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Application of DCT in image processing


Call for Papers Engineering Journal, May 2019

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Application of DCT in image processing

Application of DCT in image processing

Nitesh Agarwal

Department of Computer Science Jodhpur Institute of Engineering & Technology

Jodhpur, India niteshagarwal.234@rediffmail.com

Dr. A.M. Khan

Department Of Applied Sciences Jodhpur Institute of Engineering & Technology

Jodhpur, India arif.khan@jietjodhpur.com

AbstractDiscrete Cosine Transform (DCT) is an important technique or method to convert a signal into elementary frequency component. It is widely used in image compression techniques like in JPEG compression. It converts each pixel value of an image into its corresponding frequency value. The present paper deals with the study of transformation of an 8 bit (b/w) image into its frequency domain through DCT technique.

  1. DCT convert an image into its equivalent frequency domain by partitioning image pixel matrix into blocks of size N*N, N depends upon the type of image. For example if we used a black & white image of 8 bit then all shading of black & white color can be expressed into 8 bit hence we use N=8, similarly for color image of 24 bit we can use N=24 but using

    It is clear from (1) for u=0,

    N 1

    Cu 0 1 f x 4

    N x0

    i.e. 1st transformation coefficient is the average value of sample sequence, this coefficient known as DC coefficient & all other coefficient known as AC coefficient.

    1.2 Two Dimensional DCT

    An image is 2-D pixel matrix where each position (i,j) represents a color value for that particular point or position. Hence to transform an image into its equivalent DCT matrix we use 2-D DCT.

    2-D DCT can be defined as

    block size N=24 time complexity may increase hence we

    N1 N1

    2x1u

    2y1v

    operate DCT on individual color component for a color image.

    Cu,v u (v) f x,y cos

    cos 5

    Color image consist of 8 bit red + 8 bit green + 8 bit blue hence we apply DCT on each color component (Red, Green, Blue) using block size N=8.

    1.1 One-Dimensional DCT

    x0 y0

    for u, v = 0,1,2,,N 1.

    2N 2N

    If we have one-D sequence of signal value of length N then its equivalent DCT can be expressed as

    & inverse transformation is defined as

    N1 N1

    2x1u

    2y1v

    2x 1u

    f x,yu(v)cu,vcos

    cos

    6

    N 1

    C u u f

    x0

    x cos 2N 1

    u0 v0

    2N

    2N

    for u = 0,1,2,,N 1.

    & inverse transformation is defined as

    Where Cu, v represents frequency value for u, v &

    f x, y represents pixel color value at position ( x, y ).

    N 1

    2x 1u

    f x

    x0

    u c u cos 2N 2

    1

    N

    for

    u 0

    Where f x is signal value at point x &

    u is

    (u) 2

    7

    transform coefficient for value u.

    N

    for u 0

    1

    for

    u 0 1

    (u) N

    3

    for

    v 0

    2 (v) N

    8

    for u 0 2

    N

    N

    for v 0

    1. Implementation of DCT

      This paper describe how a b/w image is convert into equivalent frequency domain using DCT.

      Steps involved in this implementation

      1. Create pixel matrix of the image & divided it into blocks of size 8*8

      2. Apply FDCT (Forward Discrete Cosine Transform) on each 8*8 block of pixel matrix to get equivalent 8*8 DCT blocks.

      3. To get Original image we apply IDCT (Inverse Discrete Cosine Transform) on each 8*8 block DCT & get its equivalent 8*8 IDCT block.

      4. Using 8*8 IDCT blocks we create original pixel matrix to get original image.

      1. Algorithm 1

        Get_8*8_blocks (image)

        {

        n=8, k=0;

        width=width of image; height=height of image; for ( i=0;i < width/n; i++)

        {

        for ( j=0; j < height/n; j++)

        {

        xpos = i * n; ypos = j * n;

        for ( a=0; a < n; a++)

        {

        for ( b=0; b < n; b++)

        { color = color at position(xpos+a, ypos+b); block[k][a][b]=color-128;

        } //end of for loop b k=k+1;

        } // end of for loop a

      2. Algorithm 2

        FDCT (block [] [] [])

        { width=width of image, N=8; height=height of image; q=(width/8)*(height/8)

        for ( i=0;i < q; i++)

        { for ( u=0; u< N; u++)

        { for ( v=0; v < N; v++)

        { if (u==0) {

        (u) = 1

        N

        }

        else{

        (u) = 2

        N

        }

        if (v==0){

        (v) = 1

        N

        }

        else {

        (v) = 2

        N

        }

        sum=0;

        for( x=0;x<N;x++)

        { for( y=0;y<N;y++)

        { sum= sum + block[i][x][y] *

        cos 2x 1u * cos 2 y 1v ;

        2N

        2 N

        } // end of for loop j

        } // end of for loop i

        }// end of Get_8*8_blocks

        } // end of for loop y

        } // end of for loop x

        dct[i][j][k]=( (u) * (v) *sum);

        } // end of for loop v

        } // end of for loop u

        } // end of for loop i

        }//end of FDCT

      3. Algorithm 3

        IDCT(dct [] [] [])

        { width=width of image, N=8; height=height of image; q=(width/8)*(height/8)

        for ( i=0;i < q; i++)

        { for ( x=0; x< N; x++)

        { for ( y=0; y< N; y++)

        {

        sum=0;

        for( u=0;u<N;u++)

        { for( v=0;b<N;v++)

        { if (u==0) {

        (u) = 1

        N

        }

        else{

        (u) = 2

        N

        }

        if (v==0){

        (v) = 1

        N

        }

        else {

        2

        } // end of IDCT

      4. Algorithm 4

        Get_Image(pixmat [] [] [])

