GP265 / EE355 Homework 7 (Final project 1) Solutions

Transcription

1 GP265 / EE355 Homework 7 (Final project 1) Solutions We form the interferogram by multiplying image S 1 and conjugated image S 2 together: S int = S 1 S 2. The multi-look version of the complex interferogram is computed by averaging 4 range pixels and 4 azimuth pixels, so it has dimensions 3072/4 = 768 range bins by 14336/4 = 3584 azimuth bins. We plot the phase of the multi-look interferogram, which ranges from π to π. We do not see any fringes; it looks like random noise, because we have not yet resampled the second image at the correct range and azimuth offsets. 2. Calculate the offsets over the original images by cross-correlating small regions of each image. 1

2 We can use the offset program executable downloaded from the EE 355 website under Software. I ran it on cardinal.stanford.edu with the following parameters: $./offset image.1 image n This will correlate the images over 3000 small regions: 30 across (in the range direction) and 100 down (in the azimuth direction). It is important that the regions are representative of the entire image, rather than just a subset of the image. We also need to specify the approximate offset of the images for the last two parameters, which we can estimate by manually picking the same point in image 1 and image 2 (for example, the bottom corner of the triangular dark spot) and finding the pixel shift between them. I estimated a shift of 17 pixels across (in range) and -20 pixels down (in azimuth). If we do not choose good starting values for these approximate offsets, the offset program will produce junk output. The offset program outputs a file called offset.out which we can read into MATLAB for the rest of the processing. Next, I fit a least squares line to the range offset data as a function of range bin. I first removed outliers from the data, where I defined outlier as any data point outside one standard deviation of the median of the range offset data - this definitely improved the quality of the fit to the data (and this gets rid of any correlation regions that were on the water region of the image). The least squares fit (after removing outliers) results in two parameters: the slope r slope = x 10 4 and the intercept r offset = pixels of the line relating range offset p r (in pixels) to range bin i r : p r = r offset + r slope i r (1) I follow similar steps to get the azimuth offset. The least squares fit (after removing outliers) results in two parameters: the slope az slope = x 10 5 and the intercept az offset = pixels of the line relating azimuth offset p az (in pixels) to azimuth bin i az : p az = az offset + az slope i az (2) The magnitude of the azimuth slope az slope is much smaller than that of the range slope r slope, so we can approximate the azimuth offset as constant for the entire image. The average azimuth offset is pixels. 2

3 3. Baseline calculation The equation for the range offset δ r (in meters) in terms of relative range from the first bin dr = i r r is: δ r = B par + B perp dr (3) r 0 tan θ 0 We can also express the range offset by multiplying both sides of Eq. (1) by r: where r is the slant range pixel spacing: r = δ r = p r r = r offset r + r slope i r r (4) c = m/s 2f s 2( Hz) = m. (5) Comparing Eq. (4) to Eq. (3), I can see that B par = r offset r = m. Also, I can see that r slope = B perp B perp = r slope r 0 tan θ 0 (6) r 0 tan θ 0 The range to the swath center is r 0 = m. The look angle to swath center, θ 0, is computed by this equation obtained using the law of cosines for a spherical earth 3

4 geometry: ( θ0 = cos 1 r02 + z 2 + 2re z 2r0 (z + re ) ) = rad = 32.53o (7) Plugging in the results from Eq. (7) into Eq. (6), along with rslope = x 10 4, I get Bperp = m. The baseline length B is: B= 2 + B2 Bpar perp = m The orientation angle α is: ( ) Bperp π 1 α = tan + θ0 = rad = 3.89o Bpar 2 (8) (9) 4. Resample image.2 to align with image.1 and form the new interferogram. Since the Doppler centroid is fdc = 0, there is no need to shift the images to baseband before doing the resampling. I used MATLAB s meshgrid() to compute the sampling indices with the offsets included, then used interp2() to resample image 2 to align with image 1 s coordinates. I form the interferogram by multiplying image S1 and resampled image R2 together: Tint = S1 R2. I display the phase of the multilook version of the interferogram (4 range looks, 4 azimuth looks), where I am now able to see the fringes. 4

5 We could also overlay the phase image on the amplitude image: 5

6 5. Calculate multi-look (4 range x 4 azimuth) correlation images I used this equation to compute the correlation between images P 1 and P 2 : C = i P 1,iP 2,i i P 1,iP1,i i P 2,iP 2,i (10) The correlation image from Problem 1 has very low correlation values, almost zero. This is not surprising, since image 2 has not been resampled to the same coordinates as image 1. In contrast, the correlation image from Problem 4 provides useful information about the data. The correlation is low over areas of the image with water (as well as some other parts of the image), and I do not see any fringes in the interferogram at these locations. However, the correlation is high in many other parts of the image, and I do see fringes here in the interferogram. % Final project: Part1 clear;clc;close all; set(0, defaultaxesfontsize,16); %% read data tic fid=fopen( image.1 ); data=fread(fid,[3072* ], float ); fclose(fid); 6

