geometric transformations and image warpingcs6640/geometric_trans.pdfimage mosaicing • piecing...
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Geometric Transformations and Image Warping
Ross Whitaker SCI Institute, School of Computing
University of Utah
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Geometric Transformations • Greyscale transformations -> operate on range/
output • Geometric transformations -> operate on image
domain – Coordinate transformations – Moving image content from one place to another
• Two parts: – Define transformation – Resample greyscale image in new coordinates
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Geom Trans: Distortion From Optics
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Barrel Distortion Pincushion Distortion
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Geom Trans: Distortion From Optics
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Geom. Trans.: Brain Template/Atlas
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Geom. Trans.: Mosaicing
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Domain Mappings Formulation
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New image from old one
Coordinate transformation Two parts – vector valued
g is the same image as f, but sampled on these new coordinates
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Domain Mappings Formulation
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Vector notation is convenient. Bar used some times, depends on context.
T may or may not have an inverse. If not, it means that information was lost.
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Domain Mappings
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f g
f g
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No Inverse?
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f g
f g Not “one to one”
Not “onto” - doesn’t cover f
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Implementation – Two Approaches
• Pixel filling – backward mapping – T() takes you from coords in g() to coords in f() – Need random access to pixels in f() – Sample grid for g(), interpolate f() as needed
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f g
Interpolate from nearby grid points
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Interpolation: Bilinear
• Successive application of linear interpolation along each axis
12 Source: WIkipedia
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Bilinear Interpolation
• Not linear in x, y
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Binlinear Interpolation
• Convenient form – Normalize to unit grid [0,1]x[0,1]
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Implementation – Two Approaches
• Splatting – backward mapping – T-1() takes you from coords in f() to coords in g() – You have f() on grid, but you need g() on grid – Push grid samples onto g() grid and do interpolation from
unorganized data (kernel)
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f g
Nearby points are not organized – ”scattered”
?
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Scattered Data Interpolation With Kernels Shepard’s method
• Define kernel – Falls off with distance, radially symmetric
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g
?
Kernel examples
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Shepard’s Method Implementation
• If points are dense enough – Truncate kernel – For each point in f()
• Form a small box around it in g() – beyond which truncate • Put weights and data onto grid in g()
– Divide total data by total weights: B/A
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g
Data and weights accumulated here
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Transformation Examples
• Linear
• Homogeneous coordinates
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Special Cases of Linear • Translation) • Rotation • Rigid = rotation +
translation • Scaling
– Include forward and backward rotation for arbitary axis
• Skew • Reflection
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p, q < 1 : expand
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Linear Transformations
• Also called “affine” – 6 parameters
• Rigid -> 3 parameters • Invertability
– Invert matrix
• What does it mean if A is not invertible?
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Other Transformations
• All polynomials of (x,y) • Any vector valued function with 2 inputs • How to construct transformations
– Define form or class of a transformation – Choose parameters within that class
• Rigid - 3 parameters • Affine - 6 parameters
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Correspondences
• Also called “landmarks” or “fiducials”
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Transformations/Control Points Strategy
• Define a functional representation for T with k parameters (B)
• Define (pick) N correspondences • Find B so that
• If overconstrained (K < 2N) then solve
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Example: Quadratic
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Transformation
Denote
Correspondences must match
Note: these equations are linear in the unkowns
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Write As Linear System
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A – matrix that depends on the (unprimed) correspondences and the transformation x – unknown parameters of the transformation b – the primed correspondences
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Linear Algebra Background
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Simple case: A is sqaure (M=N) and invertable (det[A] not zero)
Numerics: Don’t find A inverse. Use Gaussian elimination or some kind of decomposition of A
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Linear Systems – Other Cases
• M<N or M = N and the equations are degenerate or singular – System is underconstrained – lots of solutions
• Approach – Impose some extra criterion on the solution – Find the one solution that optimizes that criterion – Regularizing the problem
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Linear Systems – Other Cases
• M > N – System is overconstrained – No solution
• Approach – Find solution that is best compromise – Minimize squared error (least squares)
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Solving Least Squares Systems
• Psuedoinverse (normal equations)
– Issue: often not well conditioned (nearly singular)
• Alternative: singular value decomposition
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Singular Value Decomposition
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Invert matrix A with SVD
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SVD for Singular Systems
