Computational Photography
Lecture 3 – filtering and frequencies
CS 590-134 (future 572) Spring 2016
Prof. Alex Berg
(Credits to many other folks on
individual slides)
Today
Today filtering and frequency analysis of images.
Slides from Derek Hoiem
Sampling and Reconstruction
Sampled representations
• How to store and compute with continuous functions?
• Common scheme for representation: samples
[FvDFH fig.14.14b / Wolberg]
– write down the function’s values at many points
© 2006 Steve Marschner • 5
Reconstruction
• Making samples back into a continuous function
[FvDFH fig.14.14b / Wolberg]
– for output (need realizable method)
– for analysis or processing (need mathematical method)
– amounts to “guessing” what the function did in between
© 2006 Steve Marschner • 6
1D Example: Audio
low
high
frequencies
Sampling in digital audio
• Recording: sound to analog to samples to
disc
• Playback: disc to samples to analog to
sound again
– how can we be sure we are filling in the gaps
correctly?
© 2006 Steve Marschner • 8
Sampling and Reconstruction
• Simple example: a sign wave
© 2006 Steve Marschner • 9
Undersampling
• What if we “missed” things between the
samples?
• Simple example: undersampling a sine
wave
– unsurprising result: information is lost
© 2006 Steve Marschner •
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Undersampling
• What if we “missed” things between the samples?
• Simple example: undersampling a sine wave
– unsurprising result: information is lost
– surprising result: indistinguishable from lower frequency
© 2006 Steve Marschner •
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Undersampling
• What if we “missed” things between the samples?
• Simple example: undersampling a sine wave
–
–
–
–
unsurprising result: information is lost
surprising result: indistinguishable from lower frequency
also, was always indistinguishable from higher frequencies
aliasing: signals “traveling in disguise” as other frequencies
© 2006 Steve Marschner •
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Aliasing in video
Slide by Steve Seitz
Aliasing in images
What’s happening?
Input signal:
Plot as image:
x = 0:.05:5; imagesc(sin((2.^x).*x))
Alias!
Not enough samples
Antialiasing
• What can we do about aliasing?
• Sample more often
– Join the Mega-Pixel craze of the photo industry
– But this can’t go on forever
• Make the signal less “wiggly”
– Get rid of some high frequencies
– Will loose information
– But it’s better than aliasing
Preventing aliasing
• Introduce lowpass filters:
– remove high frequencies leaving only safe,
low frequencies
– choose lowest frequency in reconstruction
(disambiguate)
© 2006 Steve Marschner •
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Linear filtering: a key idea
• Transformations on signals; e.g.:
– bass/treble controls on stereo
– blurring/sharpening operations in image editing
– smoothing/noise reduction in tracking
• Key properties
– linearity: filter(f + g) = filter(f) + filter(g)
– shift invariance: behavior invariant to shifting the
input
• delaying an audio signal
• sliding an image around
• Can be modeled mathematically by
18 convolution
© 2006 Steve Marschner •
Moving Average
• basic idea: define a new function by
averaging over a sliding window
• a simple example to start off: smoothing
© 2006 Steve Marschner •
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Weighted Moving Average
• Can add weights to our moving average
• Weights […, 0, 1, 1, 1, 1, 1, 0, …] / 5
© 2006 Steve Marschner •
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Weighted Moving Average
• bell curve (gaussian-like) weights […, 1, 4,
6, 4, 1, …]
© 2006 Steve Marschner •
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Moving Average In 2D
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© 2006 Steve Marschner •
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Slide by Steve Seitz
Cross-correlation filtering
• Let’s write this down as an equation. Assume the
averaging window is (2k+1)x(2k+1):
• We can generalize this idea by allowing different
weights for different neighboring pixels:
• This is called a cross-correlation operation and
written:
• H is called the “filter,” “kernel,” or “mask.”