7 reald=data(1:2:end,:); imagd=data(2:2:end,:); img1=complex(reald,imagd); img1=img1. ; fid=fopen( image.2 ); data=fread(fid,[3072* ], float ); fclose(fid); reald=data(1:2:end,:); imagd=data(2:2:end,:); img2=complex(reald,imagd); img2=img2. ; display( images loaded ) toc nr=3072; naz=14336; clear data reald imagd %% 1 form an interferogram and take looks intgram=img1.*conj(img2); figure imagesc(angle(intgram)) colormap(gray) colorbar title( interferometric phase ) xlabel( range ) ylabel( azimuth ) saveas(gcf, 1_phase, png ) % average looks nlook=4; intgram_sm=cpxlooks(intgram,nr,naz,nlook,nlook); figure subplot(1,2,1) imagesc(abs(intgram_sm)); colormap gray cax=caxis; caxis(0.4*cax) xlabel( range ) ylabel( azimuth ) title( amplitude ) 7

8 subplot(1,2,2) imagesc(angle(intgram_sm)); colormap gray xlabel( range ) ylabel( azimuth ) title( phase ) colorbar saveas(gcf, 1_withlook, png ) clear intgram %% offset determination: done in computeoffset.m %% resample image.2 to image.1 using a linear interpolater load offsets_coef ind_azimuth=1:naz; ind_range=1:nr; azimi = coefaz(2)*ind_azimuth + coefaz(1); rangei = coefrg(2)*ind_range + coefrg(1); azimi = (azimi+(1:length(azimi))); rangei = (rangei+(1:length(rangei))); azimi(azimi > naz) = naz; azimi(azimi < 1) = 1; rangei(rangei > nr) = nr; rangei(rangei < 1) = 1; % Resample image 2 with new indices [meshrange, meshazim] = meshgrid(rangei, azimi); img2_new = interp2(img2, meshrange, meshazim); display( Resampling of image.2 completed ) %% form interferogram intgram2=img1.*conj(img2_new); % average looks nlook=4; intgram_sm2=cpxlooks(intgram2,nr,naz,nlook,nlook); amp=abs(intgram_sm2); phase=angle(intgram_sm2); scalepic=mycolormap(amp,phase,floor(nr/nlook),floor(naz/nlook)); 8

9 figure imagesc(angle(intgram_sm2)) colormap(jet) colorbar saveas(gcf, 4_phase, png ) figure imagesc(scalepic) axis off xlabel( range ) ylabel( azimuth ) saveas(gcf, 4_withlook, png ) save( intgrams, intgram_sm, intgram_sm2 ) %% correlation cc1=zeros(naz/4,nr/4); cc2=zeros(naz/4,nr/4); % average looks for image.1 and image.2 img1sm=powlook(img1,nr,naz,4,4); img2sm=powlook(img2,nr,naz,4,4); img2rssm=powlook(img2_new,nr,naz,4,4); h = waitbar(0, correlation calculating ); for i=1:naz/4 amp1=abs(intgram_sm(i,:)); amp2=abs(intgram_sm2(i,:)); slc1amp=(img1sm(i,:)).^0.5; slc2amp=(img2sm(i,:)).^0.5; slc3amp=(img2rssm(i,:)).^0.5; cc1(i,:)=amp1./slc1amp./slc2amp; cc2(i,:)=amp2./slc1amp./slc3amp; waitbar(i/naz*4) end close(h) pic1=mycolormap(amp,cc1,nr/4,naz/4); pic2=mycolormap(amp,cc2,nr/4,naz/4); figure subplot(1,2,1) imagesc(pic1) title( Before SLC coregisteration ) colormap jet colorbar subplot(1,2,2) imagesc(pic2) title( After SLC coregisteration ) 9

10 colormap jet colorbar saveas(gcf, correlation.png, png ) % save correlation data as byte files cc1=255*cc1; cc2=255*cc2; fid1=fopen( corr_1, w ); fid2=fopen( corr_2, w ); fwrite(fid1,cc1, uint8 ); fwrite(fid2,cc2, uint8 ); fclose(fid1); fclose(fid2); Function to compute multilooks: cpxlooks (average complex images) % multilook of complex images function imgsm=cpxlooks(imgbg,nr,naz,nlookrg,nlookaz) imgaz=zeros(floor(naz/nlookaz),nr); imgsm=zeros(floor(naz/nlookaz),floor(nr/nlookrg)); for i=1:floor(naz/nlookaz) imgaz(i,:)=sum(imgbg((i-1)*nlookaz+1:i*nlookaz,:)); end for i=1:floor(nr/nlookrg) imgsm(:,i)=sum(imgaz(:,(i-1)*nlookrg+1:i*nlookrg),2); end return Function to compute multilooks: powlooks (average power of complex images) % average power of complex images function imgsm=powlook(imgbg,nr,naz,nlookrg,nlookaz) imgaz=zeros(floor(naz/nlookaz),nr); imgsm=zeros(floor(naz/nlookaz),floor(nr/nlookrg)); powerimg=abs(imgbg).^2; 10

