dct

Discrete cosine transform

Syntax

y = dct(x)

y = dct(x,n)

y = dct(x,n,dim)

y = dct(___,Type=dcttype)

Description

y = dct(x) returns the unitary discrete cosine transform of input array x. The output y has the same size as x. If x has more than one dimension, dct operates along the first array dimension with size greater than 1.

y = dct(x,n) zero-pads or truncates the relevant dimension of x to length n before transforming.

y = dct(x,n,dim) computes the transform along dimension dim. To input a dimension and use the default value of n, specify the second argument as empty, [].

y = dct(___,Type=dcttype) specifies the type of discrete cosine transform to compute. See Discrete Cosine Transform for details. This option can be combined with any of the previous syntaxes.

Examples

Energy Stored in DCT Coefficients

Find how many DCT coefficients represent 99% of the energy in a sequence.

x = (1:100) + 50*cos((1:100)*2*pi/40).^3;
X = dct(x);
[XX,ind] = sort(abs(X),"descend");
i = 1;
while (norm(X(ind(1:i)))/norm(X))^2 < 0.99
   i = i + 1;
end
needed = i;

Reconstruct the signal and compare it to the original signal.

X(ind(needed+1:end)) = 0;
xx = idct(X);

plot([x;xx]')
legend("Original","Reconstructed, N = " + needed, ...
       Location='SouthEast')

Image Data Compression

Load a file that contains depth measurements of a mold used to mint a United States penny. The data, taken at the National Institute of Standards and Technology, are sampled on a 128-by-128 grid. Display the data.

load penny

surf(P)
view(2)
colormap copper
shading interp
axis ij square off

Compute the discrete cosine transform of the image data. Operate first along the rows and then along the columns.

Q = dct(P,[],1);
R = dct(Q,[],2);

Find what fraction of DCT coefficients contain 99.98% of the energy in the image.

X = R(:);

[~,ind] = sort(abs(X),"descend");
coeffs = 1;
while (norm(X(ind(1:coeffs)))/norm(X))^2 < 0.9998
   coeffs = coeffs + 1;
end
disp(coeffs + " of " + numel(R) + " coefficients are sufficient")

5215 of 16384 coefficients are sufficient

Reconstruct the image using only the necessary coefficients.

R(abs(R) < abs(X(ind(coeffs)))) = 0;

S = idct(R,[],2);
T = idct(S,[],1);

Display the reconstructed image.

surf(T)
view(2)
shading interp
axis ij square off

Image Resizing

Load a file that contains depth measurements of a mold used to mint a United States penny. The data, taken at the National Institute of Standards and Technology, are sampled on a 128-by-128 grid. Display the data.

load penny

surf(P)
colormap copper
shading interp
view(2)
axis ij square off

Compute the discrete cosine transform of the image data using the DCT-1 variant. Operate first along the rows and then along the columns.

Q = dct(P,[],1,Type=1);
R = dct(Q,[],2,Type=1);

Invert the transform. Truncate the inverse so that each dimension of the reconstructed image is one-half the length of the original.

S = idct(R,size(P,2)/2,2,Type=1);
T = idct(S,size(P,1)/2,1,Type=1);

Invert the transform again. Zero-pad the inverse so that each dimension of the reconstructed image is twice the length of the original.

U = idct(R,size(P,2)*2,2,Type=1);
V = idct(U,size(P,1)*2,1,Type=1);

Display the original and reconstructed images.

surf(V)
hold on
surf(P)
surf(T)
hold off

shading interp
view(2)
axis ij equal off

Input Arguments

x — Input array

vector | matrix | N-D array

Input array, specified as a real-valued or complex-valued vector, matrix, or N-D array.

Example: sin(2*pi*(0:255)/4) specifies a sinusoid as a row vector.

Example: sin(2*pi*[0.1;0.3]*(0:39))' specifies a two-channel sinusoid.

Data Types: single | double
Complex Number Support: Yes

n — Transform length

positive integer scalar

Transform length, specified as a positive integer scalar.

Data Types: single | double

dim — Dimension to operate along

positive integer scalar

Dimension to operate along, specified as a positive integer scalar.

Data Types: single | double

dcttype — Discrete cosine transform type

2 (default) | 1 | 3 | 4

Discrete cosine transform type, specified as a positive integer scalar from 1 to 4. See Discrete Cosine Transform for the definitions of the different types of DCT.

Data Types: single | double

Output Arguments

y — Discrete cosine transform

vector | matrix | N-D array

Discrete cosine transform, returned as a real-valued or complex-valued vector, matrix, or N-D array.

More About

Discrete Cosine Transform

The discrete cosine transform (DCT) is closely related to the discrete Fourier transform. You can often reconstruct a sequence very accurately from only a few DCT coefficients. This property is useful for applications requiring data reduction.

The DCT has four standard variants. For a signal x of length N, and with δ_kℓ the Kronecker delta, the transforms are defined by:

• DCT-1:

  $$y(k)=\sqrt{\frac{2}{N-1}}{\displaystyle \sum _{n=1}^{N}x(n)\frac{1}{\sqrt{1+{\delta }_{n1}+{\delta }_{nN}}}\frac{1}{\sqrt{1+{\delta }_{k1}+{\delta }_{kN}}}\cos \left(\frac{\pi }{N-1}(n-1)(k-1)\right)}$$

• DCT-2:

  $$y(k)=\sqrt{\frac{2}{N}}{\displaystyle \sum _{n=1}^{N}x(n)\frac{1}{\sqrt{1+{\delta }_{k1}}}\cos \left(\frac{\pi }{2N}(2n-1)(k-1)\right)}$$

• DCT-3:

  $$y(k)=\sqrt{\frac{2}{N}}{\displaystyle \sum _{n=1}^{N}x(n)\frac{1}{\sqrt{1+{\delta }_{n1}}}\cos \left(\frac{\pi }{2N}(n-1)(2k-1)\right)}$$

• DCT-4:

  $$y(k)=\sqrt{\frac{2}{N}}{\displaystyle \sum _{n=1}^{N}x(n)\cos \left(\frac{\pi }{4N}(2n-1)(2k-1)\right)}$$

The series are indexed from n = 1 and k = 1 instead of the usual n = 0 and k = 0, because MATLAB^® vectors run from 1 to N instead of from 0 to N – 1.

All variants of the DCT are unitary (or, equivalently, orthogonal): To find their inverses, switch k and n in each definition. DCT-1 and DCT-4 are their own inverses. DCT-2 and DCT-3 are inverses of each other.

References

[1] Jain, A. K. Fundamentals of Digital Image Processing. Englewood Cliffs, NJ: Prentice-Hall, 1989.

[2] Oppenheim, Alan V., Ronald W. Schafer, and John R. Buck. Discrete-Time Signal Processing. 2nd Ed. Upper Saddle River, NJ: Prentice Hall, 1999.

[3] Pennebaker, W. B., and J. L. Mitchell. JPEG Still Image Data Compression Standard. New York: Van Nostrand Reinhold, 1993.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• C and C++ code generation for dct requires DSP System Toolbox™ software.

• The length of the transform dimension must be a power of two. If specified, the pad or truncation value must be constant. Expressions or variables are allowed if their values do not change.

• Inputs must be double precision.

• Only DCT-2 is allowed.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The dct function supports GPU array input with these usage notes and limitations:

• N-D input arrays are not supported.

• The dim and dcttype input arguments are not supported.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced before R2006a