# Discrete Cosine Transform (Algorithm and Program)

**Image Compression :** Image is stored or transmitted with having pixel value. It can be compressed by reducing the value its every pixel contains. Image compression is basically of two types :

**1.** Lossless compression : In this type of compression, after recovering image is exactly become same as that was before applying compression techniques and so, its quality didn’t gets reduced.

**2.** Lossy compression : In this type of compression, after recovering we can’t get exactly as older data and that’s why the quality of image gets significantly reduced. But this type of compression results in very high compression of image data and is very useful in transmitting image over network.

**Discrete Cosine Transform** is used in lossy image compression because it has very strong energy compaction, i.e., its large amount of information is stored in very low frequency component of a signal and rest other frequency having very small data which can be stored by using very less number of bits (usually, at most 2 or 3 bit).

To perform DCT Transformation on an image, first we have to fetch image file information (pixel value in term of integer having range 0 – 255) which we divides in block of 8 X 8 matrix and then we apply discrete cosine transform on that block of data.

After applying discrete cosine transform, we will see that its more than 90% data will be in lower frequency component. For simplicity, we took a matrix of size 8 X 8 having all value as 255 (considering image to be completely white) and we are going to perform 2-D discrete cosine transform on that to observe the output.

**Algorithm :** Let we are having a 2-D variable named matrix of dimension 8 X 8 which contains image information and a 2-D variable named dct of same dimension which contain the information after applying discrete cosine transform. So, we have the formula

dct[i][j] = ci * cj (sum(k=0 to m-1) sum(l=0 to n-1) matrix[k][l] * cos((2*k+1) *i*pi/2*m) * cos((2*l+1) *j*pi/2*n)

where ci= 1/sqrt(m) if i=0 else ci= sqrt(2)/sqrt(m) and

similarly, cj= 1/sqrt(n) if j=0 else cj= sqrt(2)/sqrt(n)

and we have to apply this formula to all the value, i.e., from i=0 to m-1 and j=0 to n-1

Here, **sum(k=0 to m-1)** denotes summation of values from k=0 to k=m-1.

In this code, both m and n is equal to **8** and **pi** is defined as 3.142857.

## C++

`// CPP program to perform discrete cosine transform ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` `#define pi 3.142857 ` `const` `int` `m = 8, n = 8; ` ` ` `// Function to find discrete cosine transform and print it ` `int` `dctTransform(` `int` `matrix[][n]) ` `{ ` ` ` `int` `i, j, k, l; ` ` ` ` ` `// dct will store the discrete cosine transform ` ` ` `float` `dct[m][n]; ` ` ` ` ` `float` `ci, cj, dct1, sum; ` ` ` ` ` `for` `(i = 0; i < m; i++) { ` ` ` `for` `(j = 0; j < n; j++) { ` ` ` ` ` `// ci and cj depends on frequency as well as ` ` ` `// number of row and columns of specified matrix ` ` ` `if` `(i == 0) ` ` ` `ci = 1 / ` `sqrt` `(m); ` ` ` `else` ` ` `ci = ` `sqrt` `(2) / ` `sqrt` `(m); ` ` ` `if` `(j == 0) ` ` ` `cj = 1 / ` `sqrt` `(n); ` ` ` `else` ` ` `cj = ` `sqrt` `(2) / ` `sqrt` `(n); ` ` ` ` ` `// sum will temporarily store the sum of ` ` ` `// cosine signals ` ` ` `sum = 0; ` ` ` `for` `(k = 0; k < m; k++) { ` ` ` `for` `(l = 0; l < n; l++) { ` ` ` `dct1 = matrix[k][l] * ` ` ` `cos` `((2 * k + 1) * i * pi / (2 * m)) * ` ` ` `cos` `((2 * l + 1) * j * pi / (2 * n)); ` ` ` `sum = sum + dct1; ` ` ` `} ` ` ` `} ` ` ` `dct[i][j] = ci * cj * sum; ` ` ` `} ` ` ` `} ` ` ` ` ` `for` `(i = 0; i < m; i++) { ` ` ` `for` `(j = 0; j < n; j++) { ` ` ` `printf` `(` `"%f\t"` `, dct[i][j]); ` ` ` `} ` ` ` `printf` `(` `"\n"` `); ` ` ` `} ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `int` `matrix[m][n] = { { 255, 255, 255, 255, 255, 255, 255, 255 }, ` ` ` `{ 255, 255, 255, 255, 255, 255, 255, 255 }, ` ` ` `{ 255, 255, 255, 255, 255, 255, 255, 255 }, ` ` ` `{ 255, 255, 255, 255, 255, 255, 255, 255 }, ` ` ` `{ 255, 255, 255, 255, 255, 255, 255, 255 }, ` ` ` `{ 255, 255, 255, 255, 255, 255, 255, 255 }, ` ` ` `{ 255, 255, 255, 255, 255, 255, 255, 255 }, ` ` ` `{ 255, 255, 255, 255, 255, 255, 255, 255 } }; ` ` ` `dctTransform(matrix); ` ` ` `return` `0; ` `} ` |