• If a system is singular, some of the w’s will be zero
• Properties: – Underconstrained: solution with shortest overall length – Overconstrained: least squares solution
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Warping Application: Lens Distortion
• Radial transformation – lenses are generally circularly symmetric – Optical center is known
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Correspondences
• Take picture of known grid – crossings
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Image Mosaicing
• Piecing together images to create a larger mosaic
• Doing it the old fashioned way – Paper pictures and tape – Things don’t line up – Translation is not enough
• Need some kind of warp • Constraints
– Warping/matching two regions of two different images only works when… 34
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Special Cases
• Nothing new in the scene is uncovered in one view vs another – No ray from the camera gets behind another
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1) Pure rotations–arbitrary scene 2) Arbitrary views of planar surfaces
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3D Perspective and Projection
• Camera model
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z
x y
f
Image coordinates
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Perspective Projection Properties
• Lines to lines (linear) • Conic sections to conic sections • Convex shapes to convex shapes • Foreshortening
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Image Homologies
• Images taken under cases 1,2 are perspectively equivalent to within a linear transformation – Projective relationships – equivalence is
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Transforming Images To Make Mosaics
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Linear transformation with matrix P
Perspective equivalence Multiply by denominator and reorganize terms
Linear system, solve for P
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Image Mosaicing
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4 Correspondences
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5 Correspondences
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6 Correspondences
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Mosaicing Issues
• Need a canvas (adjust coordinates/origin) • Blending at edges of images (avoid sharp
transitions) • Adjusting brightnesses • Cascading transformations
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Specifying Warps – Another Strategy
• Let the # DOFs in the warp equal the # of control points (x1/2) – Interpolate with some grid-based interpolation
• E.g. binlinear, splines
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Landmarks Not On Grid
• Landmark positions driven by application • Interpolate transformation at unorganized
correspondences – Scattered data interpolation
• How do we do scattered data interpolation? – Idea: use kernels!
• Radial basis functions – Radially symmetric functions of distance to landmark
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RBFs – Formulation
• Represent f as weighted sum of basis functions
• Need interpolation for vector-valued function, T:
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Sum of radial basis functions Basis functions centered at positions of data
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Solve For k’s With Landmarks as Constraints
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Issue: RBFs Do Not Easily Model Linear Trends
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f(x)
x
f1
f2
f3
x1 x2 x3
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RBFs – Formulation w/Linear Term
• Represent f as weighted sum of basis functions and linear part
• Need interpolation for vector-valued function, T:
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Sum of radial basis functions Linear part of transformation Basis functions centered
at positions of data
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RBFs – Solution Strategy
• Find the k’s and p’s so that f() fits at data points – The k’s can have no linear trend (force it into the p’s)
• Constraints -> linear system
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Correspondences must match
Keep linear part separate from deformation
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RBFs – Linear System
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RBF Warp – Example
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What Kernel Should We Use
• Gaussian – Variance is free parameter – controls smoothness of warp
54 σ = 2.5 σ = 2.0 σ = 1.5
From: Arad et al. 1994
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RBFs – Aligning Faces
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Mona Lisa – Target Venus – Source Venus – Warped
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RBFs – Special Case: Thin Plate Splines
• A special class of kernels
• Minimizes the distortion function (bending energy)
– No scale parameter. Gives smoothest results – Bookstein, 1989
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Application: Image Morphing
• Combine shape and intensity with time parameter t – Just blending with amounts t produces “fade”
– Use control points with interpolation in t
– Use c1, c(t) landmarks to define T1, and c2,c(t) landmarks to define T2
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Image Morphing
• Create from blend of two warped images
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T2 T1
I1 It I2
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Image Morphing
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Application: Image Templates/Atlases
• Build image templates that capture statistics of class of images – Accounts for shape and intensity – Mean and variability
• Purpose – Establish common coordinate system (for comparisons) – Understand how a particular case compares to the general
population
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Templates – Formulation
• N landmarks over M different subjects/samples
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Images
Correspondences
Mean of correspondences (template)
Transformations from mean to subjects Templated image
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Cars
62
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Car Landmarks and Warp
63
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Car Landmarks and Warp
64
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Car Mean
65
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Cars
66
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67
Cats
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Brains
68
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Brain Template
69