1. Course 10 Shading

2. 1. Basic Concepts: Light Source: Radiance: the light energy radiated from a unit area of light source (or surface) in a unit solid angle. Solid angle: If light source is a point source, the “ unit area” is omitted in above definition.

3. Illumination: light energy radiated or received on a unit area of surface. For a point light source, where is the radiance of light source in the direction ; is due to the foreshorten of effective area of the surface patch in the radiation direction.

4. Surface: Bi-directional reflectance distribution function (BRDF): BRDF of a surface is the ratio of energy radiated from a surface patch in some direction to the energy arriving at the surface from some direction. Where --- incident angle; --- emitting angle

5. Radiance (reflectance) : The is the brightness of an object surface that you look at.

6. Imaging : Assume that the irradiance at a point (X,Y) of an image plane is equal to the radiance from a corresponding surface patch, i.e., E(X,Y) = L(x,y,z) = This indicates that the terms of brightness, intensity and gray-level will have the same meaning as irradiance for an image. Note: radiance ----- out-going energy irradiance ----- in-going energy

7. 2. Surface Reflectance 1) Lambertain surface (Diffuse surface) in micro: rough in macro: smooth Def. Lambertain surface is a kind of surface that reflects incident light equally in all directions regardless the direction of incident light. A Lambertain surface is assumed not to absorb any incident illumination.

8. where I 0 ----- illumination of incident light. ----- incident angle to surface. Remark: For a Lambertain surface, its brightness percieved by a viewer do not relate to the position of the viewer. But the brightness is strongly related to the direction from which the light is illuminated to the surface.

9. 2) Specula surface (mirror) 3) Combinational surface: Surface with reflectance between Lambertain and specular surfaces: Where -----weigh factor

10. 3. Shape from shading (Horn,1970) Assumptions: i) Lambertain surface ii) Parallel light source from known direction iii) Orthographic projection for image Imaging: given the condition of light source and object surface in 3D, the intensity of each pixel in image plane can be uniquely determined. Question: If an image is given, can we evaluate the structure of the 3D surface? i.e., can we find the surface normal at each surface point? How is about the illumination of light source?

11. Surface Orientation : Let a 3D surface be z=z (x,y), then, Denote p=, q= Surface normal

12. Note: p and q are the functions of 3D point (x,y,z) at a surface. Under the assumption of orthographic projection p and q are also the functions of (X,Y) of image, i.e. p = p(x,y) = p(X,Y), q = q(x,y) = q(X,Y) Question : How to determine the two functions from image clue?

13. 1) Reflectance map Assume: Lambertain surface E(x,y) = = When image intensity is normalized by then On another hand, for a 3D surface, let be surface normal; Let be incident direction of light, then

14. Where is called reflectance map of a 3D surface. To solve for p(x,y) and q(x,y), we can use variation to minimize [I(x,y)  R(p,q)] 2 over image plane: Note that this is an ill-conditional problem, to solve for functions p = p(x,y) and q = q(x,y), we enforce smoothness constraint:

15. Thus, the minimization becomes In discrete images: Let

16. We get So,

17. Since and are computed from the value of the neighbor pixels of p(x,y), iterative method should be used to solve for p(x,y) and q(x,y). For a Lambertain surface,

18. Remark: 1) assumed incident light direction is known. 2) the choice of initial values of p(x,y) and q(x,y) is important to get a convergence at global minimum.

19. 4. Photometric stereo Assume: i) Fixed camera position ii)Light source located at 3 different positions to surface, at each position, One images is obtained. iii) Lambertain surface iv) Orthographic projection for three images. Let incident direction be :, i=1,2,3

20. Surface normal:, unknown Image: normalized image: From reflectance map of Lambertain: Write Then we have. This is a linear equation,which can be solved by

21. Further reading: In estimation of surface orientation, we need to know the direction of the illumination from light source first !! How to estimate the direction of light source from an intensity image? ---- Pentland’s method ---- Lee & Rosenfeld’s method ---- Tsai and Shah’s method ---- Zheng & Chellapa’s method All assume that the 3D surface is a part of share in the stage of estimation of illumination direction of light source.