computational photography lecture 5 – filtering and frequencies cs 590 spring 2014 prof. alex berg...
TRANSCRIPT
Computational Photographylecture 5 – filtering and frequencies
CS 590 Spring 2014
Prof. Alex Berg
(Credits to many other folks on individual slides)
Thanks for the assignments! I will post links later today. Please look over each other’s work and send me a vote for your favorite {assignment, specific image} with at least a sentence about why.
Today filtering and frequency analysis of images.
Next week, guest lecturer! Prof. Marc Niethammer, an expert in medical image analysis. Another chance to see how to get to know the pixels.
I will be at a conference – the European Conference on Computer Vision (ECCV) presenting some research, but will be thinking of you all, will be on email, and will hold office hours over google hangouts (alex.c.berg) during the same after-class time, or will announce alternatives.
Today
© 2006 Steve Marschner • 5
Sampled representations
• How to store and compute with continuous functions?• Common scheme for representation: samples
– write down the function’s values at many points
[FvD
FH
fig.1
4.1
4b /
Wolb
erg
]
© 2006 Steve Marschner • 6
Reconstruction
• Making samples back into a continuous function– for output (need realizable method)– for analysis or processing (need mathematical method)– amounts to “guessing” what the function did in between
[FvD
FH
fig.1
4.1
4b /
Wolb
erg
]
© 2006 Steve Marschner • 8
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 • 10
Undersampling
• What if we “missed” things between the samples?
• Simple example: undersampling a sine wave– unsurprising result: information is lost
© 2006 Steve Marschner • 11
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 • 12
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
What’s happening?
Input signal:
x = 0:.05:5; imagesc(sin((2.^x).*x))
Plot as image:
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
© 2006 Steve Marschner • 17
Preventing aliasing
• Introduce lowpass filters:– remove high frequencies leaving only
safe, low frequencies– choose lowest frequency in
reconstruction (disambiguate)
© 2006 Steve Marschner • 18
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 convolution
© 2006 Steve Marschner • 19
Moving Average
• basic idea: define a new function by averaging over a sliding window
• a simple example to start off: smoothing
© 2006 Steve Marschner • 20
Weighted Moving Average
• Can add weights to our moving average
• Weights […, 0, 1, 1, 1, 1, 1, 0, …] / 5
© 2006 Steve Marschner • 21
Weighted Moving Average
• bell curve (gaussian-like) weights […, 1, 4, 6, 4, 1, …]
© 2006 Steve Marschner • 22
Moving Average In 2D
What are the weights H?0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 90 90 90 90 90 0 0
0 0 0 90 90 90 90 90 0 0
0 0 0 90 90 90 90 90 0 0
0 0 0 90 0 90 90 90 0 0
0 0 0 90 90 90 90 90 0 0
0 0 0 0 0 0 0 0 0 0
0 0 90 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
Slide by Steve Seitz
© 2006 Steve Marschner • 23
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.”
Slide by Steve Seitz
Gaussian filtering
•A Gaussian kernel gives less weight to pixels further from the center of the window
•This kernel is an approximation of a Gaussian function:
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 90 90 90 90 90 0 0
0 0 0 90 90 90 90 90 0 0
0 0 0 90 90 90 90 90 0 0
0 0 0 90 0 90 90 90 0 0
0 0 0 90 90 90 90 90 0 0
0 0 0 0 0 0 0 0 0 0
0 0 90 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
1 2 1
2 4 2
1 2 1
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
© 2006 Steve Marschner • 27
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)
Salvador Dali“Gala Contemplating the Mediterranean Sea, which at 30 meters becomes the portrait of Abraham Lincoln”, 1976
A nice set of basis
This change of basis has a special name…
Teases away fast vs. slow changes in the image.
Jean Baptiste Joseph Fourier (1768-1830)
• 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
LagrangeLegendre
A sum of sines•Our building block:•
•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?
xAsin(
Fourier Transform•We want to understand the frequency w of our signal. So, let’s reparametrize the signal by w instead of x:
xAsin(
f(x) F(w)Fourier Transform
F(w) f(x)Inverse Fourier Transform
For every w from 0 to inf, F(w) holds the amplitude A and phase fof the corresponding sine
• How can F hold both? Complex number trick!
)()()( iIRF 22 )()( IRA
)(
)(tan 1
R
I
We can always go back:
Finally: Scary Math
• …not really scary:• is hiding our old friend:
• So it’s just our signal f(x) times sine at frequency w
)sin()cos( xixe xi
Q
PQPΑ
xAxQxP
122 tan
sin()sin()cos(
xAsin(
phase can be encodedby sin/cos pair
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#filteringMore: http://www.cs.unm.edu/~brayer/vision/fourier.html
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
– 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[]F[]F[ hghg
][F][F][F 111 hggh
Why does the Gaussian give a nice smooth image, but the square filter give edgy artifacts?
Gaussian Box filter
Filtering
Band-pass filtering
• Laplacian Pyramid (subband images)• Created from Gaussian pyramid by subtraction
Gaussian Pyramid (low-pass images)
Laplacian Pyramid
• How can we reconstruct (collapse) this pyramid into the original image?
Need this!
Originalimage
Project 2: Hybrid Images
http://www.cs.illinois.edu/class/fa10/cs498dwh/projects/hybrid/ComputationalPhotography_ProjectHybrid.html
Gaussian Filter!
Laplacian Filter!
Project Instructions:
• A. Oliva, A. Torralba, P.G. Schyns,
“Hybrid Images,” SIGGRAPH 2006
Gaussianunit impulse Laplacian of Gaussian
• 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
Clues from Human Perception
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
Quantization table
Filter responses
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 blocka. 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/YCbCrhttp://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
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
Where is the edge?
How to compute a derivative?
Laplacian of Gaussian
• Consider
Laplacian of Gaussianoperator
Where is the edge? Zero-crossings of bottom graph
2D edge detection filters
is the Laplacian operator:
Laplacian of Gaussian
Gaussian derivative of Gaussian
Try this in MATLAB
• 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
I will send out a pointer to the links to look at other folks’ assignment 1s. Please look over these and send me your votes for best {assignment, image} with 1-2 sentence comments.
Prof. Marc Niethammer will talk more about finding edges and regions of images next week.
I will put the next assignment online soon – image compositing with variations. It will be due September 18. In the meantime, please experiment with the matlab from the lecture slides. One of the class goals is to get familiar with manipulating images!