Image Compression Using DCT Implementing Matlab

IMAGE COMPRESSION USING DISCRETE COSINE TRANSFORM IMPLEMENTING MATLAB

SUBMITTED BY : HARISH KUMAR (07414) AMAN PUNDIR (07429) NIT HAMIRPUR, H.P.

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IMAGE COMPRESSION USING DISCRETE COSINE TRANFORM IMPLEMENTING MATLAB

ACKNOWLEDGEMENT Apart from the efforts of us, the success of this project depends largely on the encouragement encouragement and guidelines provided to us by our Digital Image Processing lecturer from time to time. We take this opportunity to express our gratitude to the person who has been instrumental in the successful completion of this project. We would like to show our greatest appreciation to

Mr. Amit Kaul (Faculty of Electrical and Electronics Engineering) Under the guidance of whom We feel motivated and encouraged every time we attend his lectures. Without his encouragement and guidance this project would not have materialized. We can’t say thank you enough for his tremendous support and help. We are grateful for his constant support and help. Thanking you for your kind anticipation in our project.

Dated:

Place:

Harish Kumar (07414) Aman Pundir (07429) NIT Hamirpur, H.P.

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IMAGE COMPRESSION USING DISCRETE COSINE TRANFORM IMPLEMENTING MATLAB

CONTENTS:

1) Introduction 2) The JPEG Process 3) The Discrete Cosine Transform a) 1D DCT b) 2D DCT 4) DCT Matrix 5) Quantization 6) Coding 7) Decompression 8) Properties of DCT 9) Conclusion 10) Matlab code 11) Output Images 12) References

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IMAGE COMPRESSION USING DISCRETE COSINE TRANFORM IMPLEMENTING MATLAB

INTRODUCTION: In today’s technological world as  our use of and reliance on computers continues to grow, so too does our need for efficient ways of storing large amounts of data and due to the bandwidth and storage limitations, images must be compressed before transmission and storage.For example, someone with a web page or online catalog that uses dozens or perhaps hundreds of images will more than likely need to use some form of image compression to store those images. This is because the amount of space required to hold unadulterated images can be prohibitively large in terms of cost. Fortunately, there are several methods of image compression available today. This fall into two general categories: lossless and lossy image compression.

However, the compression will reduce the image fidelity, especially when the images are compressed at lower bit rates. The reconstructed images suffer from blocking artifacts and the image quality will be severely degraded under the circumstance of high compression ratios. In order to have a good compression ratio without losing too much of information when the image is decompressed we use DCT. A discrete cosine transform (DCT) expresses a sequence of finitely many data points in terms of a sum of cosine functions oscillating at different frequencies . The JPEG process is a widely used form of lossy image compression that centers on the Discrete Cosine Transform . DCT and Fourier transforms convert images from time-domain to frequencydomain to decorrelate pixels. The DCT transformation is reversible .The DCT works by separating images into parts of differing frequencies. During a step called quantization, where part of compression actually occurs, the less important frequencies are discarded, hence the use of the term “lossy“. Then, only the most important frequencies that remain are used retrieve the image in the decompression process. As a result, reconstructed images contain some distortion; but as we shall soon see, these levels of distortion can be adjusted during the compression stage. The JPEG method is used for both color and blackand-white images.

THE JPEG PROCESS: The following is a general overview of the JPEG process. JPEG stands for Joint Photographic Experts Group which is a commonly used method of compression for photographic images. The degree of compression can be adjusted, allowing a selectable tradeoff between storage size and image quality. JPEG typically achieves 10:1 compression with little perceptible loss in image quality. More comprehensive understanding of the process may be acquired as such given under:

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IMAGE COMPRESSION USING DISCRETE COSINE TRANFORM IMPLEMENTING MATLAB

1. The image is broken into 8x8 blocks of pixels. 2. Working from left to right, top to bottom, the DCT is applied to each block. 3. Each block is compressed through quantization. 4. The array of compressed blocks that constitute the image is stored in a drastically reduced amount of space. 5. When desired, the image is reconstructed through decompression, a process that uses the Inverse Discrete Cosine Transform (IDCT).

THE DISCRETE COSINE TRANSFORM: Like other transforms, the Discrete Cosine Transform (DCT) attempts to decorrelate the image data. After decorrelation each transform coefficient can be encoded independently without losing compression efficiency. This section describes the DCT and some of its important properties.

