Shape from Shading Course web page: vision.cis.udel.edu/cv February 26, 2003  Lecture 6

Announcements Changed HW1 page to a PDF file to eliminate font problems (let me know if hyperlinks don’t work) Read about color in Forsyth & Ponce, Chapter 6-6.1, 6.3, 6.5 for Friday

Outline Light sources –Exitance –Point sources Photometric stereo –Shape from point sources at infinity Complications—e.g., interreflections More kinds of light sources

Light Source Exitance Just like radiosity, but generated internally –Foreshortening because off-normal viewing angles make patch look smaller

Point Light Source Model as tiny (radius ² ) light-emitting sphere with constant exitance Good approximation when light is small relative to distance to viewer (e.g., light bulb, the sun) Solid angle: Recall that d! = (dA cos®)/r 2 –For sphere, patch area is of circle and normal is always aligned with viewing direction, so d! = ¼² 2 /r 2 Radiosity scales as 1/r 2

Illumination by a Point Source at Infinity Consider a distant Lambertian object (so orthographic projection is reasonable) in camera coordinates Image brightness at a pixel (based on reflectance equation): I(x, y) = k ½(x, y) s ¢ n(x, y) where k includes the BRDF (with albedo factored out), the light’s exitance, and a photometric factor, and s is the direction of the light Let n(x, y) be normal at I(x, y) (these are all column vectors)

Normal Information from a Point Source at Infinity Suppose k and s are known, and let g(x, y) = ½(x, y) n(x, y) and v = k s, so we can write: I(x, y) = v ¢ g(x, y) Not enough information! –Image brightness constrains the polar angle of the normal at each point on the object surface, but not the azimuth—i.e., we only know that the solution is on a circle –We do know that where I(x, y) is maximal over the entire object, n(x, y) = s. This is where the highlight is on a specular object Additional sources of information –More lights (one per image—in the same image their effects sum like a single “virtual” light) –Assume normal vector varies smoothly over object

Solving for the Normals with Multiple Point Sources Need 3 circles for unique intersection point ! need 3 light sources to solve for g(x, y) Formally, we must solve a linear system at each point, which we can write as: In Matlab, solve using B = V\g Albedo is just the length of g(x, y), and n(x, y) is the result of normalizing it V B

Photometric Stereo: Example From Forsyth & Ponce Input images Recovered albedo Recovered normals

Normals from Multiple Point Sources at Infinity: Considerations Shadows –When surface patch at (x, y) is occluded, I(x, y) = 0 and the image brightness equation will be invalid there ! Zero out that row of V Intuition about error –More lights help... (least-squares solution) So that no point is illuminated by < 3 lights To reduce effects of noise –Geometry of lights matters (close together is bad)

Shape from Normals: Getting the Gradient Shape typically means depth—the z values. With these we can make a height map like Matlab’s mesh Suppose z = f(x, y) and n = (n 1, n 2, n 3 ) T. By definition, the gradient is: In these terms, n = (p, q, 1) T, so we can compute p = n 1 /n 3 and q = n 2 /n 3.

Shape from Gradient We can integrate partial derivatives along a path to get the function value at the end of the path Simple path: Starting at image origin (0, 0), follow row to x coordinate, then column to y coordinate for each point

Photometric Stereo Example: Recovered Height Map From Forsyth & Ponce

Shape Computation: Complications Usefulness of photometric stereo for specular objects inversely proportional to magnitude of diffuse component of BRDF –For mirror objects, mitigating cue is distorted reflection of environment (analogous to camera calibration) Interreflections –Light from other objects = light sources we don’t know about –Ambient illumination approximation can help—add constant to radiosity everywhere (results in extra term in photometric stereo) courtesy of P. Debevec

More Light Source Types Spotlight –Point source constrained to a small solid angle Line –For long lines, radiosity scales as 1/r Area (e.g., overcast sky) –For big areas, radiosity is uniform for nearby viewers. –Shape from shading more difficult—in the worst case, the cosine term disappears and there is no shape information at all