EECS%20274%20Computer%20Vision

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EECS 274 Computer Vision

• Sources, Shadows, and Shading

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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 Chapter5, H Chapter 11

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Radiometric properties of sources

• 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

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Radiosity due to a point sources

• small, distant sphere radius e and exitance E,
  which is far away subtends solid angle

Typo in figure, d ? r
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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

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Nearby point source model

• The angle term can be written in terms of N and S
• N is the surface normal
• ?d is diffuse albedo
• S is source vector - a vector from P to the
  source, whose length is the intensity term, e2E
• works because a dot-product is basically a cosine

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Point source at infinity

• Issue nearby point source gets bigger if one
  gets closer
• the sun doesnt for any reasonable binding of
  closer
• Assume that all points in the model are close to
  each other with respect to the distance to the
  source
• Then the source vector doesnt vary much, and the
  distance doesnt vary much either, and we can
  roll the constants together to get

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Line sources
radiosity due to line source varies with inverse
distance, if the source is long enough
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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

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Radiosity due to an area source

• ?d is albedo
• E is exitance
• r is distance between points Q and P
• Q is a coordinate on the source

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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

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Local shading models

• For point sources at infinity
• For point sources not at infinity

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Shadows cast by a point source

• A point that cant see the source is in shadow
  (self cast shadow)
• For point sources, the geometry is simple

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Area source shadows

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

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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)

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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
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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

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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
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Dealing with shadows
Known
Known
Known
Unknown
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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 r(x,y) g(x, y)
• This yields a check
• If the magnitude of g(x, y) is greater than 1,
  theres a problem
• And N(x, y) g(x, y) / r(x,y)

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Five synthetic images
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Recovered reflectance
g(x,y)?(x,y) the value should be in the range
of 0 and 1
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Recovered normal field
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Recovering a surface 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

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Recovering a surface from normals (contd)

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

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Recovered surface by integration
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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
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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
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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 fxyfyx (Georghiades,
  Belhumeur, Kriegman, PAMI, June, 2001)

3D linear 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
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Image-based rendering Cast Shadows
Single Light Source
Face Movie
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Yale face database B

• 10 Individuals
• 64 Lighting Conditions
• 9 Poses
• gt 5,760 Images

Variable lighting
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Curious experimental fact

• Prepare two rooms, one with white walls and white
  objects, one with black walls and black objects
• Illuminate the black room with bright light, the
  white room with dim light
• People can tell which is which (due to Gilchrist)
• Why? (a local shading model predicts they cant).

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Global shading models
Can adjust so that local shading model predicts
these pictures will be indistinguishable
A view of a white room with white objects. We see
a cross-section of the image intensity
corresponding to the line drawn on the image.
A view of a black room with black objects. We see
a cross-section of the image intensity
corresponding to the line drawn on the image.
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Whats going on here?

• 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

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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 cant

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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

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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