1) The One-Dimensional DCT: The most common DCT definition of a 1-D sequence of length N is

For u = 0, 1, 2, …, N −1. Similarly, the inverse transformation is defined as

For x = 0, 1, 2, …, N −1. In both equations as above, α (u) is defined as

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IMAGE COMPRESSION USING DISCRETE COSINE TRANFORM IMPLEMENTING MATLAB

It is clear from first equation that for Thus, the first transform coefficient is the average value of the sample sequence. In literature, this value is referred to as the DC Coefficient . All other transform coefficients are called the AC Coefficients.

2) The Two-Dimensional DCT: The Discrete Cosine Transform (DCT) is one of many transforms that takes its input and transforms it into a linear combination of weighted basis functions. These basis functions are commonly the frequency. The 2-D Discrete Cosine Transform is  just a one dimensional DCT applied twice, once in the x direction, and again in the y direction. One can imagine the computational complexity of doing so for a large image. Thus, many algorithms, such as the Fast Fourier Transform (FFT), have been created to speed the computation. The DCT equation (Eq.1) computes the i, jth entry of the DCT of an image.

p (x, y) is the x,yth element of the image represented by the matrix p. N is the size of the block that the DCT is done on. The equation calculates one entry (i, j th) of the transformed image from the pixel values of the original image matrix. For the standard 8x8 block that JPEG compression uses, N equals 8 and x and y range from 0 to 7. Therefore D (i, j ) would be as in Equation (3).

Because the DCT uses cosine functions, the resulting matrix depends on the horizontal and vertical frequencies. Therefore an image black with a lot of change in frequency has a very random looking resulting matrix, while an image matrix of just one color, has a resulting matrix of a large value for the first element and zeroes for the other

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IMAGE COMPRESSION USING DISCRETE COSINE TRANFORM IMPLEMENTING MATLAB

elements.

THE DCT MATRIX: To get the matrix form of Equation (1), we will use the following equation,

For an 8x8 block it results in this matrix:

The first row (i : 1) of the matrix has all the entries equal to 1/ 8 as expected from Equation (4).The columns of T form an orthonormal set, so T is an orthogonal matrix. When doing the inverse DCT the orthogonality of T is important, as the inverse of T is T’ which is easy to calculate.

DCT ON AN 8x8 BLOCK: Before we begin, it should be noted that the pixel values of a black-and-white image range from 0 to 255 in steps of 1, where pure black is represented by 0, and pure white by 255. Thus it can be seen how a photo, illustration, etc. can be accurately represented by these 256 shades of gray. Since an image comprises hundreds or even thousands of 8x8 blocks of pixels, the following description of what happens to one 8x8 block is a microcosm of the JPEG process; what is done to one block of image pixels is done to all of them, in the order earlier specified. Now, let‘s start with a block of image -

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IMAGE COMPRESSION USING DISCRETE COSINE TRANFORM IMPLEMENTING MATLAB

pixel values. This particular block was chosen from the very upper- left-hand corner of an image.

Because the DCT is designed to work on pixel values ranging from -128 to 127, the original block is “leveled off“  by subtracting 128 from each entry. This results in the following matrix.

We are now ready to perform the Discrete Cosine Transform, which is accomplished by matrix multiplication.  D = TMT’

-----(5)

In Equation (5) matrix M is first multiplied on the left by the DCT matrix T from the previous section; this transforms the rows. The columns are then transformed by multiplying on the right by the transpose of the DCT matrix. This yields the following matrix.

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IMAGE COMPRESSION USING DISCRETE COSINE TRANFORM IMPLEMENTING MATLAB

This block matrix now consists of 64 DCT coefficients, c (i, j), where i and j range from 0 to 7. The top-left coefficient, c (0, 0), correlates to the low frequencies of the original image block. As we move away from c(0,0) in all directions, the DCT coefficients correlate to higher and higher frequencies of the image block, where c(7, 7) corresponds to highest frequency. Higher frequencies are mainly represented as lower number and Lower frequencies as higher number. It is important to know that human eye is most sensitive to lower frequencies.

QUANTIZATION: Our 8x8 block of DCT coefficients is now ready for compression by quantization. A remarkable and highly useful feature of the JPEG process is that in this step, varying levels of image compression and quality are obtainable through selection of specific quantization matrices. This enables the user to decide on quality levels ranging from 1 to 100, where 1 gives the poorest image quality and highest compression, while 100 gives the best quality and lowest compression. As a result, the quality/compression ratio can be tailored to suit different needs. Subjective experiments involving the human visual system have resulted in the JPEG standard quantization matrix. With a quality level of 50, this matrix renders both high compression and excellent decompressed image quality.

