Comparison of Transient Response Reduction Methods in balanced SSFP

Transcription

1 Comparison of Transient Response Reduction Methods in balanced SSFP EE591 Magnetic Resonance Imaging and Reconstruction Final Project Report Hua Hui Dec.7 th 24

2 Introduction As a consequence of the mergence of fast gradient MRI hardware, stead-state free precession (SSFP) sequences are becoming more and more popular. Balanced SSFP sequences, which are also referred as refocused SSFP, truefisp, FIESTA and balanced FFE, have some advantages over other rapid imaging sequences. The steady states of SSFP provide the highest SNR efficiency among all the rapid imaging sequences, and show the special T2/T1 contrast.. They have inherent off-resonance effects, which limited their feasibility before rapid gradient technique. However, the establishing of the steady state has many problems: It takes long to reach steady state, and the transient response shows large fluctuations. The images reconstructed from transient signal come with artifacts. Therefore, the research of finding transient respons e reduction (catalization) methods for balanced SSFP becomes both important and necessary. Methods In this section an overview and descriptions of all the proposed methods is given in the order of complexity. Further comparison and discussion of these methods can be found in later section. Half tip angle pulse The simplest method [1] is to place a single a/2-pulse before the regular pulse sequences. It can be either a pulsea/2 around x-axis with a time interval of TR before the first regular pulse, or a pulsea/2 around (-y) axis with a time interval of TR/2. The pulse sequences are showed bellowed. 7 6 a x a x a x a x a x a x 5 Flip Angle a x / Exciton Pulse #

3 Liner flip angle pulse series A more complicated method is using a series of linear flip angle pulses instead of the singlea/2-pulse before the regular RF pulse sequence. This method is also called linear ramp up method in [2] In this methods the flip angles are increased linearly from a/(n+1) to an/(n+1) in magnitude, and the pulses are separated by TR in time. It is obvious that the half tip angle pulse is special case for N=1. The general linear flip angle sequence is shown here: 7 α α α α 6 5 Flip Angle TR Exciton Pulse # LeRoux simplified mode l In P. Le Roux s paper[6], the above linear flip angle pulse method was generalized into a category of ramp pulses catalyzing methods. Instead of the constant flip angle difference in linear ramp, the new methods use variation of the flip angle differen ce. For example, in his paper, he used Kaiser Bessel window for :

4 I ( ) is modified Bessel function of order G is the factor that guarantee, and is the control parameter. Here is a pulse sequence with 7 6 α a... 5 Flip Angle TR Exciton Pulse # C-TIDE C-TIDE is referred as Continuous Transition to Driven Equilibrium method in [ 5]. It also used modified flip angle pulse to reach the steady state. However, instead of the increasing magnitude of flip angles, it uses a decreasing flip angles. Start with and picked, C-TIDE uses the following formula to design the flip pulse sequence: Here is a sequence with = 18, N=8;

5 Flip Angle a 2 TR/2 TR Exciton Pulse # Two-stage catalyzing method The two-stage spectrally-selective method in [4] is the most complicated methods and theoretically it will give the most accurate solution. The method treats the magnitude and direction of the magnetization vector separately. First the magnitude scaling scales the equilibrium magnetization vector to approximately its steady-state length, then a direction-selection pulse rotates the magnetization so that the transient magnetization lies perpendicular to the steady-state magnetization direction and the oscillation is eliminated. In the direction-selection stage this method uses the Shinnar-Le Roux(SLR) pulse design algorithm instead of the small-angle approximation. SLR method gives the more accurate solution, but also make the methods too complicated to use. Moreover, the method is limited by its sensitivity to B1 field variation. So this method will not be implemented in the simulation. However, the idea of separation of magnitude and direction manipulation gives insight of different catalyzing methods and will be used in the analysis and comparison of other methods. Comparation Here several methods mentions are simulated for different off-resonances: Point A B C D E F Off-resonance 2 f (rad) These off-resonance frequencies are also showed the steady -state off-resonance profile:

6 .25 A.2 B.15 M xy.1 C F D Off-Resonance 2π f (xπ rad) E 2 1 F M xy -1-2 A B C D Off-Resonance 2π f (xπ rad) E The parameters for different methods are: The catalyzing flip angle series length is N=2 (except for half flip method). =18 o for C-TIDE method The actual simulating sequences are different with those mentioned in last section in the sense that the sequence is added with addition phase of p. This modulation shifts the off-resonance profile by p such that the on-resonance spin is not affected by banding effects. The signal without any catalyzing is also drawn for references.

