function c = mefwa2(x,pat,tp)% mefwa2 - 2D forward mirror-extended wave atom transform% -----------------% INPUT% --% x is a real N-by-N matrix. N is a power of 2.% --% 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% --% 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.% -----------------% 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)    %----------------------------------------------------------------------------------------------------------    N = size(x,1);    lst = freq_pat(N,pat);    E = 2^length(lst);        f = dct2(x);    f = mescatter(f,E);    f = mescatter(f',E)';    A = size(f,1);        c = cell(length(lst),1);    %------------------    for s=1:length(lst)      nw = length(lst{s});      c{s} = cell(nw,nw);      for I=0:nw-1        for J=0:nw-1          if(lst{s}(I+1)==0 & lst{s}(J+1)==0)            c{s}{I+1,J+1} = [];          else            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 = zeros(D,D);            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                res(mod(Idx,D)+1,mod(Jdx,D)+1) = res(mod(Idx,D)+1,mod(Jdx,D)+1) + conj( Icf.'*Jcf ) .* f(mod(Idx,A)+1,mod(Jdx,A)+1);              end            end            c{s}{I+1,J+1} = ifft2(res) * sqrt(prod(size(res))) / 2; %LEXING: IMPORTANT          end        end      end    end      elseif(strcmp(tp,'complex')==1)    %----------------------------------------------------------------------------------------------------------    N = size(x,1);    lst = freq_pat(N,pat);    E = 2^length(lst);        f = dct2(x);    f = mescatter(f,E);    f = mescatter(f',E)';    A = size(f,1);        c = cell(length(lst),1);    %------------------    for s=1:length(lst)      nw = length(lst{s});      c{s} = cell(nw,nw);      for I=0:nw-1        for J=0:nw-1          if(lst{s}(I+1)==0 & lst{s}(J+1)==0)            c{s}{I+1,J+1} = [];          else            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 = 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);            res(mod(Idx,D)+1,mod(Jdx,D)+1) = res(mod(Idx,D)+1,mod(Jdx,D)+1) + abs( Icf.'*Jcf ) .* f(mod(Idx,A)+1,mod(Jdx,A)+1);            c{s}{I+1,J+1} = ifft2(res) * sqrt(prod(size(res)));          end        end      end    end      elseif(strcmp(tp,'directional')==1)    %-----------------------------------------------------------------------------------------------------------    N = size(x,1);    lst = freq_pat(N,pat);    E = 2^length(lst);        f = dct2(x);    f = mescatter(f,E);    f = mescatter(f',E)';    A = size(f,1);        c1 = cell(length(lst),1);    c2 = cell(length(lst),1);    %------------------    for s=1:length(lst)      nw = length(lst{s});      c1{s} = cell(nw,nw);      c2{s} = cell(nw,nw);      for I=0:nw-1        for J=0:nw-1          if(lst{s}(I+1)==0 & lst{s}(J+1)==0)            c1{s}{I+1,J+1} = [];            c2{s}{I+1,J+1} = [];          else            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 = 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);            res(mod(Idx,D)+1,mod(Jdx,D)+1) = res(mod(Idx,D)+1,mod(Jdx,D)+1) + conj( Icf.'*Jcf ) .* f(mod(Idx,A)+1,mod(Jdx,A)+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);            res(mod(Idx,D)+1,mod(Jdx,D)+1) = res(mod(Idx,D)+1,mod(Jdx,D)+1) + conj( Icf.'*Jcf ) .* f(mod(Idx,A)+1,mod(Jdx,A)+1);            tmp1 = ifft2(res) * sqrt(prod(size(res)));            %c1{s}{I+1,J+1} = tmp1(:,end/2+1:end)/sqrt(2);            c1{s}{I+1,J+1} = tmp1(:,1:end/2)/sqrt(2);                        res = zeros(D,D);            Idx = [ceil(Ifm):floor(Ito)];      Icf = kf_rt(Idx/B*pi, I);            Jdx = [ceil(-Jto):floor(-Jfm)];      Jcf = kf_lf(Jdx/B*pi, J);            res(mod(Idx,D)+1,mod(Jdx,D)+1) = res(mod(Idx,D)+1,mod(Jdx,D)+1) + conj( Icf.'*Jcf ) .* f(mod(Idx,A)+1,mod(Jdx,A)+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);            res(mod(Idx,D)+1,mod(Jdx,D)+1) = res(mod(Idx,D)+1,mod(Jdx,D)+1) + conj( Icf.'*Jcf ) .* f(mod(Idx,A)+1,mod(Jdx,A)+1);            tmp2 = ifft2(res) * sqrt(prod(size(res)));            %c2{s}{I+1,J+1} = tmp2(:,end/2+1:end)/sqrt(2);            c2{s}{I+1,J+1} = tmp2(:,1:end/2)/sqrt(2);          end        end      end    end    c = [c1 c2];  else    error('wrong');  end