function x = meiwa2(c,pat,tp)% meiwa2 - 2D inverse mirror-extended wave atom transform% -----------------% INPUT% --% c is a cell array which contains the wave atom coefficients. If% tp=='ortho', then c{j}{m1,m2}(n1,n2) is the coefficient at scale j,% frequency index (m1,m2) and spatial index (n1,n2). If% tp=='directional', then c{j,d}{m1,m2}(n1,n2) with d=1,2 are the% coefficients at scale j, frequency index (m1,m2) and spatial index% (n1,n2). If tp=='complex', then c{j}{m1,m2)(n1,n2) is the% complex-valued coefficients at scale j, frequency index (m1,m2) and% spatial index (n1,n2). Notice thatm, for the mirror-extended wave% atoms, the spatial indices wrap around once.% --% pat specifies the type of frequency partition which satsifies% parabolic scaling relationship. pat can either be 'p' or 'q'.% --% tp is the type of tranform.% 	'ortho': frame based on the orthobasis construction of % 		the standard wave atom% 	'directional': real-valued frame with single oscillation direction% 	'complex': complex-valued frame% -----------------% OUTPUT% --% x is a real N-by-N matrix. N is a power of 2.% -----------------% Written by Lexing Ying and Laurent Demanet, 2007    if( ismember(tp, {'ortho','directional','complex'})==0 | ismember(pat, {'p','q','u'})==0 )    error('wrong');  end    if(strcmp(tp, 'ortho')==1)    %---------------------------------------------------------    T = 0;    for s=1:length(c)      nw = length(c{s});      for I=1:nw        for J=1:nw          T = T + prod(size(c{s}{I,J}));        end      end    end    N = sqrt(T/4); %redundancy of 4    lst = freq_pat(N,pat);        E = 2^length(lst);    A = 2*(N+E); %extension    f = zeros(A,A);        %------------------    for s=1:length(lst)      nw = length(lst{s});      for I=0:nw-1        for J=0:nw-1          if(~isempty(c{s}{I+1,J+1}))            B = 2^(s-1);            D = 2*B;            Ict = I*B;      Jct = J*B; %starting position in freq            if(mod(I,2)==0)              Ifm = Ict-2/3*B;        Ito = Ict+4/3*B;            else              Ifm = Ict-1/3*B;        Ito = Ict+5/3*B;            end            if(mod(J,2)==0)              Jfm = Jct-2/3*B;        Jto = Jct+4/3*B;            else              Jfm = Jct-1/3*B;        Jto = Jct+5/3*B;            end            res = fft2(c{s}{I+1,J+1}) / sqrt(prod(size(c{s}{I+1,J+1}))) / 2; %LEXING: IMPORTANT            for id=0:1              if(id==0)                Idx = [ceil(Ifm):floor(Ito)];      Icf = kf_rt(Idx/B*pi, I);              else                Idx = [ceil(-Ito):floor(-Ifm)];      Icf = kf_lf(Idx/B*pi, I);              end              for jd=0:1                if(jd==0)                  Jdx = [ceil(Jfm):floor(Jto)];      Jcf = kf_rt(Jdx/B*pi, J);                else                  Jdx = [ceil(-Jto):floor(-Jfm)];      Jcf = kf_lf(Jdx/B*pi, J);                end                f(mod(Idx,A)+1,mod(Jdx,A)+1) = f(mod(Idx,A)+1,mod(Jdx,A)+1) + ( Icf.'*Jcf ) .* res(mod(Idx,D)+1,mod(Jdx,D)+1);              end            end          end        end      end    end    f = mecombine(f',E)';    f = mecombine(f,E);        x = idct2(f);      elseif(strcmp(tp,'complex')==1)    %---------------------------------------------------------    T = 0;    for s=1:length(c)      nw = length(c{s});      for I=1:nw        for J=1:nw          T = T + prod(size(c{s}{I,J}));        end      end    end    N = sqrt(T/4); %redundancy of 4    lst = freq_pat(N,pat);        E = 2^length(lst);    A = 2*(N+E); %extension    f = zeros(A,A);        %------------------    for s=1:length(lst)      nw = length(lst{s});      for I=0:nw-1        for