7 1 (A) φ = M xy M xy (B) φ = π/ M xy M xy

8 1 (C) φ = 3π/ M xy M xy (D) φ = 7π/ M xy M xy

9 1 (E) φ = π M xy M xy (F) φ = 5π/ M xy M xy

10 From these simulation results it is not hard observe that: The half flip angle pulse only works for on-resonance spins. For off-resonance spins, it still generates wild oscillation. Linear flip angles series pulses have broader off-resonance range for least oscillation, but it generates small oscillation for on resonance ( f) spins. Raiser ramp does not have the oscillation the linear flip ramp for on resonance spins. However, both of them generate a great amount of oscillation when the off-resonance freq is very close to the banding nulls. Surprisingly, the C-TIDE method has the best performance for same catalyzing pulse series length. It does not generate much oscillation even when the off-resonance freq is very close to the banding nulls -. Unlike other methods, the transient signal of the C-TIDE method does not even have the initial signal fluctuations. The transient signal decreasing smoothly from the very beginning of the sequence. The properties makes it more attractive for the applications in which the initial waiting period is prohibited. To further prove these observations, a 2-D signal intensity mapping with respect to time and off-resonance is drawn. In these mapping, the horizontal axis represents the off-resonance frequency, the vertical axis represents excitation number and the gray scale value represents the magnitude of signal. The oscillation for a special off-resonance frequency can be found when looking down long y axis. The more wrinkles, the larger oscillation generated. Mxy, Off-Resonance 2 π f (xπ rad)

11 Half-tip x with TR delay Off-Resonance 2π f (x π rad) Off-Resonance 2 π f (xπ rad)

12 Off-Resonance 2 π f (xπ rad) Off-Resonance 2π f (x π rad)

13 Analysis Using P.Le Roux s simplified model P.Le Roux [6] proposed that the zero-order difference of the flip angles series and the perpendicular magnetization have Fourier Transform relationship: We use this idea to draw the Fourier Transform of the flip angle zero-order difference, we can find that this profile linear ramp has larger side lope compare with Kaiser Ramp, which explains the reason that Kaiser Ramp has better performance. However, the C-TIDE method shows higher side lobe, but still turns out to be better than both linear ramp and. This seems can not be explained by the model Kaiser Ramp Linear Ramp C-TIDE Using two-stage method s idea Using the idea of two-stage method, we are trying to show the transient magnetization along the direction

14 of steady -state magnetization and perpendicular to the direction of steady -state magnetization:.45 (A) φ = M along Mss direction.4.35 M along Mss direction x 1-16 (A) φ = M perpendicular to Mss direction 4 M perpendicular to Mss direction

15 .24 (B) φ = π/ M along Mss direction M along Mss direction (B) φ = π/ M perpendicular to Mss direction.4.3 M perpendicular to Mss direction

16 .12 (C) φ = 3π/ M along Mss direction.1.8 M along Mss direction (C) φ = 3π/ M perpendicular to Mss direction.3.2 M perpendicular to Mss direction

17 .6 (D) φ = 7π/ M along Mss direction.5.4 M along Mss direction (D) φ = 7π/ M perpendicular to Mss direction.2.1 M perpendicular to Mss direction

18 -3 x 1 6 (E) φ = π M along Mss direction 4 2 M along Mss direction (E) φ = π M perpendicular to Mss direction M perpendicular to Mss direction

19 .15 (F) φ = 5π/ M along Mss direction.1.5 M along Mss direction (F) φ = 5π/ M perpendicular to Mss direction.3.2 M perpendicular to Mss direction

