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: 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. wsm = conv(w,[1,1,1]); whos wsm Name Size Bytes Class Attributes wsm 1x 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

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 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)). Apr 23, 2010

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 Apr 23, 2010

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); Apr 23, 2010

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