RECONSTRUCTION OF NON- CARTESIAN DATA USING BURS/RBURS ALGORITHM

Transcription

1 EE-591 MAGNETIC RESONANCE IMAGING TERM PROJECT RECONSTRUCTION OF NON- CARTESIAN DATA USING BURS/RBURS ALGORITHM Zheng L Department of Electrcal Engneerng December 5,

2 1. INTRODUCTION There are many alternatves to 2DFT acquston methods. These nclude spral scans, radal scans, Lssajou trajectory scan and so on (Fgure 1) [1]. Many of these have specfc advantages over spn-warp, such as speed and SNR effcency. The man dsadvantage wth these methods s the dffculty of reconstructng the resultng data sets. There are many choces for non-cartesan data sets mage reconstructon. The frst approach s to collect the non-cartesan data n a way that a prevously nown reconstructon method can be appled. For example Fltered Bac Projecton (FBP) can be appled for radal scans data set. Whle ths solves the reconstructon problem, t usually requres compromses n data acquston. Second, the non-cartesan data can be demodulated pont-by-pont wth the conjugate phase reconstructon. But ths method s very slow. The most computatonally effcent method of reconstructon s to resample the data onto a Cartesan grd, whch enable the subsequent use of nverse fast Fourer transform (IFFT), and post compensaton, f necessary. b) y y K y x x K x (a) (b) (c) Fgure 1. Some alternatve acquston methods. (a) Constant angular rate spral, whch can use projecton reconstructon method. (b) Lssajou trajectory (c) Spral trajectory used n ths project In MRI, the most wdely used resamplng algorthm s grddng. Usually, the grddng methods consst of four steps: 1) pre-compensaton for varyng samplng densty; 2) convoluton wth a Kaser-Bessel wndow onto a Cartesan grd; 3) IFFT; 4) postcompensaton by dvdng the mage by the transform of the wndow. In ths paper, the Bloc Unform Re-Samplng (BURS) and regularzaton Bloc Unform Re-Samplng (rburs) are used to nterpolate the non-cartesan scan data. BURS and rburs are both 2

3 optmal/suboptmal and computatonally effcent. Comparng to the conventonal grddng, nether pre- nor post-compensaton are requred, and the results were shown to be of excellent accuracy. 2. THEORY In ths secton, the theores for BURS algorthm wll be ntroduced frst. Then the theoretcal analyss of nose for BURS wll be addressed. Fnally, one nose reducton soluton for BURS, namely regularzaton BURS (rburs), wll be provded. 2.1 BLOCK UNIFORM RESAMPLING (BURS) ALGORITHM The BURS algorthm can be summarzed as follows: 1. Intalze an N by M matrx A wth zeros (N and M represent the number of the Cartesan grd ponts and the number of the non-unformly sampled data ponts, respectvely) 2. For each Cartesan grd pont, ( = 1, L, N ) : 2.a. Select the 2.b. Select the 2.c. Form a functon. M 2.d. Compute A M non-unformly sampled ponts n a δ neghborhood of. N Cartesan grd ponts n a N, the truncated sngular value decomposton (SVD) pseudonverse matrx of A. 2.e. Transfer the row of neghborhood of. matrx A of the nterpolaton coeffcents based on the snc A correspondng to the pont to the -th row of A. 3. The unform samples are calculated as x = A b, where b s a column vector contanng the non-unform data measurements. 4. Perform an nverse Fourer transform (IFT) on the resultng unform samples. Fgure 2 llustrates how to select the δ neghborhood of and M non-unformly sampled ponts n a N Cartesan grd ponts n a neghborhood of. The and neghborhoods of the are llustrated as crcle regons n the Fgure 2. But n the mplementaton, other shapes of neghborhood maybe used. For example, square δ 3

