The Frequency Domain. Somewhere in Cinque Terre, May 2005. 15463: Computational Photography Alexei Efros, CMU, Fall 2011. Many slides borrowed from Steve Seitz. Salvador Dali “Gala Contemplating the Mediterranean Sea, which at 30 meters becomes the portrait of Abraham Lincoln ”, 1976.
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Somewhere in Cinque Terre, May 2005
15463: Computational Photography
Alexei Efros, CMU, Fall 2011
Many slides borrowed
from Steve Seitz
Salvador Dali
“Gala Contemplating the Mediterranean Sea,
which at 30 meters becomes the portrait
of Abraham Lincoln”, 1976
Salvador Dali, “Gala Contemplating the Mediterranean Sea, which at 30 meters becomes the portrait of Abraham Lincoln”, 1976
Salvador Dali, “Gala Contemplating the Mediterranean Sea, which at 30 meters becomes the portrait of Abraham Lincoln”, 1976
Teases away fast vs. slow changes in the image.
This change of basis has a special name…
had crazy idea (1807):
Any univariate function can be rewritten as a weighted sum of sines and cosines of different frequencies.
Don’t believe it?
Neither did Lagrange, Laplace, Poisson and other big wigs
Not translated into English until 1878!
But it’s (mostly) true!
called Fourier Series
there are some subtle restrictions
...the manner in which the author arrives at these equations is not exempt of difficulties and...his analysis to integrate them still leaves something to be desired on the score of generality and even rigour.
Laplace
Legendre
Lagrange
Inverse Fourier
Transform
Fourier
Transform
F(w)
f(x)
F(w)
f(x)
We can always go back:
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in Matlab, check out: imagesc(log(abs(fftshift(fft2(im)))));
Intensity Image
Fourier Image
http://sharp.bu.edu/~slehar/fourier/fourier.html#filtering
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http://sharp.bu.edu/~slehar/fourier/fourier.html#filtering
More: http://www.cs.unm.edu/~brayer/vision/fourier.html
The greatest thing since sliced (banana) bread!
The Fourier transform of the convolution of two functions is the product of their Fourier transforms
The inverse Fourier transform of the product of two Fourier transforms is the convolution of the two inverse Fourier transforms
Convolution in spatial domain is equivalent to multiplication in frequency domain!
F(sx,sy)
f(x,y)
*
h(x,y)
H(sx,sy)
g(x,y)
G(sx,sy)
Filtering
Gaussian
Box filter
Why does the Gaussian give a nice smooth image, but the square filter give edgy artifacts?
Gaussian
Box Filter
lowpass:
Highpass / bandpass:
original
smoothed (5x5 Gaussian)
smoothed – original
Gaussian Pyramid (lowpass images)
Need this!
Original
image
Gaussian
Laplacian of Gaussian
delta function
unit impulse
Laplacian of Gaussian
Gaussian
Gaussian Filter!
Laplacian Filter!
A. Oliva, A. Torralba, P.G. Schyns, “Hybrid Images,” SIGGRAPH 2006
Project Instructions:
http://www.cs.illinois.edu/class/fa10/cs498dwh/projects/hybrid/ComputationalPhotography_ProjectHybrid.html
Early processing in humans filters for various orientations and scales of frequency
Perceptual cues in the mid frequencies dominate perception
When we see an image from far away, we are effectively subsampling it
Early Visual Processing: Multiscale edge and blob filters
CampbellRobson contrast sensitivity curve

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R
G
B
Blockbased Discrete Cosine Transform (DCT)
The first coefficient B(0,0) is the DC component, the average intensity
The topleft coeffs represent low frequencies, the bottom right – high frequencies
Quantize
More coarsely for high frequencies (which also tend to have smaller values)
Many quantized high frequency values will be zero
Encode
Can decode with inverse dct
Filter responses
Quantization table
Quantized values
http://en.wikipedia.org/wiki/YCbCr
http://en.wikipedia.org/wiki/JPEG
Subsample color by factor of 2
Split into blocks (8x8, typically), subtract 128
For each block
Block size
small block
faster
correlation exists between neighboring pixels
large block
better compression in smooth regions
It’s 8x8 in standard JPEG
89k
12k
How to compute a derivative?
Where is the edge?
Look for peaks in
Where is the edge?
Laplacian of Gaussian
operator
Where is the edge?
Zerocrossings of bottom graph
Laplacian of Gaussian
is the Laplacian operator:
Gaussian
derivative of Gaussian