Transform description

This function realizes direct or inverse 1-D or N-D Discrete Cosine Transforms with shift depending on the option parameter value. For a 1-D array A of length n:

• For "dct1" the function computes the unnormalized DCT-I transform:

  

• For "dct2" the function computes the unnormalized DCT-II transform:

  

• For "dct3" the function computes the unnormalized DCT-III transform:

  

• For "dct4" the function computes the unnormalized DCT-IV transform:

  

• For "dct" the function computes the normalized DCT-II transform:

  

• For "idct" the function computes the normalized DCT-III transform:

  

The multi-dimensional DCT transforms , in general, are the separable product of the given 1d transform along each dimension of the array. For unnormalized transforms , computing the forward followed by the backward/inverse multi-dimensional transform will result in the original array scaled by the product of the dimension sizes.

The normalized multi-dimensional DCT transform of an array A with dimensions n₁, n₂, …, n_p is given by

The normalized multi-dimensional DCT inverse transform of an array A with dimensions n₁, n₂, …, n_p is given by

Syntax description

Short syntax

direct

X=dct(A,-1 [,option]) or X=dct(A [,option]) gives a direct transform according to the option value. The default is normalized DCT-II direct transform.

If A is a vector (only one dimension greater than 1) a 1-d transform is performed and in the other cases a n-dimensional transform is done.

(the -1 argument refers to the sign of the exponent..., NOT to "inverse"),

inverse

X=dct(A,1 [,option]) or X=idct(A [,option])performs the inverse transform.

If A is a vector (only one dimension greater than 1) a 1-d transform is performed and in the other cases a n-dimensional transform is done.

Long syntax for DCT along specified dimensions

• X=dct(A,sign,selection [,option]) allows to perform efficiently all direct or inverse dct of the "slices" of A along selected dimensions.

  For example, if A is a 3-D array X=dct(A,-1,2) is equivalent to:

  for i1=1:size(A,1),
    for i3=1:size(A,3),
      X(i1,:,i3)=dct(A(i1,:,i3),-1);
    end
  end

  and X=dct(A,-1,[1 3]) is equivalent to:

  for i2=1:size(A,2),
    X(:,i2,:)=dct(A(:,i2,:),-1);
  end

• X=dct(A,sign,dims,incr) is an old syntax that also allows to perform all direct or inverse dct of the slices of A along selected dimensions.

  For example, if A is an array with n1*n2*n3 elements X=dct(A,-1,n1,1) is equivalent to X=dct(matrix(A,[n1,n2,n3]),-1,1). and X=dct(A,-1,[n1 n3],[1 n1*n2]) is equivalent to X=dct(matrix(A,[n1,n2,n3]),-1,[1,3]).

Optimizing dct

Remark: function automatically stores his last parameters in memory to re-use it in a second time. This improves greatly the time computation when consecutives calls (with same parameters) are performed.

It is possible to go further in dct optimization using get_fftw_wisdom, set_fftw_wisdom functions.