4 neghborhood can be used n Cartesan coordnate for computatonal effcency and easer mplementaton. Square and crcular shapes of neghborhood are tested n our smulatons. In BURS algorthm, the selectons of values for and δ wll dramatcally affect the fnal results (whch wll be shown n the smulatons). When N > M, the pseudonverse can be computed as: A T 1 T = ( A A) A (1) When N < M, the pseudo-nverse can be computed as: T T 1 A = A ( AA ) (2) The smulaton results gve some examples of how the reconstructon results vares wth dfferent combnaton of N and M. 0.4 K y δ 0 K x Fgure 2. The llustraton for BURS algorthm. The δ and plot are defned as a crcle regons. The bg dots represents neghborhoods of the n ths M non-unformly sampled ponts n a δ neghborhood of ; the bg cross sgns represent neghborhood of. N Cartesan grd ponts n a 4

5 2.2 EFFECT OF NOISE Several papers have reported that although the BURS algorthm s very accurate, t s also senstve to the nose. As a consequence, even n the presence of a low level of measurement nose, the resultng mage s often hghly contamnated wth nose. In the grddng process, each unform output pont at locaton (=1,,N) s lnearly nterpolated usng M nown data of non-unform samples {, m = 1, L, M } whch are wthn δ neghborhood of : M m f ( ) = a f ( ) (3) m m= 1 where f ( m ) s non-unform (non-cartesan) nput data; f ) s the nterpolated unform output(cartesan) at ; a are the nterpolaton coeffcents, n BURS algorthm, these coeffcents are derved by pseudo-nverse. Assumng the nose s addtve and conssts of zero mean whte Gaussan nose wth varance derved that the nose of the nterpolated data mean and the varance : 2 σ m M 2 σ f ( ) = σ a m m= 1 ( m, usng above equaton, t can be s addtve Gaussan nose wth zero σ (4) Because the nterpolaton coeffcents vary as changes, the nose level s space dependent n -space doman, even f we assume the nose s..d n orgnal non- Cartesan -space data. In Rosenfeld s paper [4], the -space nose amplfcaton s defned as: 2 σ Ω M = σ /σ = a (5) m= 1 2 m Rosenfeld [4] tested ths nose effect of BURS by usng a four-nterleaf spral trajectory. The Ω was calculated for each unform pont. We also dd the test on our spral trajectory and get smlar results, whch are show n Fgure 3. 5

6 500 Nose Amplfcaton (a) (b) Fgure 3. Nose amplfcaton Ω usng BURS for spral trajectores. (a) the result from [4], x-axs represents the dstance from the orgn of the -plane. (b) The y = 0 based on our own spral trajectory. The x-axs represents the Ω values for the row x coordnate. Both results show that most ponts have a nose amplfcaton of about unty, however a substantal number of ponts have extreme hgh nose amplfcaton number. Ths s the reason that cause the reposted nose contamnated result for BURS algorthm. Although only small part of -space pont have very hgh nose level, after the Fourer transform, the nose wll dstrbuted across the whole mage. Equaton (5) shows that the hgh nose amplfcaton coeffcents are due to the hgh value of nterpolaton coeffcents, whch s the row of A correspondng to the pont. We now that the soluton of an nverse problem s unstable, whch means that small changes n the nput data may lead to large perturbatons n the results (ll-posed problem). So t becomes clear that, the ll-condtoned matrxes T A A cause the large perturbatons n the coeffcents and fnally result n large nose level n reconstructed mage. 2.3 REGULARIZED BLOCK UNIFORM RESAMPLING ALGORITHM The basc deal of the rburs s to stablze the matrx nverson soluton by modfyng the problem n such a way that the nverson soluton becomes less senstve to small perturbatons n the data. At the same tme, the soluton to the modfed problem 6

