## [62fbef]: wavelets / fwt.m  Maximize  Restore  History

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183``` ```function c = fwt(f,h,J,varargin) %FWT Fast Wavelet Transform % Usage: c = fwt(f,h,J); % c = fwt(f,h,J,...); % % Input parameters: % f : Input data. % h : Analysis Wavelet Filterbank. % J : Number of filterbank iterations. % % Output parameters: % c : Coefficients stored in J+1 cell-array. % % % `c=fwt(f,h,J)` computes wavelet coefficients *c* of the input signal *f* % using basis constructed from the filters *h* and the depth *J* using MRA principle. % For computing the coefficients the fast wavelet transform algorithm (or Mallat's algorithm) is % emplyed. If *f* is a matrix, the transformation is applied to each of *W* columns. % Coefficients in cell array \$c\{j\}\$ for \$j=1,\ldots,J+1\$ are ordered with inceasing central frequency % of the equivalent filter frequency response or equivalently with decreasing wavelet scale. % The number of coefficients in \$c\{j\}\$ for \$j=2,\ldots,J+1\$ is as % follows:: % % length(c{j}) = ceil(2^(j-2-J)length(f)) % % and length(c{1})=length(c{2}). % If *f* is matrix with *W* collumns, each element of the cell array \$c\{j\}\$ is a matrix % with *W* collumns with coefficients belonging to the appropriate input channel. % % The proper name for the transform is dyadic (or critically subsampled) % discrete wavelet transform which is equivalent to the (bi)orthogonal % wavelet expansion provided appropriate (bi)orthogonal wavelet filterbank is supplied. % % The transform is 2^J-shift invariant. % % The following flags are supported: % % 'dec','undec' % Type of the wavelet transform. % % 'per','zpd','sym','symw','asym','asymw','ppd','sp0' % Type of the boundary handling. % % Time-invariant wavelet tranform: % -------------------------------- % % `c=fwt(f,h,J,'undec')` computes redundant time (or shift) invariant wavelet representation of the % input signal *f* using a-trous algorithm. length(c{j})=length(f) for all *j*. % % Another names for this version of the wavelet transform are: % undecimated wavelet transform, stationary wavelet transform or even % "continuous" (as the time step is one sample) wavelet transform. % The redundancy is exactly *J+1*. % % Boundary handling: % ------------------ % % `fwt(f,h,J,ext)` computes slightly redundant wavelet % representation of the input signal *f* with the chosen boundary % extension *ext*. % % The default periodic extension at the signal boundaries can result in % "false" hight wavelet coefficients near the boundaries due to the % possible discontinuity introduced by the periodic extension. Using % different kind of the boundary extension comes with a price of a slight % redundancy of the wavelet representation. % % For the *type*='dec' option, the number of coefficients in *c{j}* for *j*=2,...,J+1 is as % follows:: % % length(c{j}) = floor(2^(j-2-J)length(f) + (1-2^(j-2-J))(length(h{1})-1)) % % and length(c{1})=length(c{2}). % % For the *type*='undec' option, the redundancy can become more than slight for the % number of coefficients in *c* grows as follows: % % .. length(c{J+1}) = length(f) + length(h{1})-1 % % and for *j*=J,...,2 % % .. length(c{j}) = length(c{j+1}) + 2^(j-J+1)*(length(h{1})-1) % % and length(c{1})=length(c{2}). % % See also: ifwt % % Demos: % % References: ma98 % AUTHOR : Zdenek Prusa. % TESTING: TEST_FWT % REFERENCE: REF_FWT if nargin<3 error('%s: Too few input parameters.',upper(mfilename)); end; if ~isnumeric(J) || ~isscalar(J) error('%s: "J" must be a scalar.',upper(mfilename)); end; if(J<1 && rem(a,1)~=0) error('%s: J must be a positive integer.',upper(mfilename)); end do_definedfb = 0; if(iscell(h)) if(length(h)<2) error('%s: h is expected to be a cell array containing two or more filters.',upper(mfilename)); end for ii=2:numel(h) if(length(h{1})~=length(h{ii})) error('%s: Wavelet filters have to have equal length.',upper(mfilename)); end end if(length(h{1})< 2) error('%s: Wavelet filters should have at least two coefficients.',upper(mfilename)); end elseif(isstruct(h)) do_definedfb = 1; elseif(ischar(h)) h = waveletfb(h); do_definedfb = 1; else error('%s: Unrecognized Wavelet filters definition.',upper(mfilename)); end %% ----- step 1 : Verify f and determine its length ------- % Change f to correct shape. [f,Ls,W,wasrow,remembershape]=comp_sigreshape_pre(f,upper(mfilename),0); if(Ls<2) error('%s: Input signal seems not to be a vector of length > 1.',upper(mfilename)); end definput.import = {'fwt'}; [flags,kv]=ltfatarghelper({},definput,varargin); if(do_definedfb) if(flags.do_type_null) flags.type = h.type; end if(flags.do_ext_null) flags.ext = h.ext; end h = h.h; else % setting defaults if(flags.do_type_null) flags.type = 'dec'; end if(flags.do_ext_null) flags.ext = 'per'; end end %% ----- step 2 : Check whether the input signal is long enough % input signal length is not restricted for expansive wavelet transform (extension type other than the default 'per') flen = length(h{1}); if(strcmp(flags.ext,'per')) if(strcmp(flags.type,'dec')) minLs = (2^J-1)*(flen-1); % length of the longest equivalent filter -1 else minLs = (2^(J-1))*(flen-1); % length of the longest upsampled filter - 1 end if Ls