Lossy Image Compression using SVD Coding Algorithm

- K M AISHWARYA- RACHANA RAMESH- PREETI M SOBARAD

PAPER ID: 149

IMAGE COMPRESSION An image – m x n matrix

Where m = number of rows = pixel height n = number of columns = pixel width

Grayscale image: Image matrix values range = 0(black) to 1(white)

Colour images: 1 pixel corresponds to 3 components: R, G, B Image Compression is an application of Data Compression

that encodes the original image with few bits. Objective: to reduce the redundancy of the image and to store

or transmit data in an efficient form.PAPER ID: 149

Why is compression needed?

To store data efficiently To transmit data efficiently To save:

MemoryBandwidthCost

PAPER ID: 149

WHY SVD? Image compression alone can experience redundancies which

can be overcome by Singular Value Decomposition (SVD) of the image matrix.

Image compression and coding techniques explore 3 types of redundancies:

Coding redundancy Interpixel redundancy Psycho-visual redundancy

PAPER ID: 149

Redundancies

Coding Redundancy

Less than optimal code words are used. Eg: Huffman coding

Interpixel Redundancy

Also called spatial redundancy. Explored by predicting a pixel value based on the

values of its neighbouring pixels

Psycho-visual Redundancy

Exists due to psychophysical

aspects of human vision which prove that the human eye does not respond

with equal sensitivity to all incoming visual

informationPAPER ID: 149

WHAT IS SVD? Basic idea:

Taking a high dimensional, highly variable set of data points and reducing it to a lower dimensional space that exposes the substructure of the original data more clearly and orders it from most variation to the least.

Definition:In linear algebra, the singular value decomposition is a factorization of a real or complex, square or non-square matrix. Consider a matrix A with m rows and n columns with rank r. Then A can be factorized into three matrices:

A = UΣVT

PAPER ID: 149

SVD - I SVD: Refactors a square/non-square matrix into 3 different components

represented as:-

Or,

U is an m × m orthogonal matrix is the conjugate transpose of the n × n orthogonal matrix Σ is an m × n diagonal matrix with non-negative real numbers on the

diagonal which are known as the singular values of A

PAPER ID: 149

SVD - II

PAPER ID: 149

The m columns of U and n rows of V are called the left-singular and right-singular vectors of A respectively.

The singular values of Σ are arranged as ≥ ≥ … ≥ ≥ 0, where the largest singular values precede the smallest and they appear on the main diagonal of Σ.

The numbers ≥ … ≥ are the eigenvalues of and .

In MATLAB: [U,W,V]=svd(A,0)

SVD - III Image matrix A:

Outer Product Expansion:

Lower rank approximation: [Truncation]

Total Storage:PAPER ID: 149

SVD - PROPERTIES The singular values ‘σ’ are unique, unlike, the matrices U and V, which

are not unique. The singular values of a rectangular matrix A are equal to the square

roots of the eigenvalues , of the matrix . Mathematically, the rank of the matrix A is the number of its non-zero

positive singular values; rank (A) = r, r ≤ m. Since = , so U diagonalizes and u(i) s are the eigenvectors of . Since = V , V diagonalizes and the v(j)s are the eigenvectors of . If A has rank ‘r’ then vi, vj, … ,vr form an orthonormal basis for range

space of , R( ), and ui, uj, … ,ur form an orthonormal basis for range space A, R(A)

PAPER ID: 149

PROPOSED FLOWCHART & ALGORITHMSTART

Read the input image

Convert integer to double data type

Calculate the required rank 'k'

Perform SVD to obtain U, ∑ , V

matrices

Apply approximation on ∑ matrices

Regenerate the matrix and remove singularity

Convert double data type to integers

Compressed image is created and displayed

STOP

1. Load the JPEG image2. Store image as array of integers3. Specify Compression Ratio4. Perform SVD on matrix components5. Regenerate decomposed matrix6. Remove singularity if any7. Collect matrix components8. Create & display the compressed image

In MATLAB: [U,W,V]=svd(A,0)

PAPER ID: 149

PERFORMANCE EVALUATION PARAMETERS - I

Compression Ratio (CR)The ratio of file sizes of the uncompressed image to that of the compressed image.