7 must reman close to the orgnal soluton. Thus the orgnal soluton x = A by the approxmate soluton x = A ρ b such that b s replaced ρ 0 ρ lm A b = A b (6) where ρ s a postve smoothng parameter. We now focus on one type of regularzaton technque, referred to as spectral wndowng. By usng equaton (1), T A b = ( A 1 T = 1 A) A b α ( v b) u (7) T T where are egenvectors of AA ; u are egenvectors of A T A ; v α, α α 1 2 L are sngular values. where W ρ A ρ b s computed as: = W 1 A ρ b ρ α ( v b) u (8) s called the wndow coeffcents. There are many dfferent defntons for these coeffcents ncludng Truncated sngular system expanson and Thonov flter whch are defned separately as: 1 < (1/ ρ) Truncated sngular system expanson : Wρ = (9) 0 otherwse T Thonov flter : α W ρ (10) α + ρ = 2 2 WhenW s defned as (10), t can be proved that x = A b can be computed as: ρ T x = A ρ b = Wρ α ( v b) u = ( A A + ρi) A b (11) In our mplementaton, equaton (11) s employed for regularzaton. 3. IMPLEMENTATION In the real system, gven the non-cartesan -space trajectory, δ and BURS algorthm descrbed n secton 2.1, the matrx 1 T ρ 1 T, by usng the can be calculated and saved pre reconstructon. Whenever the data samplng s done and reconstructon s needed, the matrx A can be reloaded and used drectly. By ths way, the computatonal tme s shortened dramatcally. But ths method needs to process the huge sze matrx A, whch maes the data handlng not so easy. In addton, n order to test the dfferent parameter A 7

8 combnatons n ths paper, the parameters δ and maes A A change from tme to tme, whch change each tme. So, n our smulaton, nstead of storng the huge matrx and nterpolatng all ponts one tme, the Cartesan pont nterpolaton s done pont by pont through the whole mage. 3.1 IMPLEMENTATION OF BURS/rBURS ALGORITHM 1. Intalze an N N matrx M wth zeros (the sze of the mage s N N) 2. For each Cartesan grd pont M, ( = 1, L, N; j = 1, L, N ) : 2.a. Select the M j j non-unformly sampled ponts n a δ neghborhood of M j. Form a M 1 column vector d usng M j j j nown non-unformly sampled data. 2.b. Select the N Cartesan grd ponts n a neghborhood of. j M j 2.c. Form a M j N j matrx A of the nterpolaton coeffcents based on the snc functon. 2.d-BURS. For BURS algorthm, A =pseudo-nverse matrx of A. 2.d-rBURS. For rburs algorthm, A T 1 T = (A A + ρ I) A. 2.e. Let a = row of A correspondng to the pont Mj, M j = aj d j 3. Perform an nverse Fourer transform (IFT) on the M. 3.2 SHAPE OF THE NEIGHBORHOOD In the real mplementaton, the neghborhood of the pont ( x, 0 ) wthn radus be defned at least n two dfferent ways: 1. Crcular Neghborhood wth radus : r C 0 y r can { neghhorhood} {( x, y) ( x, y) ( x, y0 ) r } 2. Square Neghborhood wth radus : r S = (12) 0 C { neghhorhood} {( x, y) max( x x 0, y y0 ) rs } = (13) Notce that, when the raduses have the same value, the square neghborhood has larger coverage area than that of crcular neghborhood. To mae both defntons have the same coverage area, r and r should satsfy: C S 2 2 π C S S C π r = (2r ) r = r (14) 4 8

9 Crcular neghborhood has the advantage that the closest (n the sense of norm2) ponts from the center of the neghborhood are selected. Square neghborhood wll select some ponts (n the corner of the square) not so close to the center, but square neghborhood s easer to mplement and computatonal more effectve. In the smulaton, two neghborhood defntons are tested and compared. To mae the comparson equtable, same effectve radus r s used for dfferent shapes, then r and C rs have same effectve radus For crcular neghborhood: are computed usng equaton (14). Suppose both neghborhood r, then: 4. SIMULATIONS AND RESULTS r C = r ; For square neghborhood: π r S = r (15) 4 The data set used here s a smulated phantom usng a spral acquston wth 6 nterleaves of 1536 samples. Center part of the trajectory s llustrated n Fgure 2. Four problems are studed n our smulaton: 1) How the reconstructon result changes wth and δ. Fgure 4 shows the results for =1 whle δ vares from 0.3~1. Fgure 5 shows the results for =2 whle δ vares from 0.5~1.4. Beyond these δ ranges, the results become unacceptable. 2) How the shape of neghborhood (crcular vs. square) affect the results. In the smulaton, the crcular neghborhood s always used for δ (non-cartesan), crcular AND square neghborhoods are tested for radus (Cartesan ponts). Set the effectve =1, 2, 3 respectvely, δ values are chosen such that the best reconstructon acheved for each case. Crcular and square neghborhoods are tested wth same effectve radus and δ settngs. Fgure 6 shows the results. 3) BURS vs. rburs algorthm. One mage wth hgh-snr and one wth low-snr are tested usng BURS and rburs algorthm respectvely. The low-snr mage s produced by addng Gaussan nose to K-space spral sampled data. 4) Compare the result of BURS/rBURS wth true mage. The mage reconstructed by grddng w/ Pre-Densty Compensaton & Deapodzaton s used as the orgnal 9

