2-D discrete cosine transform



B = dct2(A) returns the two-dimensional discrete cosine transform of A. The matrix B contains the discrete cosine transform coefficients B(k1,k2).

B = dct2(A,m,n) and

B = dct2(A,[m n]) pad the matrix A with 0s to size m-by-n before applying the transformation. If m or n is smaller than the corresponding dimension of A, then dct2 crops A before the transformation.


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This example shows how to remove high frequencies from an image using the two-dimensional discrete cosine transfer (DCT).

Read an image into the workspace, then convert the image to grayscale.

RGB = imread('autumn.tif');
I = rgb2gray(RGB);

Perform a 2-D DCT of the grayscale image using the dct2 function.

J = dct2(I);

Display the transformed image using a logarithmic scale. Notice that most of the energy is in the upper left corner.


Set values less than magnitude 10 in the DCT matrix to zero.

J(abs(J) < 10) = 0;

Reconstruct the image using the inverse DCT function idct2.

K = idct2(J);

Display the original grayscale image alongside the processed image.

title('Original Grayscale Image (Left) and Processed Image (Right)');

Input Arguments

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Input matrix, specified as a 2-D numeric matrix.

Number of image rows, specified as a positive integer. dct2 pads image A with 0s or truncates image A so that it has m rows. By default, m is equal to size(A,1).

Number of image columns, specified as a positive integer. dct2 pads image A with 0s or truncates image A so that it has n columns. By default, n is equal to size(A,2)

Output Arguments

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Transformed matrix using a two-dimensional discrete cosine transform, returned as an m-by-n numeric matrix.

Data Types: double

More About

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Discrete Cosine Transform

The discrete cosine transform (DCT) is closely related to the discrete Fourier transform. It is a separable linear transformation; that is, the two-dimensional transform is equivalent to a one-dimensional DCT performed along a single dimension followed by a one-dimensional DCT in the other dimension. The definition of the two-dimensional DCT for an input image A and output image B is

Bpq=αpαqm=0M1n=0N1Amncosπ(2m+1)p2Mcosπ(2n+1)q2N, 0pM10qN1


αp={1M, p=0           2M, 1pM-1


αq={1N, q=0          2N, 1qN-1

M and N are the row and column size of A, respectively.


  • If you apply the DCT to real data, the result is also real. The DCT tends to concentrate information, making it useful for image compression applications.

  • To invert the DCT transformation, use idct2.


[1] Jain, Anil K., Fundamentals of Digital Image Processing, Englewood Cliffs, NJ, Prentice Hall, 1989, pp. 150–153.

[2] Pennebaker, William B., and Joan L. Mitchell, JPEG: Still Image Data Compression Standard, Van Nostrand Reinhold, 1993.

See Also

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Introduced before R2006a