        { k=0;

        width=width of image; height=height of image;

        for ( i=0; i < width; i++) { for ( j=0; j < height; j++) {

        xpos = i * n; ypos = j * n;

        for (a=0; a < n; a++)

        { for (b=0; b < n; b++)

        {

        color=(int)pixmat[k][a][b];

        set color at position (xpos+a, ypos+b);

        }// end of loop b

        } // end of loop a k++;

        } // end of loop j

        } // end of loop i

        }// end of Get_Image

    1. Outputs

8*8 block 1

8*8 block 2

8*8 block 3

8*8 block 4

  1. Convert pixel matrix into blocks of size 8*8

    =

    (v) =

    N

    }

    sum= sum + (u) * (v) *dct[i][u][v] *

    Input Image of size 16*16

    Output blocks of size 8*8

    cos 2x 1u * cos 2 y 1v ;

  2. Transform Input image into equivalent DCT image

    2N

    2 N

    } // end of for loop v

    } // end of for loop u idct[i][j][k]=sum;

    } // end of for loop y

    } // end of for loop x

    } // end of for loop i

    Input Image of size 16*16

    FDCT

    Output DCT Image of size 16*16

  3. Get original image from DCT image

    IDCT

    DCT rather than 8*8 DCT the time complexity of DCT is increases in a very large amount.

    For example

    For an image of size 48*48

    1. If 8*8 DCT used

Total no of blocks q=(48/8)*(48/8)=36 For FDCT

for ( i=0;i < q; i++) // loop runs 36 times

Input DCT Image of size 16*16

Output Image of size 16*16

{ for ( u=0; u< 8; u++) // loop runs 36* times

{ for ( v=0; v < 8; v++)// loop runs 36*8*8 times

{

3. Modification in original DCT

    1. Using sin operator rather than cos

      There is a difference of /2 between sin & cos operator hence using sin rather than cos operator in DCT may loss some pixel data

      }

      for( x=0;x<8;x++)// loop runs 36*8*8*8 times

      { for( y=0;y<8;y++) // loop runs 36*8*8*8*8 times

      Input Image of size 16*16

      Input Image of size 16*16

    2. Change in block size

FDCT

Using sin operator

IDCT

Using sin operator

Output DCT Image of size 16*16

Output Image of size 16*16

{

} // end of for loop y

} // end of for loop x

} // end of for loop v

} // end of for loop u

} // end of for loop i

Total no. of iteration = 36*8*8*8*8= 147456

2. If 24*24 DCT used

Total no of blocks q=(48/24)*(48/24)=4 For FDCT

for ( i=0;i < q; i++) // loop runs 4 times

{ for ( u=0; u< 24; u++) // loop runs 4*24 times

{ for ( v=0; v < 24; v++)// loop runs 4*24*24 times

{

}

for( x=0;x<24;x++)// loop runs 4*24*24*24 times

All shading of black & white image can be expressed in 8 bit of blocks hence we use block size 8*8 to perform DCT on it. But in color image each color value of a pixel can be expressed into 24 bit of block which contain 8 bit red + 8 bit green + 8 bit blue. To transform a color image into its equivalent DCT format we extract each 8 bit color component from 24 bit of block & then perform 8*8 DCT on each color component rather than using 24*24 DCT for 24 bit block. The main reason is that if use 24*24

{

for( y=0;y<24;y++) // loop runs 4*24*24*24*24 times

{

} // end of for loop y

} // end of for loop x

} // end of for loop v

} // end of for loop u

} // end of for loop i

Total no. of iteration =4*24*24*24*24= 1327104

Hence 24 * 24 DCT required 1327104-147456=1179648 extra iteration to preform DCT which increases time complexity in large amount hence DCT used with block size 8*8.

4. Conclusion

The result presented in this document shows that

  1. It is very easy to implement DCT rather than other transformation on image.

  2. If DCT used with sin operator rather than cos some pixel data may lose. But if we use DCT with sin operator as

    N1 N1

    2x1u

    2y1v

    Cu,vu(v) f x, ysin 2 2N

    sin 2 2N

    x0 y0

    & its inverse as

    N1 N 1

    2x1u

    2y1v

    f x, y u(v)cu, vsin 2 2N

    sin 2 2N

    u0 v0

    then there is no loss of pixel data because it is equivalent to DCT with cos operator.

  3. If DCT used with block size 24*24 rather than block size 8*8 then time complexity of DCT is increases in very large amount.

  1. N.Ahmed, T.Natatarajan, and K.R. Rao, Discrete Cosine Transform, IEEE Transactions on Computers, vol. C-32, pp. 90-93, Jan. 1974.

  2. Maneesha Gupta and Dr.Amit Kumar Garg, Analysis Of Image Compression Algorithm Using DCT International Journal of Engineering Research and Applications (IJERA), vol.2, pp. 515-521,

    Jan-Feb 2012

  3. Andrew B. Watson, Image Compression Using Discrete Cosine Transform, NASA Ames Research Centre, 4(1), pp. 81-88,1994.

  4. Anjali Kapoor and Dr. Renu Dhir, Image Compression Using Fast 2-D DCT Technique, International Journal on Computer Science and Engineering (IJCSE), vol. 3 pp. 2415-2419, 6 June 2011.

  5. Harley R. Myler and Arthur R. Weeks The Pocket Handbook of Image Processing Algorithms in C, ISBN 0-13-642240-3 Prentice Hall P T R Englewood Cliffs , New Jercy 07632.

  6. Iain E.G. Richardson H.264 and MPEG-4 Video Compression: Video Coding for Next-generation Multimedia, ISBN 0470848375, 9780470848371, Wiley,2003.

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