*chevron_right*

*filter_none*

## Java

`// Java program to perform discrete cosine transform ` ` ` `import` `java.util.*; ` ` ` `class` `GFG ` `{ ` ` ` `public` `static` `int` `n = ` `8` `,m = ` `8` `; ` ` ` `public` `static` `double` `pi = ` `3.142857` `; ` ` ` ` ` `// Function to find discrete cosine transform and print it ` ` ` `static` `strictfp` `void` `dctTransform(` `int` `matrix[][]) ` ` ` `{ ` ` ` `int` `i, j, k, l; ` ` ` ` ` `// dct will store the discrete cosine transform ` ` ` `double` `[][] dct = ` `new` `double` `[m][n]; ` ` ` ` ` `double` `ci, cj, dct1, sum; ` ` ` ` ` `for` `(i = ` `0` `; i < m; i++) ` ` ` `{ ` ` ` `for` `(j = ` `0` `; j < n; j++) ` ` ` `{ ` ` ` `// ci and cj depends on frequency as well as ` ` ` `// number of row and columns of specified matrix ` ` ` `if` `(i == ` `0` `) ` ` ` `ci = ` `1` `/ Math.sqrt(m); ` ` ` `else` ` ` `ci = Math.sqrt(` `2` `) / Math.sqrt(m); ` ` ` ` ` `if` `(j == ` `0` `) ` ` ` `cj = ` `1` `/ Math.sqrt(n); ` ` ` `else` ` ` `cj = Math.sqrt(` `2` `) / Math.sqrt(n); ` ` ` ` ` `// sum will temporarily store the sum of ` ` ` `// cosine signals ` ` ` `sum = ` `0` `; ` ` ` `for` `(k = ` `0` `; k < m; k++) ` ` ` `{ ` ` ` `for` `(l = ` `0` `; l < n; l++) ` ` ` `{ ` ` ` `dct1 = matrix[k][l] * ` ` ` `Math.cos((` `2` `* k + ` `1` `) * i * pi / (` `2` `* m)) * ` ` ` `Math.cos((` `2` `* l + ` `1` `) * j * pi / (` `2` `* n)); ` ` ` `sum = sum + dct1; ` ` ` `} ` ` ` `} ` ` ` `dct[i][j] = ci * cj * sum; ` ` ` `} ` ` ` `} ` ` ` ` ` `for` `(i = ` `0` `; i < m; i++) ` ` ` `{ ` ` ` `for` `(j = ` `0` `; j < n; j++) ` ` ` `System.out.printf(` `"%f\t"` `, dct[i][j]); ` ` ` `System.out.println(); ` ` ` `} ` ` ` `} ` ` ` ` ` `// driver program ` ` ` `public` `static` `void` `main (String[] args) ` ` ` `{ ` ` ` `int` `matrix[][] = { { ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `}, ` ` ` `{ ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `}, ` ` ` `{ ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `}, ` ` ` `{ ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `}, ` ` ` `{ ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `}, ` ` ` `{ ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `}, ` ` ` `{ ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `}, ` ` ` `{ ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `, ` `255` `} }; ` ` ` `dctTransform(matrix); ` ` ` `} ` `} ` ` ` `// Contributed by Pramod Kumar ` |

*chevron_right*

*filter_none*

Output:

2039.999878 -1.168211 1.190998 -1.230618 1.289227 -1.370580 1.480267 -1.626942 -1.167731 0.000664 -0.000694 0.000698 -0.000748 0.000774 -0.000837 0.000920 1.191004 -0.000694 0.000710 -0.000710 0.000751 -0.000801 0.000864 -0.000950 -1.230645 0.000687 -0.000721 0.000744 -0.000771 0.000837 -0.000891 0.000975 1.289146 -0.000751 0.000740 -0.000767 0.000824 -0.000864 0.000946 -0.001026 -1.370624 0.000744 -0.000820 0.000834 -0.000858 0.000898 -0.000998 0.001093 1.480278 -0.000856 0.000870 -0.000895 0.000944 -0.001000 0.001080 -0.001177 -1.626932 0.000933 -0.000940 0.000975 -0.001024 0.001089 -0.001175 0.001298

This article is contributed by Aditya Kumar. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

## Recommended Posts:

- Discrete logarithm (Find an integer k such that a^k is congruent modulo b)
- Booth’s Multiplication Algorithm
- Doolittle Algorithm : LU Decomposition
- Stein's Algorithm for finding GCD
- Saddleback Search Algorithm in a 2D array
- Midpoint ellipse drawing algorithm
- Computer Organization | Booth's Algorithm
- Banker's Algorithm in Operating System
- Greedy Algorithm for Egyptian Fraction
- Euclid's Algorithm when % and / operations are costly
- Pollard's Rho Algorithm for Prime Factorization
- New Algorithm to Generate Prime Numbers from 1 to Nth Number
- Convex Hull | Set 1 (Jarvis's Algorithm or Wrapping)
- Find HCF of two numbers without using recursion or Euclidean algorithm
- Hungarian Algorithm for Assignment Problem | Set 1 (Introduction)