10 mage. Then we compare the best results produced by BURS and rburs wth the orgnal mage. Fgure 8 shows the mages and the dfference mages. Fgure 9. shows the profle of the mages. (a) δ =0.3, =1; M =3, N =5@(64, 64) (b) δ =0.4, =1; M =4, N =5@(64, 64) (c) δ =0.5, =1; M =5, N =5@(64, 64) (d) δ =0.7, =1; M =12, N =5@(64, 64) (e) δ =0.9, =1; M =35, N =5@(64, 64) (f) δ =1, =1; M =37, N =5@(64, 64) Fgure 4. Comparng dfferent used for and δ. Fx =1, and δ combnatons for BURS algorthm. Crcular neghborhoods are δ value s changed from 0.3 to 1. M and N ( x, y ) =(64, 10

11 64) are provded for each case. It shows that when bad underdetermned case ( M >> N ) occurs, some artfacts wll appear n the reconstructed mage. (a) δ =0.5, =2; M =5, N =13@(64, 64) (b) δ =0.7, =2; M =12, N =13@(64, 64) (c) δ =0.9, =2; M =35, N =13@(64, 64) (d) δ =1, =2; M =37, N =13@(64, 64) (e) δ =1.2, =2; M =43, N =13@(64, 64) (f) δ =1.4, =2; M =51, N =13@(64, 64) Fgure 5.. Comparng dfferent and used for and, ) ( x y δ. Fx =2, δ selectons for BURS algorthm. Crcular neghborhoods are δ value s changed from 0.4 to 1.4. M and N =(64, 64) are provded for each case. It shows that when bad underdetermned case ( M >> N ) occurs, some artfacts wll appear n the reconstructed mage. 11

12 (a) δ =0.6, =1, Crcular δ =0.6, =1, Square δ =0.9, =2, Crcular δ =0.9, =2, Square δ =1.3, =3, Crcular δ =1.3, =3, Square Fgure 6. Comparng crcular neghborhood wth square neghborhood for BURS algorthm. Crcular neghborhoods are always used for δ ; crcular and square neghborhoods are tested for. Fx the effectve neghborhood radus =1, 2 and 3, δ values are selected such that best reconstructon result s acheved for each case. The results show that dfferent shapes of neghborhood have some but lmted effect (crcular neghbor s lttle bt better) on reconstructed mage. 12

13 (1a) Orgnal w/ Hgh SNR (2a) Orgnal w/ Low SNR (1b) BURS result w/ Hgh SNR (2b) BURS result w/ Low SNR (1c) rburs result w/ Hgh SNR (2c) rburs result w/ Low SNR Fgure 7. Compare BURS wth rburs algorthm. Left column s for Hgh SNR case, rght column s for Low SNR case. The orgnal mage s produced by grddng wth Pre-Densty Compensaton & Deapodzaton. Low SNR mage s produced by addng Gaussan nose n K-space. For all BURS/rBURS reconstructons, set δ =1.5, =3. Regularzaton smoothng parameter ρ=

14 (a) Orgnal Image (b) BURS (c) Dfference Image between BURS and orgnal mage (d) rburs (e) Dfference Image for rburs between rburs and orgnal mage Fgure 8. Compare the best BURS and best rburs results wth orgnal mage. The orgnal mage s produced by grddng wth Pre-Densty Compensaton & Deapodzaton. The results shown here for BURS and rburs are the best results we get durng the smulaton. The dfference mage shown on the rght s the dfference between BURS/rBURS wth the orgnal mage. 14