20 From these results, we could observed that almost all of the magnetizations have smooth decay along the direction of steady-state. The effectiveness of a catalyzing methods depends on the smoothness of the perpendicular magnetization. This further prove the methods in [4] Referrence [1]. R. F. Busse and S. J. Riederer, "Steady-state preparation for spoiled gradient echo imaging," Magn Reson Med, vol. 45, pp , 21. [2]. V. S. Deshpande, Y. C. Chung, Q. Zhang, S. M. Shea, and D. Li, "Reduction of transient signal oscillations in true-fisp using a linear flip angle series magnetization preparation," Magn Reson Med, vol. 49, pp , 23. [3]. F. Fuchs, G. Laub, and K. Othomo, "TrueFISP--technical considerations and cardiovascular applications," Eur J Radiol, vol. 46, pp , 23. [4]. B. A. Hargreaves, S. S. Vasanawala, J. M. Pauly, and D. G. Nishimura, "Characterization and reduction of the transient response in steady-state MR imaging," Magn Reson Med, vol. 46, pp , 21. [5]. J. Hennig, O. Speck, and K. Scheffler, "Optimization of signal behavior in the transition to driven equilibrium in steady-state free precession sequences," Magn Reson Med, vol. 48, pp. 81-9, 22. [6]. P. Le Roux, "Simplified model and stabilization of SSFP sequences, " J Magn Reson, vol. 163, pp , 23. [7]. W. R. Overall, D. G. Nishimura, and B. S. Hu, "Steady-state sequence synthesis and its application to efficient fat-suppressed imaging," Magn Reson Med, vol. 5, pp. 55-9, 23. [8]. K. Scheffler and S. Lehnhardt, "Pr inciples and applications of balanced SSFP techniques," Eur Radiol, vol. 13, pp , 23. [9]. K. Scheffler and J. Hennig, "Is TrueFISP a gradient-echo or a spin-echo sequence?,"

21 APPENDIX (MATLAB CODES): clear; close all; TR=1; T1=12; TE=TR/2; T2=25; alpha=7*pi/18; N=2; modulation=pi*1; ismax=;% Maximize the plot, only works under 124X768 % Set it to under other resolutions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%% % Print the result to word for the project report %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% isprint2word_=;% - not print / 1-print to word filename = 'D:\My Documents \Course\EE591\Project\Result_part_1.doc'; if isprint2word_ word = actxserver('word.application'); op = invoke(word.documents,'add'); end_of_doc = get(word.activedocument.content,'end'); set(word.application.selection,'start',end_of_doc); set(word.application.selection,'end',end_of_doc); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%% % Different Mode of the program %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% ndebug=1 switch ndebug case 1 % show the transient signal for 6 different off-resonance freq. phi=[,pi/2,3*pi/4,7*pi/8,pi,5*pi/4]; index_phi='a':'g'; str_phi=[' '; '\pi/2 '; '3\pi/4'; '7\pi/8'; ' \pi '; '5\pi/4']; case 2 % show transient signal vs. off-resonance freq phi=-2*pi:.2*pi:2*pi; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%

22 % % plot the off-resonance profile %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% % if ndebug==1 off=-2*pi:pi/16:2*pi; for n=1:length(off) Rflip =zrot(modulation)*xrot(alpha); [Atr,Btr] = freeprecess(tr -TE,T1,T2,off(n)/(2*pi*(TR/1))); [Ate,Bte] = freeprecess(te,t1,t2,off(n)/(2*pi*(tr/1))); % Mss = Ate*Rflip*Atr*Mss + (Ate*Rfl ip*btr+bte) Mss(:,n) = inv(eye(3)-ate*rflip*atr) * (Ate*Rflip*Btr+Bte); Mss_xy=Mss(1,:)+i*Mss(2,:); h=figure; subplot(2,1,1); plot(off/pi,abs(mss_xy)); for k=1:length(phi) hold on; inde=find(off==phi(k)); stem(off(inde)/pi,abs(mss_xy(inde)),'k','linewidth',1,'markeredgecolor','k', 'MarkerFaceColor',[ ],'MarkerSize',1); % stem(off(inde),abs(mss_xy(inde)),'-mo','linewidth',2,'markeredgecolor','k', 'MarkerFaceColor',[ ],'MarkerSize',12); text(off(inde)/pi,abs(mss_xy(inde))+.2, index_phi(k), 'clipping', 'off'); xlabel('off-resonance 2\pi \Deltaf (x\pi rad)'); ylabel(' M_{xy} '); hold off; subplot(2,1,2); plot(off/pi,angle(mss_xy)); hold on; for k=1:length(phi) hold on; inde=find(off==phi(k)); stem(off(inde)/pi,angle(mss_xy(inde)),'k','linewidth',1,'markeredgecolor','k', 'MarkerFaceColor',[ ],'MarkerSize',1); text(off(inde)/pi+.1,angle(mss_xy(inde))-.4, index_phi(k), 'clipping', 'off'); xlabel('off-resonance 2\pi \Deltaf (x\pi rad)'); ylabel('\anglem_{xy}'); if ismax==1 set(gcf,'units','normalized','position',[ ]); hold off; if isprint2word_ print(h,'-dmeta'); invoke(word.selection,'pastespecial',,,,,3); invoke(op,'saveas',filename,1);