© 2006 Steve Marschner •
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Slide by Steve Seitz
Gaussian filtering
•A Gaussian kernel gives less weight to
pixels
further
from
the0 center
of the window
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•This kernel is an approximation of a
Gaussian function:
Slide by Steve Seitz
Mean vs. Gaussian filtering
Slide by Steve Seitz
Convolution
•
cross-correlation:
•
A convolution operation is a cross-correlation where the filter is flipped
both horizontally and vertically before being applied to the image:
•
It is written:
•
Suppose H is a Gaussian or mean kernel. How does convolution differ from
cross-correlation?
Slide by Steve Seitz
Convolution is nice!
•
•
Notation:
Convolution is a multiplication-like operation
–
–
–
–
–
•
•
commutative
associative
distributes over addition
scalars factor out
identity: unit impulse e = […, 0, 0, 1, 0, 0, …]
Conceptually no distinction between filter and signal
Usefulness of associativity
– often apply several filters one after another: (((a * b1) * b2) * b3)
– this is equivalent to applying one filter: a * (b1 * b2 * b3)
© 2006 Steve Marschner •
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Tricks with convolutions
10
20
=
30
40
50
60
10
20
30
40
50
60
Slides from Alexei Eros and Steve Seitz
Salvador Dali
“Gala Contemplating the Mediterranean Sea,
which at 30 meters becomes the portrait
of Abraham Lincoln”, 1976
A nice set of basis
Teases away fast vs. slow changes in the image.
This change of basis has a special name…
Jean Baptiste Joseph Fourier (1768-1830)
...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
Any univariate function can be be desired on the score of generality and even
rigour.
rewritten as a weighted sum of
• had crazy idea
(1807):
•
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
Laplace
Legendre
Lagrange
A sum of sines
•Our building block:
•
Asin( x   
•Add enough of them to
get any signal f(x) you
want!
•How many degrees of
freedom?
•What does each control?
•Which one encodes the
coarse vs. fine structure of
the signal?
Fourier Transform
•We want to understand the frequency  of our signal. So,
let’s reparametrize the signal by  instead of x:
f(x)
F()
Fourier
Transform
For every  from 0 to inf, F() holds the amplitude A and phase  of
the corresponding sine
• How can F hold both? Complex number trick!
Asin( x   
F ( )  R( )  iI ( )
2
2
1 I ( )
A   R( )  I ( )
  tan
R( )
We can always go back:
F()
Inverse Fourier
Transform
f(x)
Time and Frequency
• example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t)
Time and Frequency
• example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t)
=
+
Frequency Spectra
• example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t)
=
+
Frequency Spectra
• Usually, frequency is more interesting than the phase
Frequency Spectra
=
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+
Frequency Spectra
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Frequency Spectra
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Frequency Spectra
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Frequency Spectra
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Frequency Spectra

1
= A sin(2 kt )
k 1 k
Frequency Spectra
FT: Just a change of basis
M * f(x) = F()
*
.
.
.
=
IFT: Just a change of basis
M-1 * F() = f(x)
*
.
.
.
=
Finally: Scary Math
Finally: Scary Math
• …not really scary: eix
• is hiding our old friend:
phase can be encoded
by sin/cos pair
 cos(x )  i sin( x )
Asin( x   
P cos( x )  Q sin( x )  A sin( x   
Α   P2  Q2
P
Q
  tan 1  
• So it’s just our signal f(x) times sine at frequency 
Extension to 2D
in Matlab, check out: imagesc(log(abs(fftshift(fft2(im)))));
Fourier analysis in images
Intensity Image
Fourier Image
http://sharp.bu.edu/~slehar/fourier/fourier.html#filtering
Signals can be composed
+
=
http://sharp.bu.edu/~slehar/fourier/fourier.html#filtering
More: http://www.cs.unm.edu/~brayer/vision/fourier.html
Man-made Scene
Can change spectrum, then reconstruct
Low and High Pass filtering
The Convolution Theorem
•The greatest thing since sliced (banana) bread!
– The Fourier transform of the convolution of two
functions is the product of their Fourier transforms
F[ g  h]  F[ g ] F[h]
– The inverse Fourier transform of the product of two
Fourier transforms is the convolution of the two
inverse Fourier transforms
1
1
1
F [ gh]  F [ g ]  F [h]
– Convolution in spatial domain is equivalent to
multiplication in frequency domain!