15 (a) Profle for Orgnal row x= (b) Profle for BURS row x= (c) Profle for rburs row x=78 Fgure 9. The profles for dfferent mages shown n Fgure CONCLUSIONS 15

16 (1) Effect of neghborhood radus δ and. Fgure 4 and 5 show that () Ifδ s too small ( δ <0.3), then no matter how large the s, we can not get very good result. () Keep fxed, when δ ncreases from a very small number (around 0.3), the result wll become better frst, then become worse. For the tested cases, when δ /1.5 ~ /. 0, the BURS produces best result. () When 2 δ fxed, ncreasng the value of, the result becomes better. If we chec the BURS algorthm more carefully, we wll fnd that although δ, wll affect the result, they are not the root of the reason. In fact, t s who really affect the result! In order to produce good results, M and N values M should NOT exceed N too much. If M >> N occurs for some ponts (often occurs around the orgn n -plane, because our spral data s more dense around the orgn whch maes M acheve t s maxmum value around the orgn), we can stll get the result, however, there wll be some low frequency artfacts n the mages (see Fgure 4e, 4f, 5e, 5f ). Now we can explan the ()~() lsted above based on too small. () M and N values. () N and M should not be M can not exceed N too much all the tme, otherwse the result wll have some low-frequency artfact. () the bgger the N and M, the better the result. (2) Effect of the shape of the neghborhood. Our results show that BURS wth crcular neghborhood wll produce a lttle bt better results than that of square neghborhood, but the dfferences are small (Fgure 6). (3) BURS vs. rburs. BURS s senstve to the hgh level of nose as well as underdetermned case (Fgure 7-1b, 2b). On the contrary, the rburs s robust to the hgh level of nose as well as underdetermned case (Fgure 7-1c, 2c). rburs s also robust n the case of combnaton of hgh nose and underdetermned matrx. Even n ths worst case, the result of rburs (Fgure 7-2c) s stll very close to the orgnal mage (Fgure 7-2a), whch s produced usng grddng wth Pre-Densty Compensaton & Deapodzaton. 16

17 (4) Fdelty of BURS/rBURS By checng the reconstructed mages, dfference mages (Fgure 8) and the profles of the reconstructed mages (Fgure 9), we can conclude that (I) The best results produced by BURS and rburs are very close to the orgnal mage. (II) There are some small errors occur n hgh frequency components,.e. some errors around the edges. REFERENCES [1] John Pauly. Image Reconstructon Textboo (n progress), Chaper 5: Reconstructon of non-cartesan Data. [2] Rosenfeld D. An optmal and effcent new grddng algorthm usng sngular value decomposton. Magnetc Resonance n Medcne 1998; 40:12-23 [3] Morguch H, Wendt M, Duer JL. Applyng the unform resamplng (URS) algorthm a a Lssajous trajectory: fast mage reconstructon wth optmal grddng. Magnetc Resonance n Medcne 2000; 44: [4] Rosenfeld, Danel. New Approach to Grddng usng Regularzaton and Estmaton Theory, Magnetc Resonance n Medcne 2002; 48: APPENDIX MATLAB CODES %%%%%%%%%%%%%%%%%%%%%%%%%%%% % EE591 MRI % % Term Project BURS & rburs % % Zheng L, Dec % %%%%%%%%%%%%%%%%%%%%%%%%%%%% clear; close all; n=128; load rt_spral.mat; %load nose_spral; %d=nd+d; %{d: data; : samplng ernel; w: weght} %{nd: addtve Gausan nose} load same random nose data each tme erc=3; % (effectve) radus of delta- neghborhood n Cartesan coordnate r=1.3; % radus of delta- neghborhood n Non-Cartesan coordnate shape='c'; % shape of the neghorhood, 'c'-->crcular; 's'-->square % for sqare neghorhood, rc=effectve r * sqrt(p/4) f sequal(shape, 's') rc=erc*sqrt(p/4); dsp(strcat('square Neghborhood, Radus=', num2str(rc))); % for crcular neghorhood, rc=effectve r f sequal(shape, 'c') rc=erc; dsp(strcat('crcular Neghborhood, Radus=', num2str(rc))); f (0) %BURS [MB, OMB]= grdburs(d,,n, r, rc, shape); % call BURS grddng functon; mgb=ft(mb); fgure; magesc(abs(mgb)); axs square; colormap('gray'); colormenu; axs off; else % rburs 17

