From be8985cfbc9a0aade9e42b02408273ab2fb4fe51 Mon Sep 17 00:00:00 2001 From: Bryan Steinbach Date: Thu, 16 Sep 2010 17:45:01 -0700 Subject: [PATCH] import week2 HW --- week2/problem1/fits.png | Bin 0 -> 79176 bytes week2/problem1/problem1.py | 83 ++++++++++++++++++++++++++++++++ week2/problem2/filtered.png | Bin 0 -> 220377 bytes week2/problem2/moonlanding.png | Bin 0 -> 216481 bytes week2/problem2/problem2.py | 107 +++++++++++++++++++++++++++++++++++++++++ week2/problem2/resid.png | Bin 0 -> 720193 bytes week2/problem3/problem3.py | 59 +++++++++++++++++++++++ week2/problem3/resid-hist.png | Bin 0 -> 24532 bytes week2/readme | 8 +++ 9 files changed, 257 insertions(+) create mode 100644 week2/problem1/fits.png create mode 100644 week2/problem1/problem1.py create mode 100644 week2/problem2/filtered.png create mode 100644 week2/problem2/moonlanding.png create mode 100644 week2/problem2/problem2.py create mode 100644 week2/problem2/resid.png create mode 100644 week2/problem3/problem3.py 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Mask out the first few individually, then throw the rest out whole sale when they get dense. Image power follows a 1/f^a model, so we don't lose much signal doing this. +''' + +import numpy as np +import pylab as pl +from scipy import weave + +imgfn = 'moonlanding.png' + +img = pl.imread(imgfn)*1.0 # Read and convert to doubles + +print "shape: ",img.shape +# (474,630) +# So this is already a gray scale image + +''' +pl.imshow(img,cmap='gray') +pl.show() +''' + +# Compute 2d fourier transform, normalized 2d amplitude spectral density, and 1d radial asd +fimg = np.fft.fft2(img) +fimg_asd = np.abs(fimg) +fimg_psd = fimg_asd**2 + +fimg_asd_norm = fimg_asd/np.max(fimg_asd) + +fimg_asd_db = 10*np.log10(fimg_asd_norm) + +# Strategy - make a signal and noise model, and a matched filter to maximize one and reject the other + +''' +pl.imshow(fimg_asd_db,cmap='gray') +pl.colorbar() +pl.show() +''' + +nx = fimg.shape[0] +ny = fimg.shape[1] + +# 2d versions of fftfreq +kx = np.tile(np.fft.fftfreq(nx,1),(ny,1)).transpose() +ky = np.tile(np.fft.fftfreq(ny,1),(nx,1)) +kr = np.sqrt(kx**2+ky**2) # Radial frequencies + +# Make a 1d radial power spectrum by binning +krmin = 0.0 +krmax = np.max(kr) + +nbins = 256 +psd1d = np.zeros(nbins) +psd1d_hits = np.zeros(nbins,dtype=np.int) + +# Radial frequency bins of fourier transformed image +kimg = np.int_((nbins-1)*(kr-krmin)/(krmax-krmin)) + +# Radial frequencies of output spectrum +kbin = np.linspace(krmin,krmax,nbins) + +# From looking at this plot I find these masks +maskout = np.zeros(fimg.shape,dtype=np.bool) +maskout |= (kr > 0.148) +maskout |= (kr > 0.096) & (kr < 0.1) +maskout |= (kr > 0.113) & (kr < 0.118) +maskout |= (kr > 0.133) & (kr < 0.140) + +fimg_psd[maskout] = 0. + +# Build mask by looking for spikes in the radial frequency spectrum +def build_mask(): + # I'd like to write + # bins[kbin] += fimg_psd + # But the += semantics are wrong for that - behaves like bins[kbin] = bins[kbin] + fimg_psd and assigns with out updating + # Previously I've hacked around this using sparse arrays, but for a single small image there's no need to optimize + for i in range(fimg.shape[0]): + for j in range(fimg.shape[1]): + psd1d[kimg[i,j]] += fimg_psd[i,j] + psd1d_hits[kimg[i,j]] += 1 + + ok = psd1d_hits > 0 + psd1d[ok] /= psd1d_hits[ok] + + asd1d = np.sqrt(psd1d) + + ok = asd1d > 0 + + pl.loglog(kbin[ok][1:],asd1d[ok][1:]) + pl.show() + +#build_mask() + +fimg[maskout] = 0. + +filtered_img = np.fft.ifft2(fimg).real + +delta_img = filtered_img - img + +#pl.imshow(filtered_img,cmap='gray') +pl.imsave('filtered.png',filtered_img,cmap='gray') +pl.imsave('resid.png',delta_img,cmap='gray') + + diff --git a/week2/problem2/resid.png b/week2/problem2/resid.png new file mode 100644 index 0000000000000000000000000000000000000000..e0923661613756b9052acef528163a8b034a50d3 GIT binary patch literal 720193 zcwRJo2RN7C|Gp+4QAyc*L`KL=Av?;57KLm{MwF2eMcGnB*|U<&qEbRulpRWD36U+C z|NDG@*T3uP>QcR*=RB`-zwU9)dEUYLx|+1qEYuVf6tr4L)eI;oHeREk*zlEVEB+?j zV)Hcq*zBsJb&?9dys512;OGCGj~cn+PXY3KgV29>^(ZK~D74g+PkJP{m3f~3G|IZZ 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Fortunately the problem statement doesn't specify that we need to implement Gauss elimination + # Use scipy to build the LU decomposition, then extract the matrix inverse by solving for the basis vectors + lu,piv = linalg.lu_factor(mat) + + matinv = np.zeros((n,n)) + + for i in range(n): + v = np.zeros(n) + v[i] = 1 + + matinv[:,i] = linalg.lu_solve((lu,piv),v) + + return matinv + +# Test with a 200x200 random matrix +nt = 200 + +m = np.random.randn(nt*nt).reshape((nt,nt)) + +mi = inv(m) + +id = np.dot(m,mi) + +resid = (id - np.eye(nt)).flatten() + +def rms(x): return np.sqrt(np.dot(x,x)/x.size) +rmsresid = rms(resid) +print "RMS residual: ",rmsresid + +pl.hist(resid,bins=100,log=True) +pl.ylabel('Count') +pl.xlabel('Residual') +pl.title('Residual histogram of $I - M \cdot M^{-1}$, RMS=%0.3g'%(rmsresid)) +pl.savefig('resid-hist.png') diff --git a/week2/problem3/resid-hist.png b/week2/problem3/resid-hist.png new file mode 100644 index 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