Mean Square Error (MSE)Difference between pixel value of original image and the corresponding pixel value of the compressed image averaged over the entire image.

(x,y) - (x,y))

PAPER ID: 149

PERFORMANCE EVALUATION PARAMETERS - II

Signal to Noise Ratio (SNR)

Peak Signal and Noise Ratio (PSNR)Ratio between the maximum possible power of a signal and the power of corrupting noise that affects the fidelity of its representation.

Compressed image Bytes Storage space occupied by the compressed image.

PAPER ID: 149

SVD v/s MEMORY Without rank-k approximation of image I:

Non-compressed image, I, requires:

With rank-k approximation of I:Originally U m×m matrix, but only first k columns are rquired. Hence,

UM = mk Similarly,

VM = nk.

AM = UM+ VM+∑ M

AM = mk + nk + k

AM = k(m + n + 1)PAPER ID: 149

LOWEST RANK-k APPROXIMATIONS For SVD to save memory, there are important limits on k:

AM ≤IMk(m + n + 1) < mn

Necessary condition: Hence k is limit bound to ensure SVD compressed image occupies lesser

space than original image. If SVD of A:

For k = {1,2,…r}, A becomes:

PAPER ID: 149

RESULT ANALYSIS - I Original image:

Rank of image = 202

PAPER ID: 149

RESULT ANALYSIS - II

SVD compression for different Singular Values ‘k’

PAPER ID: 149

RESULT ANALYSIS - III

No. of Singular

Values used ‘k’

Performance Evaluation Parameters

Compression Ratio (CR)

MSE (in dB)

PSNR (in dB)

Bytes of Compressed Image (B)

2 61.046 40.61 25.04 3079612 23.588 39.42 28.17 3167922 12.101 38.09 30.32 3169952 5.72 38.047 32.39 31781112 2.01 37.44 33.46 31815

R=202 0.580 -20.93 83.88 31840251 0.564 -20.72 86.42 31862260 0.552 -20.69 86.437 31905262 0.544 -20.66 86.453 31926264 0.536 -20.66 86.453 32124

PAPER ID: 149

Performance measurement of SVD for different values of ‘k’ [Bytes of input image: 124358]

RESULT ANALYSIS - III

PAPER ID: 149

Plot of SVD Coefficients v/s Compression Ratio

Plot of SVD Coefficients v/s MSE

Plot of SVD Coefficients v/s PSNR PAPER ID:

149

INFERENCE The following inference can made:

Compression Ration v/s Singular values: Decreasing exponential function Fewer Singular Values implies smaller resulting file size Less storage

space Compression Ratio approaches 1 when SV = Rank of the image matrix Increase in SV’s Decrease in MSE Increase in SV’s Increase in PSNR Computation Time recorded to be the same

PAPER ID: 149

CONCLUSIONIt is necessary to strike a balance between storage space required and image quality for good image compression. From the above observations, it is found that optimum compression results are obtained when MSE of the compressed image is just less than or equal to 38dB (i.e. MSE ≤ 38dB). In our case, this is obtained when value of k is 52.

When k is equal to the rank of the image matrix (202 here), the reconstructed image is almost same as the original one.

PAPER ID: 149

APPLICATIONS Data Compression Noise Reduction Pseudo-inverse EOF Analysis: Pattern Extraction – Mid Ocean topography Face Recognition

PAPER ID: 149

FUTURE SCOPE

Wavelet Difference Reduction (WDR) offers high compression of the overall system

Adaptively Scanned Wavelet Difference Reduction (ASWDR) adapts the scanning procedure used by WDR in order to predict

locations of the significant transform values at half thresholds

PAPER ID: 149

Thank you.