induction to wavelet transform and image compression

cung cấp kiến thức về biến đổi wavelet trong xử lý ảnh

Introduction to Wavelet Transform and Image

Compression

Student: Kang-Hua Hsu 徐康華 Advisor: Jian-Jiun Ding 丁建均 E-mail: r96942097@ntu.edu.tw Graduate Institute of Communication Engineering National Taiwan University, Taipei, Taiwan, ROC

Introduction(1)-WT v.s FT

Bases of the

• FT: time-unlimited weighted sinusoids with different

frequencies No temporal information.

• WT: limited duration small waves with varying

frequencies, which are called wavelets WTs contain the

temporal time information

Thus, the WT is more adaptive

Introduction(2)-WT v.s TFA

• Temporal information is related to the time-frequency

analysis

• The time-frequency analysis is constrained by the

Heisenberg uncertainty principal

• Compare tiles in a time-frequency plane (Heisenberg cell):

• MRA is just a concept, and the wavelet-based

transformation is one method to implement it.

MRA-Subband Coding(1)

• Since the bandwidth of the resulting subbands is smaller than that of the original image, the subbands can be

downsampled without loss of information

input can be perfectly reconstructed

MRA-Subband Coding(2)

• Biorthogonal filter bank:

• Orthonormal (it’s also biorthogonal) filet bank:

: time-reversed relation

,where 2K denotes the number of coefficients in each filter.

• The other 3 filters can be obtained from one prototype filter

MRA-Subband Coding(3)

• 1-D to 2-D: 1-D two-band subband coding to the rows and then to the columns of the original image.

• Where a is the approximation (Its histogram is scattered, and

thus lowly compressible.) and d means detail (highly

compressible because their histogram is centralized, and thus

easily to be modeled).

Multiresolution Expansions(1)

• , : the real-valued expansion coefficients.

, : the real-valued expansion functions

• Scaling function : span the approximation of the

Multiresolution Expansions(2)

• 4 requirements of the scaling function:

 The scaling function is orthogonal to its integer translates

 The subspaces spanned by the scaling function at low scales are nested within those spanned at higher scales

 The only function that is common to all is

 Any function can be represented with arbitrary coarse

resolution, because the coarser portions can be represented

by the finer portions

j

V f x    0 V 

Multiresolution Expansions(4)

: wavelet function coefficients

• Relation between the scaling coefficients and the wavelet coefficients:

This is similar to the relation between the impulse response

of the analysis and synthesis filters in page 11 There is

time-reverse relation in both cases

• The definition of the CWT is

• Continuous input to a continuous output with 2 continuous variables, translation and scaling

This is still the continuous case If we change the integral to

summation, the DWT is then developed

1 ( , ) M ( ) j k( )

The coefficients measure the similarity (in linear algebra,

the orthogonal projection) of with basis functions

By the 2 relations we mention in subband coding,

We can then have

2 , 0

n k k m

 

When the input is the samples of a function or an image, we can exploit the relation of the adjacent scale coefficients to obtain all

of the scaling and wavelet coefficients without defining the

scaling and wavelet functions

2 , 0

n k k m

FWT resembles the two-band subband coding scheme!

1 :

FWT 

•The constraints for perfect reconstruction is the same

as in the subband coding

2-D WT(2)

Note that the upmost-leftmost subimage is similar to the

original image due to the energy of an image is usually

distributed around lower band

Wavelet Packets

A wavelet packet is a more flexible decomposition.

Goal: To convey the same information with

least amount of data (bits).

Fundamentals of Image

Compression(2)

• Image compression can be classified to

 Lossless(error-free, without distortion after

reconstructed)

 Lossy

• Information theory is an important tool

• Data Information : information is “carried” by the data.

Fundamentals of Image

Compression(3)

1 2

R

n C

n



1 1

Relative data redundancy :

•Evaluation of the lossy compression:

root-mean-square (rms) error

1 2

Coding Redundancy

• We can obtain the probable information from the histogram

of the original image

• Variable-length coding: assign shorter codeword to more

probable gray level

If there is a set of codeword to represent the

original data with less bits, the original data is

said to have coding redundancy.

Interpixel Redundancy(1)

• Because the value of any given pixel can be

reasonably predicted from the value of its

neighbors, the information carried by individual

pixels is relatively small.