23 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%% % compute the catalyzing flip angle series %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% K=2; %liner flip angle series length % % Kaiser Bessel Window beta=5; % a+b=-1 a*k+b=1 a=2/(k-1); b=-1-a; k=1:k; tau=a*k+b; [x I]=ik1a(beta*sqrt(1-tau.^2),[],[],[],[],[],[],[],[]); delta=i; flip_kaiser=delta; for l=2:k flip_kaiser(l)=flip_kaiser(l-1)+flip_kaiser(l); G=alpha/sum(delta); delta=delta*g; flip_kaiser=g*flip_kaiser; % % flip angle flip_ctide(1)=pi; dflip=(flip_ctide(1)-alpha)/(k-3/2); k=2:k; temp=flip_ctide(1)*ones(size(k))-(2*k-3)/2*dflip; flip_ctide=[flip_ctide(1)/2 flip_ctide(1) temp]; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%% % compute transient signal... %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% % for m=1:length(phi) M(:,:,m)= zeros(3,n+1); [Atr,Btr] = freeprecess(tr -TE,T1,T2,phi(m)/(2*pi*(TR/1))); [Ate,Bte] = freeprecess(te,t1,t2,phi(m)/(2*pi*(tr/1))); % Mn+1 = Ate*Rflip*Atr*Mn+ (Ate*Rflip*Btr+Bte) Rflip = zrot(modulation)*xrot(alpha); A=Ate*Rflip*Atr; b=ate*rflip*btr+bte; % steady state ss = inv(eye(3)-ate*rflip*atr) * (Ate*Rflip*Btr+Bte); % Non-catalyzing M(:,1,m)=Ate*Rflip*[,,1 ]'+Bte;

24 for n=2:n+1 M(:,n,m)=A*M(:,n-1,m)+b; Mxy(:,m)=squeeze(M(1,:,m)+i*M(2,:,m)); % Half-tip catalyzing (around -y with TR/2 delay) Halftip = zrot(modulation)*yrot(-alpha/2); M_halftip(:,1,m)=Ate*Halftip*[,,1]'; M_halftip(:,2,m)=Ate*Rflip*Ate*M_halftip(:,1,m); for n=3:n+1 M_halftip(:,n,m)=A*M_halftip(:,n -1,m)+b; Mxy_halftip(:,m)=squeeze(M_halftip(1,:,m)+i*M_halftip(2,:,m)); %% Half-tip catalyzing (around x with TR delay) Halftip2 = zrot(modulation)*xrot(alpha/2); M_halftip2(:,1,m)=Ate*Halftip2*[,,1]'; M_halftip2(:,2,m)=Ate*Rflip*Atr*M_halftip2(:,1,m); for n=3:n+1 M_halftip2(:,n,m)=A*M_halftip2(:,n -1,m)+b; Mxy_halftip2(:,m)=squeeze(M_halftip2(1,:,m)+i*M_halftip2(2,:,m)); % catalyzing Rmod = zrot(modulation); M_LFA(:,1,m)=Ate*xrot(alpha/K)*[,,1]'+Bte; for k=2:k Rflip_LFA=Rmod*xrot(k*alpha/K); M_LFA(:,k,m)=Ate*Rflip_LFA*Atr*M_LFA(:,k-1,m)+ Ate*Rflip_LFA*Btr+Bte; end for n=k+1:n+1 M_LFA(:,n,m)=A*M_LFA(:,n -1,m)+b; Mxy_LFA(:,m)=squeeze(M_LFA(1,:,m)+i*M_LFA(2,:,m)); % % Kaiser Bessel Window Rmod = zrot(modulation); M_KaiserRamp(:,1,m)=Ate*xrot(flip_Kaiser(1))*[,,1]'+Bte; for k=2:k Rflip_KaiserRamp=Rmod*xrot(flip_Kaiser(k)); M_KaiserRamp(:,k,m)=Ate*Rflip_KaiserRamp*Atr*M_KaiserRamp(:,k -1,m)+ Ate*Rflip_KaiserRamp*Btr+Bte; end for n=k+1:n+1 M_KaiserRamp(:,n,m)=A*M_KaiserRamp(:,n -1,m)+b; Mxy_KaiserRamp(:,m)=squeeze(M_KaiserRamp(1,:,m)+i*M_KaiserRamp(2,:,m)); % : Countinous Transition into driven equilibrium Rmod = zrot(modulation);

