Physics 114: Lecture 20 2D Data Analysis Dale E. Gary NJIT Physics Department.

Physics 114: Lecture 20 2D Data Analysis

Dale E. Gary

NJIT Physics Department

Reminder1D Convolution and Smoothing Let’s create a noisy sine wave:

u = -5:.1:5; w = sin(u*pi)+0.5*randn(size(u)); plot(u,w)

We can now smooth the data by convolvingit with a vector [1,1,1], which does a 3-pointrunning sum.

wsm = conv(w,[1,1,1]); whos wsm

Name Size Bytes Class Attributes wsm 1x103 824 double

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: plot(u,w) hold on; plot(u,wsm(2:102)/3,’r’,’linewidth’,2)

Apr 23, 2010

-5 0 5-2

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2D Convolution 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

zsm = conv2(ztest-z,k)/9.; imagesc(x,y,zsm(2:102,2:102))

Now we can see the effect of missing thevalue of cx by 0.05 due to our limitedsearch range.

There are other uses for convolution, suchas 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)).

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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:

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 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:

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)

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):

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

fuzzy = conv2(stars,Z); colormap(gray(256)); imagesc(stars); axis square imagesc(fuzzy(51:150,51:150)); axis square

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Deconvolution 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)

[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

Now let’s sharpen it again. MatLAB has a family of deconvolution routines. The standard one is deconvreg():

image(deconvreg(fuzzy,Z))

The image is dark, because we have to normalize the convolving function: image(deconvreg(fuzzy,Z)*sum(sum(Z)))

This looks pretty good, but note the edge effects. Try another routine image(deconvlucy(fuzzy,Z)*sum(sum(Z)))

This one looks almost perfect. However, if you compare images you do see differences

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.

With some time average, the above pattern smears out into an equivalent gaussian. The equivalent gaussian to

Zsinc = sinc(sqrt(X.^2 + Y.^2)*5).^2;

is Zgaus = exp(-(X.^2 + Y.^2)*(5*1.913)^2);

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Incorrect PSF Let’s convolve the image with the Gaussian (i.e. instrument plus

atmospheric turbulence), creating a larger PSF Zgaus = exp(-(X.^2 + Y.^2)*(3*1.913)^2); % Note use of 3 to enlarge Gaussian

Convolve with this blurred PSF fuzzy = conv2(double(img),Zgaus)/sum(sum(Zgaus)); image(fuzzy)

Now deconvolve with the instrumental PSF 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.

Apr 23, 2010