we can “see through”. Opaque surfaces, we cannot see through. –some transparent objects are translucent --- light is transmitted diffusely in all directions through the material –translucent materials make the object viewed through them blurry

26. Transparent/Translucent surfaces Terminology index of refraction of a material –is defined to be the ratio of the speed of light in a vacuum to the speed of light in the material. Snell's Law is a relationship between angle of incidence and refraction and indices of refraction.

28. Transparent/Translucent surfaces To determine the direction of the refracted ray, the angle ( theta r ) off of -N, we need to know several things –the direction of the incoming ray, the angle ( theta i ) of incidence –the index of refraction (eta i ) of the material the ray is coming from –the index of refraction (eta r ) of the material the ray is entering

29. Transparent/Translucent surfaces Snell's law states that sin ( theta r ) eta i -------------- = ----- sin ( theta i ) eta r which can be written as: sin ( theta r ) = ((eta i ) / (eta r )) * sin ( theta i )‏

30. Transparent/Translucent surfaces Assuming all of our vectors are unit vectors, using Snell's law, according to our textbook we can compute the unit refracted ray, T, to be T = (((eta i ) / (eta r )) cos ( theta r ) – cos ( theta r )) N – ((eta i ) / (eta r ))L See page 578 in our text. This assumes L is in the direction shown in the diagram of figure 10-30. Verifying that the equation above is correct is left as an exercise to the reader. It may appear on a hw assignment.

31. Transparent/Translucent surfaces Table 10-1 on page 578 in our text shows average indices of refraction for common materials, such as Vacuum (1.0), Ordinary Crown Glass (1.52), Heavy Crown Glass (1.61), Ordinary Flint Glass (1.61), Heavy Flint Glass (1.92), Rock Salt (1.55), Quartz (1.54), Water (1.33), Ice (1.31)‏ Different frequencies of light travel at different speeds through the same material. Therefore, each frequency has its own index of refraction. The indices of refraction above are averages.

32. Surface Rendering Now that we have an illumination model built, we can use it in different ways to do surface rendering aka shading.

33. Shading Models Constant shading (aka flat shading, aka faceted shading) is the simplest shading model. It is accurate when a polygon face of an object is not approximating a curved surface, all light sources are far and the viewer is far away from the object. Far meaning sufficiently far enough away that N L and R V are constant across the polygon. If we use constant shading but the light and viewer are not sufficiently far enough away then we need to choose a value for each of the dot products and apply it to the whole surface. This calculation is usually done on the centroid of the polygon. Advantages: fast & simple Disadvantages: inaccurate shading if the assumptions are not met.

34. Shading Models Interpolated shading model –Assume we're shading a polygon. The shading is interpolated linearly. That is, we calculate a value at two points and interpolate between them to get the value at the points in between --- this saves lots of computation (over computing at each position) and results in a visual improvement over flat shading. –This technique was created for triangles e.g. compute the color at the three vertices and interpolate to get the edge colors and then interpolate across the triangle's surface from edge to edge to get the interior colors. –Gouraud shading is an interpolation technique that is generalized to arbitrary polygons. Phong shading is another interpolation technique but doesn't interpolate intensities.

35. Shading Models Gouraud shading (aka Gouraud Surface Rendering) is a form of interpolated shading. –How to calculate the intensity at the vertices? we have normal vectors to all polygons so, consider all the polygons that meet at that vertex. Drawing on the board. Suppose for example, n polygons all meet at one vertex. We want to approximate the normal of the actual surface at the vertex. We have the normals to all n polygons that meet there. So, to approximate the normal to the vertex we take the average of all n polygon normals. What good is knowing the normal at the vertex? Why do we want to know that?

36. Shading Models Gouraud shading. –What good is knowing the normal at the vertex? Why do we want to know that? so we can calculate the intensity at that vertex from the illumination equations. –Calculate the intensity at each vertex using the normal we estimate there. –Then linearly interpolate the intensity along the edges between the vertices. –Then linearly interpolate the intensity along a scan line for the intensities of interior points of the polygon. –Example on the board.

37. Shading Models Advantages: easy to implement if already doing a scan line algorithm. Disadvantages: –Unrealistic if the polygon doesn't accurately represent the object. This is typical with polygon meshes representing curved surfaces. –Mach banding problem when there are discontinuities in intensities at polygon edges, we can sometimes get this unfortunate effect. let's see an example. –The shading of the polygon depends on its orientation. If it rotates, we will see an unwanted effect due to the interpolation along a scanline. Example on board. –Specular reflection is also averaged over polygons and therefore doesn't accurately show these reflections.

38. Shading Models Phong shading (aka Phong Surface Rendering). –Assume we are using a polygon mesh to estimate a surface. –Compute the normals at the vertices like we did for Gouraud. –Then instead of computing the intensity (color) at the vertices and interpolating the intensities, we interpolate the normals. –So, given normals at two points of a polygon, we interpolate over the surface to estimate the normals between the two points. –Example picture on the board. –Then we have to apply the illumination equation at all the points in between to compute intensity. We didn't have to do this for Gouraud. –Problems arise since the interpolation is done in the 3d world space and the pixel intensities are in the image space. –More accurate intensity value calculations, more realistic surface highlights and reduced Mach banding.

39. Shading Models Recap –Flat Shading –Gouraud Shading --- interpolates intensities –Phong Shading --- interpolates normal vectors (before calculating illumination.)‏ –For these three methods, as speed/efficiency decreases, realism increases (as you'd expect.)‏ –They all have the problem of silhouetting. Edges of polygons on the visual edge of an object are apparent. –Let's see examples of Gouraud and Phong shading.