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EECS 274 Computer Vision

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### 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 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

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

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.

x

x

1

2

n

The illumination coneWhat 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

1

2

n

N-dimensional

Image Space

The illumination coneWhat 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

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

- Add a constant to the radiosity at every point in the scene to account for brighter shadows than predicted by point source model

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