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ECE 472/572 - Digital Image Processing
Lecture 6 – Geometric and Radiometric Transformation 09/27/11
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Roadmap
¬ Introduction – Image format (vector vs. bitmap) – IP vs. CV vs. CG – HLIP vs. LLIP – Image acquisition
¬ Perception – Structure of human eye – Brightness adaptation and Discrimination – Image resolution
¬ Image enhancement – Enhancement vs. restoration – Spatial domain methods
• Point-based methods – Log trans. vs. Power-law – Contrast stretching vs. HE – Gray-level vs. Bit plane slicing – Image averaging (principle)
• Mask-based methods - spatial filter – Smoothing vs. Sharpening filter – Linear vs. Non-linear filter – Smoothing (average vs. Gaussian vs. median) – Sharpening (UM vs. 1st vs. 2nd derivatives)
– Frequency domain methods • Understanding Fourier transform • Implementation in the frequency domain • Low-pass filters vs. high-pass filters vs. homomorphic filter
¬ Geometric correction – Affine vs. Perspective
transformation • Homogeneous coordinates • Inverse vs. forward transform • Composite transformation
– General transformation • Model distortion with polynomial • Least square solution
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Questions
¬ Affine transformation vs. Perspective transformation
¬ Forward transformation vs. Inverse transformation
¬ Composite transformation vs. Sequential transformation
¬ Homogeneous coordinate ¬ General geometric transformations
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Usage
¬ Image correction ¬ Color interpolation ¬ Forensic analysis ¬ Entertainment effect
http://w3.impa.br/~morph/ http://www.mpi-sb.mpg.de/resources/FAM/demos.html
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Affine transformations
¬ Preserve lines and parallel lines ¬ Homogeneous coordinates ¬ General form
¬ Special matrices – R: rotation, S: scaling, T: translation, H: shear
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R =
cosθ −sinθ 0sinθ cosθ 00 0 1
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,S =
sx 0 00 sy 00 0 1
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,T =
1 0 tx0 1 ty0 0 1
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,H =
1 hx 0hy 1 00 0 1
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uv1
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=
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⋅
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Composite vs. Sequential transformation
Original Image (f(x,y))
R H T S
Transformed Image (g(u, v))
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uv1
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' ' '
= S ⋅T ⋅H ⋅ R ⋅xy1
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RHTSC ⋅⋅⋅=
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Forward vs. Inverse transforms
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uv1
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= C ⋅xy1
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Forward transform
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C−1
uv1
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=
xy1
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Inverse transform
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Examples - Shear
hy = 0.2 hx = 0.2 hx = hy = 0.2
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Examples – Translation + Rotation
theta = PI/4 theta = PI/4 tx = -140, ty = 60
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Perspective transformation
¬ Preserve parallel lines only when they are parallel to the projection plane. Otherwise, lines converge to a vanishing point
¬ General form
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" u " v " w
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xy1
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,u =" u " w ,v =
" v " w
0,0 3231 ≠≠ aa
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Determine the coefficients
8 unknowns, 4-point least squares
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,u =" u " w ,v =
" v " w
(0,0) (0,255)
(255,0) (255,255)
(0,0) (0,255)
(255,0)
(511,511)
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Example - PT
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General approaches
¬ Find tiepoints ¬ Spatial transformation
9/27/11 15
x-ray sensitive scintillator
fiber optics
CCD array 1242 x 1152
misalignment greater than 50 micron
Example – CCD butting
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9/27/11 16
Sources of distortions
¬ defects in the production of fiber-optic tapers
¬ imperfect compression and cutting ¬ different light transfer efficiency across the
whole surface
9/27/11 17
),(),( yxvuP =
),(),( yxvuT =
approximation
interpolation
),( yx ),( vu
control point
map exactly
map close
Geometric correction
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Spatial transformation
¬ Bilinear equation
¬ n-th degree polynomial
¬ Use information from tiepoints to solve coefficients – Exact solution – Least square solution
€
ˆ x = r u,v( ) = a1u + a2v + a3uv + a4
ˆ y = s u,v( ) = b1u + b2v + b3uv + b4
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ˆ x iˆ y i
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Px (ui,vi)Py (ui,vi)"
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akrsuirvi
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bkrsuirvi
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r+s= k∑
k= 0
d
∑
∑−
=
−+−=1
0
22, ])ˆ()ˆ[(minm
iiiiiba yyxxε
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How is it applied?
¬ Step 1: Choose a set of tie points – (xi,yi): coordinates of tie points in
the original (or distorted) image – (ui,vi): coordinates of tie points in
the corrected image ¬ Step 2: Decide on which degree of
polynomial to use to model the inverse of the distortion, e.g.,
¬ Step 3: Solve the coefficients of the polynomial using least-squares approach
¬ Step 4: Use the derived polynomial model to correct the entire original image
€
X = x0 x1 x24[ ]T
Y = y0 y1 y24[ ]T
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ˆ x = r u,v( ) = a1u + a2v + a3uv + a4
ˆ y = s u,v( ) = b1u + b2v + b3uv + b4
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W =
u0 v0 u0v0 1u1 v1 u1v1 1 u24 v24 u24v24 1
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' ' ' '
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A = a1 a2 a3 a4[ ]T
B = b1 b2 b3 b4[ ]T
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A =W −1X,B =W −1YW −1 = (W TW )−1W T
For each (u,v) in the corrected image, find the corresponding (x,y) in the original image and use its intensity as the intensity at (u,v).
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Example – Geometric correction
¬ Geometric correction of images from butted CCD arrays
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Example - Color correction
Tiepoints are colors (R, G, B), instead of spatial coordinates
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Example - Image warping
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Image warping
¬ Two-pass mesh warping by Douglas Smythe
¬ Reference: G. Wolberg, Digital Image Warping, 1990
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Example 1
From Joey Howell and Cory McKay, ECE472, Fall 2000
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Example 2
From Adam Miller, Truman Bonds, Randal Waldrop, ECE472, Fall 2000