Physics 114: Lecture 20
2D Data Analysis
Dale E. Gary
NJIT Physics Department
Reminder
1D Convolution and Smoothing
Let’s create a noisy sine wave:
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u = -5:.1:5;
w = sin(u*pi)+0.5*randn(size(u));
plot(u,w)
We can now smooth the data by convolving
it with a vector [1,1,1], which does a 3-point
running sum.
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2
wsm = conv(w,[1,1,1]);
whos wsm
Name
wsm
Size
1x103
Bytes Class
Attributes
824 double
w
wsm
1.5
1
0.5
w
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0
-0.5
-1
-1.5
-2
-5
0
u
Notice wsm is now of length 103. That means we cannot plot(u,wsm),
but we can plot(u,wsm(2:102)). Now we see another problem.
 Try this:
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plot(u,w)
hold on; plot(u,wsm(2:102)/3,’r’,’linewidth’,2)
Apr 23, 2010
5
2D Convolution
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To do 2D smoothing, we will use a 2D kernel k = ones(3,3), and use the
conv2() function. So to smooth the residuals of our fit, we can use
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zsm = conv2(ztest-z,k)/9.;
imagesc(x,y,zsm(2:102,2:102))
Now we can see the effect of missing the
value of cx by 0.05 due to our limited
search range.
 There are other uses for convolution, such
as edge detection. For example, we can
convolve with a kernel k = [1,1,1,1,1,1].
 Or a kernel k = [1,1,1,1,1,1]’. Or even
a kernel k = eye(6). Or k = rot90(eye(6)).
-5
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0
5
-5
0
Apr 23, 2010
5
Convolution and Resolution
Convolution can be used for smoothing data, but it is also something that
happens naturally whenever measurements are made, due to instrumental
resolution limitations.
 For example, an optical system (telescope or microscope, etc.) has an
aperture size that limits the resolution due to diffraction (called the
diffraction limit). Looking at a star with a telescope, assuming no other
effects like atmospheric turbulence, results in a star image of a certain size,
surrounded by an “airy disk” with diffraction rings.
 This shape is mathematically just the sinc() function we introduced last
time:
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x = -5:0.1:5; y = -5:0.1:5;
[X, Y] = meshgrid(x,y);
Z = sinc(sqrt(X.^2 + Y.^2));
imagesc(x,y,Z);
In fact, this is the electric field pattern, and to get the intensity we need to
square the electric field: imagesc(x,y,Z.^2)
Apr 23, 2010
Point Spread Function
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To show this point better, consider a “perfect” instrument that perhaps has
noise, but shows stars as perfect point sources. Let’s generate an image
of some stars:
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To see the effect of observing such a star pattern with an instrument,
convolve the star image with the sinc function representing the diffraction
pattern of the instrument (the point spread function or PSF):
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stars = randn(size(X))*0.1;
stars(50,50) = 1; stars(20,37) = 4; stars(87,74) = 2; stars(45,24) = 0.5;
imagesc(stars)
Z = sinc(sqrt(X.^2 + Y.^2)*5).^2; % the *5 makes it smaller/sharper
imagesc(conv2(stars,Z))
You see that the result is twice as large due to the way convolution works.
Try
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fuzzy = conv2(stars,Z); colormap(gray(256));
imagesc(stars); axis square
imagesc(fuzzy(51:150,51:150)); axis square
Apr 23, 2010
Deconvolution
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It is actually possible to do the inverse of convolution, called deconvolution.
Let’s read in an image and fuzz it up (download fruit.gif from course web pg)
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Now let’s sharpen it again. MatLAB has a family of deconvolution routines.
The standard one is deconvreg():
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image(deconvreg(fuzzy,Z)*sum(sum(Z)))
This looks pretty good, but note the edge effects. Try another routine
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image(deconvreg(fuzzy,Z))
The image is dark, because we have to normalize the convolving function:
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[img map] = imread(‘fruit.gif’);
fuzzy = conv2(single(img),Z)/sum(sum(Z);
image(img) % Original image—observe the sharpness
image(fuzzy(51:515,51:750)) % fuzzy image
image(deconvlucy(fuzzy,Z)*sum(sum(Z)))
This one looks almost perfect. However, if you compare images you do see
differences
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sharp = deconvlucy(fuzzy,Z)*sum(sum(Z))
imagesc(sharp(51:515,51:750) – single(img))
Apr 23, 2010
Deconvolution Problems
Any time you do an inversion of data, the result can be unstable. Success
depends critically on having the correct point spread function.
 The deconvolution we just did was after convolving the image with a “perfect”
instrument and neglecting atmospheric turbulence. Further blurring by the
atmosphere acts to increase the size of the “airy disk” and smear out the
diffraction rings.
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With some time average, the above pattern smears out into an equivalent
gaussian. The equivalent gaussian to
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Zsinc = sinc(sqrt(X.^2 + Y.^2)*5).^2;
is
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Zgaus = exp(-(X.^2 + Y.^2)*(5*1.913)^2);
Apr 23, 2010
Incorrect PSF
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Let’s convolve the image with the Gaussian (i.e. instrument plus atmospheric
turbulence), creating a larger PSF
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Convolve with this blurred PSF
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fuzzy = conv2(double(img),Zgaus)/sum(sum(Zgaus));
image(fuzzy)
Now deconvolve with the instrumental PSF
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Zgaus = exp(-(X.^2 + Y.^2)*(3*1.913)^2); % Note use of 3 to enlarge Gaussian
dconl = deconvlucy(fuzzy,Zsinc)*sum(sum(Zsinc));
image(dconl)
We see that we cannot recover the original instrumental resolution. The clarity
is lost due to atmospheric turbulence.
However, if we measure the PSF of instrument plus atmosphere, we CAN
recover the blurring due to the atmosphere.
Apr 23, 2010
Laser Guide Stars
Astronomers now use a laser to
create a bright, nearby “guide star”
near the region of the sky of
interest.
 By imaging the laser scintillation
pattern instantaneously, they can
freeze the atmosphere and correct
the images in real time.
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Apr 23, 2010