J=0:nw-1          if(~isempty(c{s}{I+1,J+1}))            B = 2^(s-1);            D = 2*B;            Ict = I*B;      Jct = J*B; %starting position in freq            if(mod(I,2)==0)              Ifm = Ict-2/3*B;        Ito = Ict+4/3*B;            else              Ifm = Ict-1/3*B;        Ito = Ict+5/3*B;            end            if(mod(J,2)==0)              Jfm = Jct-2/3*B;        Jto = Jct+4/3*B;            else              Jfm = Jct-1/3*B;        Jto = Jct+5/3*B;            end            res = fft2(c{s}{I+1,J+1}) / sqrt(prod(size(c{s}{I+1,J+1}))); %res = zeros(D,D);            Idx = [ceil(Ifm):floor(Ito)];      Icf = kf_rt(Idx/B*pi, I);            Jdx = [ceil(Jfm):floor(Jto)];      Jcf = kf_rt(Jdx/B*pi, J);            f(mod(Idx,A)+1,mod(Jdx,A)+1) = f(mod(Idx,A)+1,mod(Jdx,A)+1) + abs( Icf.'*Jcf ) .* res(mod(Idx,D)+1,mod(Jdx,D)+1);          end        end      end    end    f = mecombine(f',E)';    f = mecombine(f,E);        x = idct2(f);      elseif(strcmp(tp,'directional')==1)    %---------------------------------------------------------    c1 = c(:,1);    c2 = c(:,2);        T = 0;    for s=1:length(c1)      nw = length(c1{s});      for I=1:nw        for J=1:nw          T = T + prod(size(c1{s}{I,J})) + prod(size(c2{s}{I,J}));        end      end    end    N = sqrt(T/4);    lst = freq_pat(N,pat);        E = 2^length(lst);    A = 2*(N+E); %extension    f = zeros(A,A);    %------------------    for s=1:length(lst)      nw = length(lst{s});      for I=0:nw-1        for J=0:nw-1          if(~isempty(c1{s}{I+1,J+1}))            B = 2^(s-1);            D = 2*B;            Ict = I*B;      Jct = J*B; %starting position in freq            if(mod(I,2)==0)              Ifm = Ict-2/3*B;        Ito = Ict+4/3*B;            else              Ifm = Ict-1/3*B;        Ito = Ict+5/3*B;            end            if(mod(J,2)==0)              Jfm = Jct-2/3*B;        Jto = Jct+4/3*B;            else              Jfm = Jct-1/3*B;        Jto = Jct+5/3*B;            end            [a,b] = size(c1{s}{I+1,J+1});            tmp = zeros(a,2*b);          tmp(:,1:end/2) = c1{s}{I+1,J+1}/sqrt(2);            res = fft2(tmp) / sqrt(prod(size(tmp)));            Idx = [ceil(Ifm):floor(Ito)];      Icf = kf_rt(Idx/B*pi, I);            Jdx = [ceil(Jfm):floor(Jto)];      Jcf = kf_rt(Jdx/B*pi, J);            f(mod(Idx,A)+1,mod(Jdx,A)+1) = f(mod(Idx,A)+1,mod(Jdx,A)+1) + ( Icf.'*Jcf ) .* res(mod(Idx,D)+1,mod(Jdx,D)+1);            Idx = [ceil(-Ito):floor(-Ifm)];      Icf = kf_lf(Idx/B*pi, I);            Jdx = [ceil(-Jto):floor(-Jfm)];      Jcf = kf_lf(Jdx/B*pi, J);            f(mod(Idx,A)+1,mod(Jdx,A)+1) = f(mod(Idx,A)+1,mod(Jdx,A)+1) + ( Icf.'*Jcf ) .* res(mod(Idx,D)+1,mod(Jdx,D)+1);                        [a,b] = size(c2{s}{I+1,J+1});            tmp = zeros(a,2*b);          tmp(:,1:end/2) = c2{s}{I+1,J+1}/sqrt(2);            res = fft2(tmp) / sqrt(prod(size(tmp)));            Idx = [ceil(Ifm):floor(Ito)];      Icf = kf_rt(Idx/B*pi, I);            Jdx = [ceil(-Jto):floor(-Jfm)];      Jcf = kf_lf(Jdx/B*pi, J);            f(mod(Idx,A)+1,mod(Jdx,A)+1) = f(mod(Idx,A)+1,mod(Jdx,A)+1) + ( Icf.'*Jcf ) .* res(mod(Idx,D)+1,mod(Jdx,D)+1);            Idx = [ceil(-Ito):floor(-Ifm)];      Icf = kf_lf(Idx/B*pi, I);            Jdx = [ceil(Jfm):floor(Jto)];      Jcf = kf_rt(Jdx/B*pi, J);            f(mod(Idx,A)+1,mod(Jdx,A)+1) = f(mod(Idx,A)+1,mod(Jdx,A)+1) + ( Icf.'*Jcf ) .* res(mod(Idx,D)+1,mod(Jdx,D)+1);          end        end      end    end    f = mecombine(f',E)';    f = mecombine(f,E);        x = idct2(f);  else    %---------------------------------------------------------    error('wrong');  end