2D convolution theorem example
|F(sx,sy)|
f(x,y)
*
h(x,y)
|H(sx,sy)|
g(x,y)
|G(sx,sy)|
Filtering
Why does the Gaussian give a nice smooth
image, but the square filter give edgy artifacts?
Gaussian
Box filter
Gaussian
Box Filter
Fourier Transform pairs
Low-pass, Band-pass, High-pass filters
low-pass:
High-pass / band-pass:
Edges in images
What does blurring take away?
original
What does blurring take away?
smoothed (5x5 Gaussian)
High-Pass filter
smoothed – original
Band-pass filtering
Gaussian Pyramid (low-pass images)
• Laplacian Pyramid (subband images)
• Created from Gaussian pyramid by subtraction
Laplacian Pyramid
Need this!
Original
image
• How can we reconstruct (collapse) this
pyramid into the original image?
Why Laplacian?
Gaussian
delta function
Laplacian of Gaussian
Project 2: Hybrid Images
Gaussian Filter!
•
A. Oliva, A. Torralba, P.G. Schyns,
“Hybrid Images,” SIGGRAPH 2006
Laplacian Filter!
Project Instructions:
unit impulse
Gaussian Laplacian of Gaussian
http://www.cs.illinois.edu/class/fa10/cs498dwh/projects/hybrid/ComputationalPhotography_ProjectHyb
Clues from Human Perception
• 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: Multi-scale edge and blob filters
Frequency Domain and Perception
Campbell-Robson contrast sensitivity curve
Da Vinci and Peripheral Vision
Leonardo playing with peripheral vision
Unsharp Masking
100
200
-
300
=
400
500
200
400
+a
600
800
=
Freq. Perception Depends on Color
R
G
B
Lossy Image Compression (JPEG)
Block-based Discrete Cosine Transform (DCT)
Using DCT in JPEG
• The first coefficient B(0,0) is the DC
component, the average intensity
• The top-left coeffs represent low
frequencies, the bottom right – high
frequencies
Image compression using DCT
• 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
JPEG Compression Summary
Subsample color by factor of 2
–
People have bad resolution for color
Split into blocks (8x8, typically), subtract 128
For each block
a. Compute DCT coefficients for
b. Coarsely quantize
•
Many high frequency components will become zero
c. Encode (e.g., with Huffman coding)
http://en.wikipedia.org/wiki/YCbCr
http://en.wikipedia.org/wiki/JPEG
Block size in JPEG
• Block size
– small block
• faster
• correlation exists between neighboring pixels
– large block
• better compression in smooth regions
– It’s 8x8 in standard JPEG
JPEG compression comparison
89k
12k
Image gradient
• The gradient of an image:
•
The gradient points in the direction of most rapid change in intensity
The gradient direction is given by:
• how does this relate to the direction of the edge?
The edge strength is given by the gradient magnitude
Effects of noise
• Consider a single row or column of the image
– Plotting intensity as a function of position gives a signal
How to compute a derivative?
Where is the edge?
Solution: smooth first
Where is the edge?
Look for peaks in
Derivative theorem of convolution
• This saves us one operation:
Laplacian of Gaussian
• Consider
Laplacian of Gaussian
operator
Where is the edge?
Zero-crossings of bottom graph
2D edge detection filters
Laplacian of Gaussian
Gaussian
derivative of Gaussian
is the Laplacian operator:
Try this in MATLAB
•
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•
g = fspecial('gaussian',15,2);
imagesc(g); colormap(gray);
surfl(g)
gclown = conv2(clown,g,'same');
imagesc(conv2(clown,[-1 1],'same'));
imagesc(conv2(gclown,[-1 1],'same'));
dx = conv2(g,[-1 1],'same');
imagesc(conv2(clown,dx,'same'));
lg = fspecial('log',15,2);
lclown = conv2(clown,lg,'same');
imagesc(lclown)
imagesc(clown + .2*lclown)
Find your own image, clown doesn’t seem to be a default image anymore 
For next class
Read over the slides and the textbook, Chapter 3.