11 for i=1:floor(naz/nlookaz) imgaz(i,:)=sum(powerimg((i-1)*nlookaz+1:i*nlookaz,:)); end for i=1:floor(nr/nlookrg) imgsm(:,i)=sum(imgaz(:,(i-1)*nlookrg+1:i*nlookrg),2); end return Function to overlay phase image on the amplitude image function pic=mycolormap(amp,phase,nr,naz) amplow=prctile(amp(:),5); amphi=prctile(amp(:),95); amp(amp<amplow)=amplow; amp(amp>amphi)=amphi; scale=(amp-amplow)/(amphi-amplow); phlow=prctile(phase(:),1); phhi=prctile(phase(:),99); % create a color table colormap jet; map=colormap; stack=max(phase,phlow); stack=min(phase,phhi); colorstack=round((stack-phlow)/(phhi-phlow)*64); colorstack=max(colorstack,1); colorstack=min(colorstack,64); for k=1:naz for kk=1:nr red(k,kk)=map(colorstack(k,kk),1); green(k,kk)=map(colorstack(k,kk),2); blue(k,kk)=map(colorstack(k,kk),3); end end pic(:,:,1)=red.*scale; pic(:,:,2)=green.*scale; pic(:,:,3)=blue.*scale; Function to compute offset coefficients: % Estimate offsets between image.1 and image.2 11

12 clear; close all; clc; set(0, defaultaxesfontsize, 16); %% radar parameters c=3e8; fs=16e6; hgt=696000; re=6378e3; deltar=c/2/fs; rmid=844768; costheta0=((hgt+re)^2+rmid^2-re^2)/2/rmid/(hgt+re); theta0=acos(costheta0); nr=3072; naz=14336; %% calculate the offsets % use the offset.f program to get the offsets load offset.out rgpos=offset(:,1); rgoffset=offset(:,3); azpos=offset(:,2); azoffset=offset(:,4); meanazoffset=mean(azoffset); display(meanazoffset, Average azimuth offset is, ) meanrgoffset=mean(rgoffset); display(meanrgoffset, Average range offset is, ) % plot range offset vs. range figure subplot(1,2,1) dr=rgpos; scatter(dr,rgoffset) xlabel( range bins ) ylabel( range bin offsets ) subplot(1,2,2) daz=azpos; scatter(daz,azoffset) xlabel( azimuth bins ) ylabel( azimuth bin offsets ) saveas(gcf, offsetdata, tiff ) % get rid of offliers varrg=std(rgoffset); varaz=std(azoffset); num=length(rgpos); 12

13 index=find(rgoffset<meanrgoffset+0.5*varrg & rgoffset>meanrgoffset-0.5*varrg); newrgpos=rgpos(index); newrgoffset=rgoffset(index); index=find(azoffset<meanazoffset+0.5*varaz & azoffset>meanazoffset-0.5*varaz); newazpos=azpos(index); newazoffset=azoffset(index); figure subplot(1,2,1) dr=newrgpos; scatter(dr,newrgoffset) xlabel( range bins ) ylabel( range bin offsets ) subplot(1,2,2) daz=newazpos; scatter(daz,newazoffset) xlabel( azimuth bins ) ylabel( azimuth bin offsets ) % saveas(gcf, offsetdata.filtered, png ) %% estimate range and azimuth offsets as a liner function % get rid of more outliers % least square method YY=newrgoffset; dr=newrgpos; GG=[ones(size(newrgpos)), dr]; coefrg=gg\yy; est_rg=gg*coefrg; YY=newazoffset; daz=newazpos; GG=[ones(size(newazpos)), daz]; coefaz=gg\yy; est_az=gg*coefaz; % plot the estimated offsets figure subplot(1,2,1) scatter(rgpos,rgoffset) hold on plot(dr,est_rg, linewidth,2) hold off xlabel( range bins ) ylabel( range bin offsets ) subplot(1,2,2) scatter(azpos,azoffset) 13

14 hold on plot(daz,est_az, linewidth,2) hold off xlabel( azimuth bins ) ylabel( azimuth bin offsets ) saveas(gcf, offsetdata.filtered.fit.png, png ) %% calculate the baseline components B_par=coefrg(1)*deltar; B_perp=coefrg(2)*rmid*tan(theta0); B=sqrt(B_par^2+B_perp^2); sindiff=b_perp/b; alpha=atan(b_perp/b_par)+theta0-pi/2; alpha_dg=alpha/pi*180; save( offsets_coef, coefrg, coefaz ) 14