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IMAGE COMPRESSION USING DISCRETE COSINE TRANFORM IMPLEMENTING MATLAB

If, however, another level of quality and compression is desired, scalar multiples of the JPEG standard quantization matrix may be used. For a quality level greater than 50 (less compression, higher image quality), the standard quantization matrix is multiplied by (100-quality level)/50. For a quality level less than 50 (more compression, lower image quality), the standard quantization matrix is multiplied by 50/quality level. The scaled quantization matrix is then rounded and clipped to have positive integer values ranging from 1 to 255. For example, the following quantization matrices yield quality levels of 10 and 90.

Quantization is achieved by dividing each element in the transformed image matrix D by

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IMAGE COMPRESSION USING DISCRETE COSINE TRANFORM IMPLEMENTING MATLAB

corresponding element in the quantization matrix, and then rounding to the nearest integer value. For the following step, quantization matrix Q50 is used.

Recall that the coefficients situated near the upper-left corner correspond to the lower frequencies œ to which the human eye is most sensitive œ of the image block. In addition, the zeros represent the less important, higher frequencies that have been discarded, giving rise to the lossy part of compression. As mentioned earlier, only the remaining nonzero coefficients will be used to reconstruct the image. It is also interesting to note the effect of different quantization matrices; use of Q10 would give C significantly more zeros, while Q90 would result in very few zeros.

CODING: The quantized matrix C is now ready for the final step of compression. Before storage, all coefficients of C are converted by an encoder to a stream of binary data (01101011...). In-depth coverage of the coding process is beyond the scope of this article. However, we can point out one key aspect that the reader is sure to appreciate. After quantization, it is quite common for most of the coefficients to equal zero. JPEG takes advantage of this by encoding quantized coefficients in the zig-zag sequence shown in Figure as under. The advantage lies in the consolidation of relatively large runs of zeros, which compress very well. The sequence in Figure 1(4x4) continues for the entire 8x8 block.

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IMAGE COMPRESSION USING DISCRETE COSINE TRANFORM IMPLEMENTING MATLAB

DECOMPRESSION: Reconstruction of our image begins by decoding the bit stream representing the Quantized matrix C. Each element of C is then multiplied by the corresponding element of the quantization matrix originally used R i, j = Q i, j × C i, j

The IDCT is next applied to matrix R, which is rounded to the nearest integer. Finally, 128 is added to each element of that result, giving us the decompressed JPEG version N of our original 8x8 image block M.

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IMAGE COMPRESSION USING DISCRETE COSINE TRANFORM IMPLEMENTING MATLAB

N = round (T’RT) + 128

PROPERTIES OF DCT: Some properties of the DCT which are of particular value to image processing applications:

a) Decorrelation: The principle advantage of image transformation is the removal of redundancy between neighboring pixels. This leads to uncorrelated transform coefficients which can be encoded independently. It can be inferred that DCT exhibits excellent decorrelation properties. Efficacy of a transformation scheme can be directly gauged by its ability to pack input data into as few coefficients as possible. This allows the quantizer to discard coefficients with relatively small amplitudes without introducing visual distortion in the reconstructed image. DCT exhibits excellent energy compaction for highly correlated images. b ) Energy Compaction:

c) Separability: The DCT transform equation can be expressed as

This property, known as separability, has the principle advantage that D (i, j) can be computed in two steps by successive 1-D operations on rows and columns of an image. The arguments presented can be identically applied for the inverse DCT computation.

d) Symmetry :  Another look at the row and column operations in above Equation reveals that these operations are functionally identical. Such a transformation is called a symmetric transformation. A separable and symmetric transform can be expressed in the form  D = TMT’ 

where M is an N ×N symmetric transformation matrix This is an extremely useful property since it implies that the transformation matrix can be precomputed offline and then applied to the image thereby providing orders

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IMAGE COMPRESSION USING DISCRETE COSINE TRANFORM IMPLEMENTING MATLAB

of magnitude improvement in computation efficiency.

COMPARISON OF MATRICES: Let us now see how the JPEG version of our original pixel block compares,

CONCLUSION: If we look at the above two matrices, this is a remarkable result, considering that nearly 70% of the DCT coefficients were discarded prior to image block decompression/reconstruction. Given that similar results will occur with the rest of the

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IMAGE COMPRESSION USING DISCRETE COSINE TRANFORM IMPLEMENTING MATLAB

blocks that constitute the entire image, it should be no surprise that the JPEG image will be scarcely distinguishable from the original. Remember, there are 256 possible shades of gray in a black-and-white picture, and a difference of, say, 10, is barely noticeable to the human eye. DCT takes advantage of redundancies in the data by grouping pixels with similar frequencies together. And moreover if we observe as the resolution of the image is very high, even after sufficient compression and decompression there is very less change in the original and decompressed image. Thus, we can also conclude that at the same compression ratio the difference between original and decompressed image goes on decreasing as there is increase in image resolution.