25 M_cTIDE(:,1,m)=Rmod*xrot(flip_cTIDE(1))*[,,1]'; for k=2:k+1 Rflip_cTIDE=Rmod*xrot(flip_cTIDE(k)); M_cTIDE(:,k,m)=Ate*Rflip_cTIDE*Atr*M_cTIDE(:,k-1,m)+ Ate*Rflip_cTIDE*Btr+Bte; end for n=k+2:n+1 M_cTIDE(:,n,m)=A*M_cTIDE(:,n-1,m)+b; Mxy_cTIDE(:,m)=squeeze(M_cTIDE(1,:,m)+i*M_cTIDE(2,:,m)); % % 'k','linewidth',1,'markeredgecolor','k', 'MarkerFaceColor',[ ],'MarkerSize',1); if ndebug==1 str=str_phi(m); str=['(',index_phi(m),') \phi = ',str_phi(m,:)]; h=figure; plot(abs(mxy(:,m)),':'); % hold on; % plot(abs(mxy_halftip(:,m)),'r^-'); hold on; plot(abs(mxy_halftip2(:,m)),'g.--'); hold on; plot(abs(mxy_lfa(:,m)),'r*-'); hold on; plot(abs(mxy_kaiserramp(:,m)),'ko-','markersize',5); hold on; plot(abs(mxy_ctide(:,m)),'k','linewidth',3); legend('','',' ',' ',' '); %legend('','half-tip -y with TR/2 delay','half-tip x with TR delay','linear Flip Angle ',' ',' '); title(str); ylabel(' M_{xy} '); xlabel(''); title([str,' M_{xy} ']); if ismax==1 set(gcf,'units','normalized','position',[ ]); if isprint2word_ print(h,'-dmeta'); invoke(word.selecti on,'pastespecial',,,,,3); hold off; [Malong Mperpend]=Mprojection(squeeze(M(:,:,m)),ss); [Malong_halftip2 Mperpend_halftip2]=Mprojection(squeeze(M_halftip2(:,:,m)),ss); [Malong_LFA Mperpend_LFA]=Mprojection(squeeze(M_LFA(:,:,m)),ss); [Malong_KaiserRamp Mperpend_KaiserRamp]=Mprojection(squeeze(M_KaiserRamp(:,:,m)),ss); [Malong_cTIDE Mperpend_cTIDE]=Mprojection(squeeze(M_cTIDE(:,:,m)),ss); h=figure; plot((malong),':'); % hold on; % plot((malong_halftip),'r^-'); hold on; plot((malong_halftip2),'g.--');

26 hold on; plot((malong_lfa),'r*-'); hold on; plot((malong_kaiserramp),'ko-','markersize',5); hold on; plot((malong_ctide),'k','linewidth',3); legend('','',' ',' ',' '); %legend('','half-tip -y with TR/2 delay','half-tip x with TR delay','linear Flip Angle ',' ',' '); title(str); ylabel('m along Mss direction'); xlabel(''); title([str ' M along Mss direction']); if ismax==1 set(gcf,'units','normalized','position',[ ]); if isprint2word_ print(h,'-dmeta'); invoke(word.selection,'pastespecial',,,,,3); hold off; h=figure; plot((mperpend),':'); % hold on; % plot((mperpend_halftip),'r^-'); hold on; plot((mperpend_halftip2),'g.--'); hold on; plot((mperpend_lfa),'r*-'); hold on; plot((mperpend_kaiserramp),'ko-','markersize',5); hold on; plot((mperpend_ctide),'k','linewidth',3); legend('','',' ',' ',' '); %legend('','half-tip -y with TR/2 delay','half-tip x with TR delay','linear Flip Angle ',' ',' '); title(str); ylabel('m perpendicular to Mss direction'); xlabel(''); title([str ' M perpendicular to Mss direction']); if ismax==1 set(gcf,'units','normalized','position',[ ]); if isprint2word_ print(h,'-dmeta'); invoke(word.selection,'pastespecial',,,,,3); invoke(op,'saveas',filename,1); hold off; % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%% % Plot transient signal vs. off-resonance %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

