EECS 274 Computer Vision

EECS 274 Computer Vision Sources, Shadows, and Shading

Surface brightness • Depends on local surface properties (albedo), surface shape (normal), and illumination • Shading model: a model of how brightness of a surface is obtained • Can interpret pixel values to reconstruct its shape and albedo • Reading: FP Chapter 2, H Chapter 11

Radiometric properties • How bright (or what color) are objects? • One more definition: Exitance of a light source is • the internally generated power (not reflected) radiated per unit area on the radiating surface • Similar to radiosity: a source can have both • radiosity, because it reflects • exitance, because it emits • Independent of its exit angle • Internally generated energy radiated per unit time, per unit area • But what aspects of the incoming radiance will we model? • Point, line, area source • Simple geometry

Radiosity due to a point sources

Radiosity due to a point source • As r is increased, the rays leaving the surface patch and striking the sphere move closer evenly, and the collection changes only slightly, i.e., diffusive reflectance, or albedo • Radiosity due to source

Nearby point source model • The angle term, can be written in terms of N and S • N: surface normal • ρd: diffuse albedo • S: source vector - a vector from P to the source, whose length is the intensity term, ε2E • works because a dot-product is basically a cosine

Point source at infinity • Issue: nearby point source gets bigger if one gets closer • the sun doesn’t for any reasonable assumption • Assume that all points in the model are close to each other with respect to the distance to the source • Then the source vector doesn’t vary much, and the distance doesn’t vary much either, and we can roll the constants together to get:

Line sources Infinitely long narrow cylinder with constant exitance radiosity due to line source varies with inverse distance, if the source is long enough

Area sources • Examples: diffuser boxes, white walls • The radiosity at a point due to an area source is obtained by adding up the contribution over the section of view hemisphere subtended by the source • change variables and add up over the source

Radiosity due to an area source • ρd is albedo • E is exitance • r is distance between points Q and P • Q is a coordinate on the source

Shading models • Local shading model • Surface has radiosity due only to sources visible at each point • Advantages: • often easy to manipulate, expressions easy • supports quite simple theories of how shape information can be extracted from shading • Global shading model • Surface radiosity is due to radiance reflected from other surfaces as well as from surfaces • Advantages: • usually very accurate • Disadvantage: • extremely difficult to infer anything from shading values

Local shading models • For point sources at infinity: • For point sources not at infinity

Shadows cast by a point source • A point that can’t see the source is in shadow (self cast shadow) • For point sources, the geometry is simple (i.e., the relationship between shape and shading is simple) • Radiosity is a measurement of one component of the surface normal Analogous to the geometry of viewing in a perspective camera

Area source shadows • Are sources do not produce dark • shadows with crisp boundaries • Out of shadow • Penumbra (“almost shadow”) • Umbra (“shadow”)

Photometric stereo • Assume: • A local shading model • A set of point sources that are infinitely distant • A set of pictures of an object, obtained in exactly the same camera/object configuration but using different sources • A Lambertian object (or the specular component has been identified and removed)

Monge patch Projection model for surface recovery - Monge patch In computer vision, it is often known as height map, depth map, or dense depth map

Image model • For each point source, we know the source vector (by assumption) • We assume we know the scaling constant of the linear camera (i.e., intensity value is linear in the surface radiosity) • Fold the normal and the reflectance into one vector g, and the scaling constant and source vector into another Vj • Out of shadow: • g(x,y): describes the surface • Vj: property of the illumination and of the camera • In shadow:

From many views • From n sources, for each of which Vi is known • For each image point, stack the measurements • Solve least squares problem to obtain g One linear system per point

Dealing with shadows Known Known Known Unknown

Recovering normal and reflectance • Given sufficient sources, we can solve the previous equation (e.g., least squares solution) for g(x, y) • Recall that g(x, y) =r (x,y) N(x, y) , and N(x, y) is the unit normal • This means that alberdo r(x,y) =||g(x, y)|| • This yields a check • If the magnitude of g(x, y) is greater than 1, there’s a problem • And N(x, y) = g(x, y) / r(x,y)

Five synthetic images Generated from a sphere in a orthographic view from the same viewing position

Recovered reflectance ||g(x,y)||=ρ(x,y): the alberdo value should be in the range of 0 and 1

Recovered normal field For viewing purpose, vector field is shown for every 16th pixel in each direction

Parametric surface tangents v u

Shape from normals • Recall the surface is written as • Parametric surface • This means the normal has the form: • If we write the known vector g as • Then we obtain values for the partial derivatives of the surface:

Shape from normals • Recall that mixed second partials are equal --- this gives us a check. We must have: (or they should be similar, at least) • Known as integrability test • We can now recover the surface height at any point by integration along some path, e.g.

Recovered surface by integration

x x x 1 2 n The illumination cone What is the set of n-pixel images of an object under all possible lighting conditions (at fixed pose)? (Belhuemuer and Kriegman IJCV 99) Single light source image N-dimensional Image Space

x x x 1 2 n N-dimensional Image Space The illumination cone What is the set of n-pixel images of an object under all possible lighting conditions (but fixed pose)? Proposition:Due to the superposition of images, the set of images is a convex polyhedral cone in the image space. Illumination Cone 2-light source image Single light source images: Extreme rays of cone

Generating the illumination cone For Lambertian surfaces, the illumination cone is determined by the 3D linear subspace B(x,y),where When no shadows, then Use least-squares to find 3D linear subspace, subject to the constraint fxy=fyx(Georghiades, Belhumeur, Kriegman, PAMI, June, 2001) 3Dlinear subspace a(x,y)fx(x,y) fy(x,y) albedo (surface normals) Surface. f(x,y) (albedo textured mapped on surface) Original (Training) Images

Image-based rendering: Cast shadows Single Light Source Face Movie

Yale face database B • 10 Individuals • 64 Lighting Conditions • 9 Poses • => 5,760 Images Variable lighting

Limitation • Local shading model is a poor description of physical processes that give rise to images • because surfaces reflect light onto one another • This is a major nuisance; the distribution of light (in principle) depends on the configuration of every radiator; big distant ones are as important as small nearby ones (solid angle) • The effects are easy to model • It appears to be hard to extract information from these models

Interreflections - a global shading model • Other surfaces are now area sources - this yields: • Vis(x, u) is 1 if they can see each other, 0 if they can’t

What do we do about this? • Attempt to build approximations • Ambient illumination • Study qualitative effects • reflexes • decreased dynamic range • smoothing • Try to use other information to control errors

Ambient illumination • Two forms • Add a constant to the radiosity at every point in the scene to account for brighter shadows than predicted by point source model • Advantages: simple, easily managed (e.g. how would you change photometric stereo?) • Disadvantages: poor approximation (compare black and white rooms • Add a term at each point that depends on the size of the clear viewing hemisphere at each point • Advantages: appears to be quite a good approximation, but jury is out • Disadvantages: difficult to work with

EECS 274 Computer Vision