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Transcript of Image Pyramids and Blending - Computer graphicsImage Pyramids Known as a Gaussian Pyramid [Burt and...
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Image Pyramids and Blending
15-463: Computational PhotographyAlexei Efros, CMU, Fall 2005
© Kenneth Kwan
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Image Pyramids
Known as a Gaussian Pyramid [Burt and Adelson, 1983]• In computer graphics, a mip map [Williams, 1983]• A precursor to wavelet transform
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A bar in the big images is a hair on the zebra’s nose; in smaller images, a stripe; in the smallest, the animal’s nose
Figure from David Forsyth
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What are they good for?Improve Search
• Search over translations– Like homework– Classic coarse-to-fine strategy
• Search over scale– Template matching– E.g. find a face at different scales
Precomputation• Need to access image at different blur levels• Useful for texture mapping at different resolutions (called
mip-mapping)
Image Processing• Editing frequency bands separately• E.g. image blending…
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Gaussian pyramid construction
filter mask
Repeat• Filter• Subsample
Until minimum resolution reached • can specify desired number of levels (e.g., 3-level pyramid)
The whole pyramid is only 4/3 the size of the original image!
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Image sub-sampling
Throw away every other row and column to create a 1/2 size image
- called image sub-sampling
1/4
1/8
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Image sub-sampling
1/4 (2x zoom) 1/8 (4x zoom)
Why does this look so bad?
1/2
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Sampling
Good sampling:•Sample often or,•Sample wisely
Bad sampling:•see aliasing in action!
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Really bad in video
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Alias: n., an assumed name
Picket fence recedingInto the distance willproduce aliasing…
Input signal:
x = 0:.05:5; imagesc(sin((2.^x).*x))
Matlab output:
WHY?
Aj-aj-aj:Alias!
Not enough samples
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Gaussian pre-filtering
G 1/4
G 1/8
Gaussian 1/2
Solution: filter the image, then subsample• Filter size should double for each ½ size reduction. Why?
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Subsampling with Gaussian pre-filtering
G 1/4 G 1/8Gaussian 1/2
Solution: filter the image, then subsample• Filter size should double for each ½ size reduction. Why?• How can we speed this up?
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Compare with...
1/4 (2x zoom) 1/8 (4x zoom)1/2
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Image Blending
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Feathering
01
01
+
=Encoding transparency
I(x,y) = (αR, αG, αB, α)
Iblend = Ileft + Iright
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Affect of Window Size
0
1 left
right0
1
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Affect of Window Size
0
1
0
1
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Good Window Size
0
1
“Optimal” Window: smooth but not ghosted
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What is the Optimal Window?To avoid seams
• window >= size of largest prominent feature
To avoid ghosting• window <= 2*size of smallest prominent feature
Natural to cast this in the Fourier domain• largest frequency <= 2*size of smallest frequency• image frequency content should occupy one “octave” (power of two)
FFT
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What if the Frequency Spread is Wide
Idea (Burt and Adelson)• Compute Fleft = FFT(Ileft), Fright = FFT(Iright)• Decompose Fourier image into octaves (bands)
– Fleft = Fleft1 + Fleft
2 + …• Feather corresponding octaves Fleft
i with Frighti
– Can compute inverse FFT and feather in spatial domain• Sum feathered octave images in frequency domain
Better implemented in spatial domain
FFT
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What does blurring take away?
original
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What does blurring take away?
smoothed (5x5 Gaussian)
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High-Pass filter
smoothed – original
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Band-pass filtering
Laplacian Pyramid (subband images)Created from Gaussian pyramid by subtraction
Gaussian Pyramid (low-pass images)
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Laplacian Pyramid
How can we reconstruct (collapse) this pyramid into the original image?
Need this!
Originalimage
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Pyramid Blending
0
1
0
1
0
1
Left pyramid Right pyramidblend
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Pyramid Blending
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laplacianlevel
4
laplacianlevel
2
laplacianlevel
0
left pyramid right pyramid blended pyramid
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Laplacian Pyramid: BlendingGeneral Approach:
1. Build Laplacian pyramids LA and LB from images A and B2. Build a Gaussian pyramid GR from selected region R3. Form a combined pyramid LS from LA and LB using nodes
of GR as weights:• LS(i,j) = GR(I,j,)*LA(I,j) + (1-GR(I,j))*LB(I,j)
4. Collapse the LS pyramid to get the final blended image
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Blending Regions
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Horror Photo
© prof. dmartin
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Season Blending (St. Petersburg)
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Season Blending (St. Petersburg)
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Simplification: Two-band BlendingBrown & Lowe, 2003
• Only use two bands: high freq. and low freq.• Blends low freq. smoothly• Blend high freq. with no smoothing: use binary mask
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Low frequency (λ > 2 pixels)
High frequency (λ < 2 pixels)
2-band Blending
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Linear Blending
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2-band Blending
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Gradient DomainIn Pyramid Blending, we decomposed our
image into 2nd derivatives (Laplacian) and a low-res image
Let us now look at 1st derivatives (gradients):• No need for low-res image
– captures everything (up to a constant)• Idea:
– Differentiate– Blend– Reintegrate
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Gradient Domain blending (1D)
Twosignals
Regularblending
Blendingderivatives
bright
dark
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Gradient Domain Blending (2D)
Trickier in 2D:• Take partial derivatives dx and dy (the gradient field)• Fidle around with them (smooth, blend, feather, etc)• Reintegrate
– But now integral(dx) might not equal integral(dy)• Find the most agreeable solution
– Equivalent to solving Poisson equation– Can use FFT, deconvolution, multigrid solvers, etc.
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Comparisons: Levin et al, 2004
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Perez et al., 2003
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Perez et al, 2003
Limitations:• Can’t do contrast reversal (gray on black -> gray on white)• Colored backgrounds “bleed through”• Images need to be very well aligned
editing
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Don’t blend, CUT!
So far we only tried to blend between two images. What about finding an optimal seam?
Moving objects become ghosts
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Davis, 1998Segment the mosaic
• Single source image per segment• Avoid artifacts along boundries
– Dijkstra’s algorithm
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Input texture
B1 B2
Random placement of blocks
block
B1 B2
Neighboring blocksconstrained by overlap
B1 B2
Minimal errorboundary cut
Efros & Freeman, 2001
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min. error boundary
Minimal error boundary
overlapping blocks vertical boundary
__ ==22
overlap error
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GraphcutsWhat if we want similar “cut-where-things-
agree” idea, but for closed regions?• Dynamic programming can’t handle loops
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Graph cuts (simple example à la Boykov&Jolly, ICCV’01)
n-links
s
t a cuthard constraint
hard constraint
Minimum cost cut can be computed in polynomial time(max-flow/min-cut algorithms)
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Kwatra et al, 2003
Actually, for this example, DP will work just as well…
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Lazy Snapping (Li el al., 2004)
Interactive segmentation using graphcuts
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Putting it all togetherCompositing images
• Have a clever blending function– Feathering– blend different frequencies differently– Gradient based blending
• Choose the right pixels from each image– Dynamic programming – optimal seams– Graph-cuts
Now, let’s put it all together:• Interactive Digital Photomontage, 2004 (video)
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