18 [MrB, OMrB]=grdrBURS(d,, n, r, rc, 0.01, shape); % call rburs grddng functon mgrb=ft(mrb); fgure; magesc(abs(mgrb)); axs square; colormap('gray'); colormenu; axs off; %%%%%%%%%%%%%%%%%%%%%%% % functon BURS % %%%%%%%%%%%%%%%%%%%%%%% functon [M, OM] = grdburs(d,,n,r,rc,shape) % functon [M, OM] = grdburs(d,,n,r,rc) % Bloc Unform ReSamplng method for grddng % d -- -space data % -- -trajectory, scaled -0.5 to 0.5 % n -- mage sze % r-- non-cartesan ernel radus % rc-- cartesan ernel radus % shape-- choose crcle (=='c') neghborhood % or square neghborhood (=='s') for Cartesan ponts % % M -- K-space nterpolated data % OM-- nose amplfcaton (defned n Rosenfeld 2002 Magn Reson Med) % % Zheng L, Nov % convert to sngle column d=d(:); =(:); % convert -space samples to matrx ndces nx=(n+1)/2 + (n-1)*real(); ny=(n+1)/2 + (n-1)*mag(); % ntalze the output matrx M=zeros(n,n); OM=zeros(n,n); % change the cartesan coordnate to one column, so that we can fnd % the cartesan pont wthng "rc" easly. [mxc, myc]=meshgrd(1:n, 1:n); mxc=mxc(:); myc=myc(:); % man loop, compute the BURS grddng value for each pont for xc=cel(1+rc):floor(n-rc) for yc=cel(1+rc):floor(n-rc) f shape=='s' [mxdc, mydc]=meshgrd(cel(xc-rc):floor(xc+rc), cel(yc-rc):floor(yc+rc)); % get the Cartesan ponts n "square" neghborhood of (xc,yc) xyc=mxdc(:)+*mydc(:); ndc=ones(length(xyc),1); % just to mae to!=[] elsef shape=='c' % fnd the ndex of the Cartesan ponts n the "rc" neghborhood of (xc+yc*) ndc=fnd( ((mxc-xc).^2 + (myc-yc).^2) <=rc^2+eps ); f ~(sempty(ndc)) % get the Cartesan ponts n "crcular" neghborhood of (xc,yc) xyc=mxc(ndc)+*(myc(ndc)); % fnd the ndex of the Non-Cartesan ponts n the "c" neghborhood of (xc+yc*) nd=fnd( ((nx-xc).^2 + (ny-yc).^2) <=r^2+eps ); % prnt the N ( of Cartesan ponts), and M ( of Non-Cartesan ponts) several postons along y=0 axs. f (yc==64) & (mod(xc, 16)==0) dsp(strcat('n=', num2str(length(ndc)),... ', M=', num2str(length(nd)),... num2str(xc), ',', num2str(yc), ')')); f ~(sempty(nd)) & ~(sempty(ndc)) % get the Non-Cartesan ponts n "crcular" neghborhood of (xc,yc) 18

