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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]) zero-pads or crops the matrix A to size m-by-n before applying the transformation.


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

RGB = imread('autumn.tif');
I = im2gray(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.

colormap parula

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. Rescale the values to the range [0, 1] expected of images of data type double.

K = idct2(J);
K = rescale(K);

Display the original grayscale image alongside the processed image. The processed image has fewer high frequency details, such as in the texture of the trees.

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.

Extended Capabilities

Version History

Introduced before R2006a

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See Also

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