# Documentation

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# dct2

2-D discrete cosine transform

## Syntax

```B = dct2(A) B = dct2(A,m,n) B = dct2(A,[m n]) ```

## Description

`B = dct2(A)` returns the two-dimensional discrete cosine transform of `A`. The matrix `B` is the same size as `A` and contains the discrete cosine transform coefficients B(k1,k2).

`B = dct2(A,m,n)` pads the matrix `A` with 0's to size `m`-by-`n` before transforming. If `m` or `n` is smaller than the corresponding dimension of `A`, `dct2` truncates `A`.

`B = dct2(A,[m n])` same as above.

## Class Support

`A` can be numeric or logical. The returned matrix `B` is of class `double`.

## Examples

collapse all

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.

```figure imshow(log(abs(J)),[]) colormap(gca,jet(64)) colorbar```

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.

```figure imshowpair(I,K,'montage') title('Original Grayscale Image (Left) and Processed Image (Right)');```

## Algorithms

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

where

and

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.

This transform can be inverted using `idct2`.

## References

[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.