19 xy=nx(nd)+*(ny(nd)); A=nterp2snc(xyc, xy); pa=pnv(a); % pnv can handle over/under determned cases automatcally nd0=fnd( (xyc==xc+yc*) ); M(xc, yc)=pa(nd0, :)*d(nd); OM(xc, yc)=sqrt( sum(pa(nd0, :).^2) ); %end functon [M, OM] = grdrburs(d,,n,r,rc,r,shape) %%%%%%%%%%%%%%%%%%%%%%%%%%%% % functon rburs % %%%%%%%%%%%%%%%%%%%%%%%%%%%% functon [M, OM] = grdrburs(d,,n,r,rc,r,shape) % regularzed Bloc Unform ReSamplng method for grddng % d -- -space data % -- -trajectory, scaled -0.5 to 0.5 % n -- mage sze % r-- non-cartesan ernel radus % rc-- cartesan ernel radus % shape-- choose crcle (=='c') neghborhood % or square neghborhood (=='s') for Cartesan ponts % r -- regularzaton smoothng parameter % % M -- K-space nterpolated data % OM-- nose amplfcaton (defned n Rosenfeld 2002 Magn Reson Med) % % Zheng L, Nov % convert to sngle column d=d(:); =(:); % convert -space samples to matrx ndces nx=(n+1)/2 + (n-1)*real(); ny=(n+1)/2 + (n-1)*mag(); % ntalze the output matrx M=zeros(n,n); OM=zeros(n,n); % change the cartesan coordnate to one column, so that we can fnd % the cartesan pont wthng "rc" easly. [mxc, myc]=meshgrd(1:n, 1:n); mxc=mxc(:); myc=myc(:); % man loop, compute the BURS grddng value for each pont for xc=cel(1+rc):floor(n-rc) for yc=cel(1+rc):floor(n-rc) f shape=='s' [mxdc, mydc]=meshgrd(cel(xc-rc):floor(xc+rc), cel(yc-rc):floor(yc+rc)); % get the Cartesan ponts n "square" neghborhood of (xc,yc) xyc=mxdc(:)+*mydc(:); ndc=ones(length(xyc),1); % just to mae to!=[] elsef shape=='c' % fnd the ndex of the Cartesan ponts n the "rc" neghborhood of (xc+yc*) ndc=fnd( ((mxc-xc).^2 + (myc-yc).^2) <=rc^2+eps ); f ~(sempty(ndc)) % get the Cartesan ponts n "crcular" neghborhood of (xc,yc) xyc=mxc(ndc)+*(myc(ndc)); % fnd the ndex of the Non-Cartesan ponts n the "c" neghborhood of (xc+yc*) nd=fnd( ((nx-xc).^2 + (ny-yc).^2) <=r^2+eps ); % prnt the N ( of Cartesan ponts), and M ( of Non-Cartesan ponts) several postons along y=0 axs. 19

20 f (yc==64) & (mod(xc, 16)==0) dsp(strcat('n=', num2str(length(ndc)),... ', M=', num2str(length(nd)),... num2str(xc), ',', num2str(yc), ')')); f ~(sempty(nd)) & ~(sempty(ndc)) % get the Non-Cartesan ponts n "crcular" neghborhood of (xc,yc) xy=nx(nd)+*(ny(nd)); A=nterp2snc(xyc, xy); % compute the regularzed pseduo-nverse usng "Thonov" wndow coeffcents f length(nd)<=length(ndc) % underdetermned case pa=a'*nv(a*a'+r*eye(length(nd))); else % overdetermned case pa=nv(a'*a+r*eye(length(ndc)))*a'; nd0=fnd( (xyc==xc+yc*) ); M(xc, yc)=pa(nd0, :)*d(nd); OM(xc, yc)=sqrt( sum(pa(nd0, :).^2) ); %end %%%%%%%%%%%%%%%%%%%%%%% % functon nterp2snc % %%%%%%%%%%%%%%%%%%%%%%% functon A=nterp2snc(xyc, xy); % functon A=nterp2snc(xyc, xy) % 2D nterpolaton usng snc functon. % xyc: the column vector of Cartesan pont postons, % each poston s represented by a complex number. % xy: the column vector of Non-Cartesan pont poston. % A: the lnear transform matrx s.t. the (DATA@xy)=A*(DATA@xyc); % % Zheng L, Nov % pxy* s the postons n Cartesan(c) and Non-cartan() [pxyc, pxy]=meshgrd(xyc, xy); % the dstances between each ponts of Cartesan and Non-Cartesan dst=pxyc-pxy; A=snc(real(dst)).*snc(mag(dst)); % end 20