Extended Gaussian Images

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Extended Gaussian Images

• Berthold K. P. Horn

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Outline

• Discrete Case Convex Polyhedra
• Continuous Case Smoothly Curved Objects
• Discrete Approximation Needle Maps
• Tessellation of the Gaussian sphere Orientation
  Histograms
• Solids of Revolution

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Outline

• Discrete Case Convex Polyhedra
• Continuous Case Smoothly Curved Objects
• Discrete Approximation Needle Maps
• Tessellation of the Gaussian sphere Orientation
  Histograms
• Solids of Revolution

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Gaussian Image

• What is a Gaussian Image?

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Gaussian Image

• What is a Gaussian Image?
• A mapping of surface normals of an object onto
  the unit sphere (Gaussian sphere.)
• Tail lies at the center of the Gaussian sphere
• Head lies on the surface of the Gaussian sphere.

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Example
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Example
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Extended Gaussian Image

• Extended to include area of each face.
• Place point mass on the sphere surface equal to
  area of face at the head of each normal.
• Alternate representation Scale the normal
  proportional to area.

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Properties

• Face position information is lost.
• Which faces are connected?
• As long as the polyhedron is convex, its extended
  Gaussian image is unique.
• Algorithms exist that extract a polyhedron from
  an extended Gaussian Image.

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Properties

• Unaffected by translation
• Rotation of the object causes an equal rotation
  of the extended Gaussian image.
• Center of mass lies at the origin of the sphere.

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Center of mass property proof

• Imagine viewing a convex polyhedron from a great
  distance
• is a vector from the object in the direction
  of the viewer.
• is the unit normal at each face.
• is the surface area of each face.
• A face is visible if
• The apparent size due to foreshortening of a face
  is

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Center of mass property proof

• The apparent area of the visible surface is
• When viewed from the opposite direction

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Center of mass property proof

• Since the above two equations are equal, we have
• Since this is true for all
• That is, the center of mass of the extended
  Gaussian image is at the origin.

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Reconstructing a Tetrahedron

• Goal Find the offset from the center of mass for
  each face. (a, b, c, d)
• Given The surface normals
• and face areas A B C D.

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Reconstructing a Tetrahedron

• Tetrahedron property the distance from the
  center of mass to a face is ¼ the distance from
  that face to its opposite vertex.
• Start by finding a formula for distance from face
  of area D to opposite vertex d. The desired
  distance, d, with be ¼ that.
• Repeat for each vertex after doing a cyclical
  permutation of the variables.

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Reconstructing a Tetrahedron

• Assume D is at the origin, we have

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Reconstructing a Tetrahedron

• Find distance from origin to D face by dot
  product of A, B, or C vector and
• Area of the Face

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Reconstructing a Tetrahedron

• Consider the four distinct triple products of the
  normals

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Reconstructing a Tetrahedron

• Multiply these formulae

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Reconstructing a Tetrahedron

• Compute cyclic permutations of each face and use
  the previous equation to find the 4 distances of
  each face from the center of mass.
• With the normals and distances from the center of
  mass, create 4 planes.
• The intersection of the planes creates the
  tetrahedron.

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Outline

• Discrete Case Convex Polyhedra
• Continuous Case Smoothly Curved Objects
• Discrete Approximation Needle Maps
• Tessellation of the Gaussian sphere Orientation
  Histograms
• Solids of Revolution

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Gaussian Image

• Create the same mapping between object normal and
  Gaussian sphere.
• Consider a small patch on the object.
• Each point in patch maps to a point on Gaussian
  sphere.
• Creates a patch on the sphere

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Gaussian Curvature

• Gaussian Curvature
• K lt 1 curves in (like a crater)
• K 1 plane
• K gt 1 curves out (like a sphere)
• Integrate K over the object
• Integrate 1/K over the sphere
• Use 1/K in definition of extended Gaussian Image.

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Extended Gaussian Image

• Associate 1/K of a point on the surface with
  corresponded point on Gaussian sphere.
• Let u, v be parameters to identify points on the
  object.
• Let ?, ? be parameters to identify points on the
  sphere.
• Extended Gaussian Image

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Properties

• Center of mass at origin.
• Similar proof to discrete case
• Total mass of extended Gaussian Image equals
  surface area of object.
• Unaffected by translation
• Rotation of the object causes an equal rotation
  of the extended Gaussian image.