MATLAB CODE: >>o = imread ('e:\img10.jpg'); w = size (o, 2); samplesHalf = floor(w / 2); samplesQuarter = floor(w / 4); samplesEighth = floor(w / 8); ci2 = []; ci4 = []; ci8 = []; for k=1:3 % all color layers: RGB for i=1:size(o, 1) % all rows rowDCT = dct(double(o(i,:,k))); ci2(i,:,k) = idct(rowDCT(1:samplesHalf), w); ci4(i,:,k) = idct(rowDCT(1:samplesQuarter), w); ci8(i,:,k) = idct(rowDCT(1:samplesEighth), w); end end h = size(o, 1); samplesHalf = floor(h / 2); samplesQuarter = floor(h / 4); samplesEighth = floor(h / 8); ci2f = []; ci4f = []; ci8f = []; for k=1:3 % all color layers: RGB for i=1:size(o, 2) % all columns

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IMAGE COMPRESSION USING DISCRETE COSINE TRANFORM IMPLEMENTING MATLAB

columnDCT2=dct(double(ci2(:,i,k))); columnDCT4=dct(double(ci4(:,i,k))); columnDCT8=dct(double(ci8(:,i,k))); ci2f(:,i,k) = idct(columnDCT2(1:samplesHalf), h); ci4f(:,i,k) = idct(columnDCT4(1:samplesQuarter), h); ci8f(:,i,k) = idct(columnDCT8(1:samplesEighth), h); end end subplot(2,2,1), image(uint8(o)), title('Original Image'); subplot(2,2,2), image(uint8(ci2)), title('Compression Factor 2'); subplot(2,2,3), image(uint8(ci4)), title('Compression Factor 4'); subplot(2,2,4), image(uint8(ci8)), title('Compression Factor 8'); figure subplot(2,2,1), image(uint8(o)), title('Original Image'); subplot(2,2,2), image(uint8(ci2f)), title('Compression Factor 2 * 2'); subplot(2,2,3), image(uint8(ci4f)), title('Compression Factor 4 * 4'); subplot(2,2,4), image(uint8(ci8f)), title('Compression Factor 8 * 8');

THE OUTPUT IMAGES: 1) Image of size 352 × 352:

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IMAGE COMPRESSION USING DISCRETE COSINE TRANFORM IMPLEMENTING MATLAB

a) DCT to rows only:

Original Image

Compression Factor 2

100

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Compression Factor 4

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Compression Factor 8

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IMAGE COMPRESSION USING DISCRETE COSINE TRANFORM IMPLEMENTING MATLAB

b) DCT to rows*column:

Original Image

Compression Factor 2 * 2

100

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Compression Factor 4 * 4

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Compression Factor 8 * 8

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IMAGE COMPRESSION USING DISCRETE COSINE TRANFORM IMPLEMENTING MATLAB

2) Image of size 375 × 276 ×3(Low Resolution Image) a) DCT to rows only:

Original Image

Compression Factor 2

50

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Compression Factor 4 50

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Compression Factor 8

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IMAGE COMPRESSION USING DISCRETE COSINE TRANFORM IMPLEMENTING MATLAB

b) DCT to rows*column:

Original Image

Compression Factor 2 * 2

50

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250 100

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Compression Factor 4 * 4 50

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Compression Factor 8 * 8

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IMAGE COMPRESSION USING DISCRETE COSINE TRANFORM IMPLEMENTING MATLAB

3) Image of size 1920×1200×3(High Resolution Image): a) DCT to rows only:

Original Image

Compression Factor 2

200

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Compression Factor 4 200

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Compression Factor 8

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IMAGE COMPRESSION USING DISCRETE COSINE TRANFORM IMPLEMENTING MATLAB

b) DCT to rows*column:

Original Image

Compression Factor 2 * 2

200

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Compression Factor 4 * 4 200

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Compression Factor 8 * 8

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IMAGE COMPRESSION USING DISCRETE COSINE TRANFORM IMPLEMENTING MATLAB

REFERENCES: 1) Digital Image Processing by “Gonzalez and Woods”. 2) IEEE Journals and Research papers. 3) Wikipedia.org. 4) Mathworks Matlab Central. 5) fip-Contents.html

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