27 %%%%%%%%% % if ndebug==2 str_phi=[' -1'; '-.5'; ' '; '.5'; ' 1']; iptsetpref('imshowaxesvisible','on') h=figure; imshow(flipud(abs(mxy)),[ 1],'notruesize'); title([' Mxy,']); set(gca,'xlim',[ length(phi)]); set(gca,'xtick',[:length(phi)/4:length(phi)]); set(gca,'xticklabels',str_phi); xlabel('off-resonance 2\pi \Deltaf (x\pi rad)'); ylabel(''); if ismax==1 set(gcf,'units','normalized','position',[ ]); if isprint2word_ print(h,'-dmeta'); invoke(word.selection,'pastespecial',,,,,3); % h=figure; % imshow(flipud(abs(mxy_halftip)),[ 1],'notruesize'); % title(['half-tip -y with TR/2 delay']); % if ismax==1 set(gcf,'units','normalized','position',[ ]); % set(gca,'xlim',[ length(phi)]); % set(gca,'xtick',[:length(phi)/4:length(phi)]); % set(gca,'xticklabels',str_phi); % xlabel('off-resonance 2\pi \Deltaf (x\pi rad)'); % ylabel(''); % if isprint2word_ % print(h,'-dmeta'); % invoke(word.selection,'pastespecial',,,,,3); % h=figure; imshow(flipud(abs(mxy_halftip2)),[ 1],'notruesize'); title(['half-tip x with TR delay']); if ismax==1 set(gcf,'units','normalized','position',[ ]); set(gca,'xlim',[ length(phi)]); set(gca,'xtick',[:length(phi)/4:length(phi)]); set(gca,'xticklabels',str_phi); xlabel('off-resonance 2\pi \Deltaf (x\pi rad)'); ylabel(''); if isprint2word_ print(h,'-dmeta'); invoke(word.selection,'pastespecial',,,,,3);

28 h=figure; imshow(flipud(abs(mxy_lfa)),[ 1],'notruesize '); title([' ']); if ismax==1 set(gcf,'units','normalized','position',[ ]); set(gca,'xlim',[ length(phi)]); set(gca,'xtick',[:length(phi)/4:length(phi)]); set(gca,'xticklabels',str_phi); xlabel('off-resonance 2\pi \Deltaf (x\pi rad)'); ylabel(''); if isprint2word_ print(h,'-dmeta'); invoke(word.selection,'pastespecial',,,,,3); h=figure; imshow(flipud(abs(mxy_kaiserramp)),[ 1],'notruesize'); title([' ']); if ismax==1 set(gcf,'units','normalized','position',[ ]); set(gca,'xlim',[ length(phi)]); set(gca,'xtick',[:length(phi)/4:length(phi)]); set(gca,'xticklabels',str_phi); xlabel('off-resonance 2\pi \Deltaf (x\pi rad)'); ylabel(''); if isprint2word_ print(h,'-dmeta'); invoke(word.selection,'pastespecial',,,,,3); h=figure; imshow(flipud(abs(mxy_ctide)),[ 1],'notruesize'); title([' ']); if ismax==1 set(gcf,'units','normalized','position',[ ]); set(gca,'xlim',[ length(phi)]); set(gca,'xtick',[:length(phi)/4:length(phi)]); set(gca,'xticklabels',str_phi); xlabel('off-res onance 2\pi \Deltaf (x\pi rad)'); ylabel(''); if isprint2word_ print(h,'-dmeta'); invoke(word.selection,'pastespecial',,,,,3); if isprint2word_ invoke(op,'saveas',filename,1); invoke(op,'close'); invoke(word,'quit'); close all; disp(' '); disp(' done!'); disp(' ');