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Outline

• Discrete Case Convex Polyhedra
• Continuous Case Smoothly Curved Objects
• Discrete Approximation Needle Maps
• Tessellation of the Gaussian sphere Orientation
  Histograms
• Solids of Revolution

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Discrete Approximation

• Break the surface into small patches.
• Create a surface normal for each patch.
• Consider the polyhedron formed by the
  intersection of the planes tangent to each of
  these normals.
• Approximation of the original surface.

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Discrete Approximation

• How to break into small patches?
• Parameterize the model u, v
• Find the normal
• ru and rv are found by differentiating the
  parametric form of the equation for the surface.
• The area of a patch
• No need to compute K

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Needle Map
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Outline

• Discrete Case Convex Polyhedra
• Continuous Case Smoothly Curved Objects
• Discrete Approximation Needle Maps
• Tessellation of the Gaussian sphere Orientation
  Histograms
• Solids of Revolution

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Tessellation

• Need a way to represent in a computer
• Tessellate the Gaussian sphere
• Create a histogram of orientations

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Tessellation Properties

• All cells should have the same area.
• All cells should have the same shape.
• The cells should have regular shapes that are
  compact.
• The division should be fine enough to provide
  good angular resolution.
• For some rotation, the cells should be brought
  into coincidence with themselves.

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Simple Case

• Latitude and Longitude
• Easy to Compute which cell a normal belongs to
• No uniform shape or area
• Alignments only through rotations about its axis.

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Regular Polyhedra based Tessellation

• Project a regular polyhedron onto the unit
  sphere.
• Cells have the same shape and area.
• Sampling is too coarse.
• Icosahedron is only 20 regular cells.

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Semi-Regular Polyhedra

• Regular polygons for faces, but not the same
  polygon.
• Higher polygon count
• Area difference between the differing polygons
  could be significant.

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Finer Subdivision

• Each polygon can be sub-divided into triangles.
• How fine do we need?
• Lower bound for angular spread with n cells
• Example from paper n240, ?11.5

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Geodesic Domes

• Divide Triangular cells into four smaller
  triangles.
• All faces not same area or shape.
• Allows for arbitrary fineness.
• Start with pentakis dodecahedron or truncated
  icosahedron.
• Each triangle edge is divided into f sections to
  create f2 triangles.
• Desire that f is a power of 2.

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Geodesic Domes
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Geodesic Domes

• Create Histogram by assigning each normal to a
  cell on the tessellated Gaussian sphere.
• Which cell does a normal belong to?
• Find dot product between given vector and normals
  from the original regular polyhedron the dome is
  based on.
• Normal belongs to the face giving the greatest
  result.
• Determine which triangle within the face
• Use the second greatest normal from above.
• Proceed as above hierarchically.
• Binary search through the triangles
• In practice a lookup table is used

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Orientation Histogram

• Compute the number of normals in each cell.
• Count them
• Create the histogram
• Store in computer as simple array.
• Display graphically with a normal for each cell
  or as a grayscale image where the brightness of
  each cell is determined by the count.

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Outline

• Discrete Case Convex Polyhedra
• Continuous Case Smoothly Curved Objects
• Discrete Approximation Needle Maps
• Tessellation of the Gaussian sphere Orientation
  Histograms
• Solids of Revolution

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Solids of Revolution

• Rotate a curve about an axis to create a surface.
• Compute the EGI
• r is the distance from axis to arc.
• s is distance along arc.
• ? is the angle around the axis.
• ?, ? are the longitude and latitude on the
  sphere.
• ? corresponds to ?

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Gaussian Curvature

• Consider a small patch on the solid and the
  corresponding patch on the sphere, their areas
  are
• Sphere
• Solid
• The Gaussian curvature is the limit of the ratio
  of the areas as they approach zero

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Gaussian Curvature

• The curvature of the generating curve is the rate
  of change of direction with arc length. Giving
• Plug that in to the previous equation

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Gaussian Curvature

• From the figure
• Differentiate with respect to s
• Substitute

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Gaussian Curvature

• Sometimes we want to express r as a function of
  z.
• From the figure
• Differentiate with respect to s
• From the figure

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Gaussian Curvature

• Substituting
• More Substituting

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Conclusion

• How does it do as a shape descriptor?
• Concise to store
• Quick to compute
• Efficient to match
• Discriminating
• Invariant to transformations
• Invariant to deformations
• Insensitive to noise
• Insensitive to topology
• Robust to degeneracies

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Conclusion

• How does it do as a shape descriptor?
• J Concise to store
• J Quick to compute
• J Efficient to match
• L Discriminating
• L Invariant to transformations
• L Invariant to deformations
• L Insensitive to noise
• J Insensitive to topology
• L Robust to degeneracies