Interpixel redundancy is resulted from the

correlation between neighboring pixels.

Interpixel Redundancy(2)

• To reduce interpixel redundancy, the original image will be transformed to a more efficient and nonvisual format This transformation is called mapping.

• Run-length coding Ex 10000000 1,111

Psychovisual Redundancy

• For example, the edges are more noticeable for us

• Information loss!

• We truncate or coarsely quantize the gray levels (or

coefficients) that will not significantly impair the perceived image quality

• The animation take advantage of the persistence of vision to reduce the scanning rate

Humans don’t respond with equal importance

to every pixel

Image Compression Model

•The quantizer is not necessary

•The mapper would

1.reduce the interpixel redundancy to compress directly,

such as exploiting the run-length coding

or

2.make it more accessible for compression in the later

stage, for example, the DCT or the DWT coefficients are

Lossless Compression

• No quantizer involves in the compression procedure

• Generally, the compression ratios range from 2 to 10

• Trade-off relation between the compression ratio and the

computational complexity

It can be reconstructed without distortion.

Variable-Length Coding

• It merely reduces the coding redundancy

• Ex Huffman coding

It assigns fewer bits to the more probable gray levels than

to the less probable ones

Bit-plane Coding

A monochrome or colorful image is decomposed into a series of binary images (that is, bit planes), and then they are compressed

by a binary compression method

•It reduces the interpixel redundancy

Lossless Predictive Coding

• It reduces the interpixel redundancies of closely spaced

pixels

• The ability to attack the redundancy depends on the

predictor

It encodes the difference between the actual and predicted

value of that pixel

Lossy Compression

• It exploits the quantizer

• Its compression ratios range from 10 to 100 (much more than the lossless case’s)

• Trade-off relation between the reconstruction accuracy and

compression performance

It can not be reconstructed without distortion

due to the sacrificed accuracy.

Lossy Predictive Coding

• It exploits the quantizer

• Its compression ratios range from 10 to 100 (much more

than the lossless case’s)

• The quantizer is designed based on the purpose for

minimizing the quantization error

• Trade-off relation between the quantizer complexity and less quantization error

• Delta modulation (DM) is an easy example exploiting the

oversampling and 1-bit quantizer

It is just a lossless predictive coding containing

a quantizer.

Transform Coding(1)

Most of the information is included among a small number

of the transformed coefficients Thus, we truncate or coarsely quantize the coefficients including little information

•The goal of the transformation is to pack as much information

as possible into the smallest number of transform coefficients

•Compression is achieved during the quantization of the

Transform Coding(2)

•More truncated coefficients Higher compression ratio, but the rms error between the reconstructed image and the original one would also increase

•Every stage can be adapted to local image content

•Choosing the transform:

Information packing ability

Computational complexity needed

Information packing ability Best Not good Good

Computational complexity High Lowest Low

Practical!

Transform Coding(3)

•Disadvantage: Blocking artifact when highly compressed (this causes errors) due to subdivision

•Size of the subimage:

Size increase: higher compression ratio, computational

complexity, and bigger block size

How to solve the blocking artifact problem? Using the WT!

Wavelet Coding(1)

• No subdivision due to:

 Computationally efficient (FWT)

 Limited-duration basis functions

Avoiding the blocking artifact!

Wavelet coding is not only the transforming coding

exploiting the wavelet transform -No subdivision!

Wavelet Coding(2)

• We only truncate the detail coefficients.

• The decomposition level: the initial decompositions would draw out the majority of details Too many decompositions is just wasting time.

Wavelet Coding(3)

• Quantization with dead zone threshold: set a threshold to

truncate the detail coefficients that are smaller than the

threshold

The WT is a powerful tool to analyze signals There are many applications of the WT, such as image

compression However, most of them are still not

adopted now due to some disadvantage Our future

work is to improve them For example, we could

improve the adaptive transform coding, including the shape of the subimages, the selection of transformation, and the quantizer design They are all hot topics to be studied.

[1] R.C Gonzalez, R.E Woods, Digital Image

Processing, 2nd edition, Prentice Hall, 2002.

[2] J.C Goswami, A.K Chan, Fundamentals of Wavelets,

John Wiley & Sons, New York, 1999.

[3] Contributors of the Wikipedia, “Arithmetic coding”, available in