ImprovingImprovingImprovingImproving Compression TimeCompression TimeCompression TimeCompression Time in in in in ZeroZeroZeroZero----Tree Based Image Tree Based Image Tree Based Image Tree Based Image Coding ProceduresCoding ProceduresCoding ProceduresCoding Procedures

Jonas Valantinas1 and Deividas Kančelkis2

1 Kaunas University of Technology, Department of Applied Mathematics, Studentų St 50-325c, LT-51368 Kaunas, Lithuania,

e-mail: jonas.valantinas@ktu.lt,

2 Kaunas University of Technology, Department of Applied Mathematics, Studentų St 50-325c, LT-51368 Kaunas, Lithuania,

e-mail: deividas.kancelkis@stud.ktu.lt

Abstract—In this paper, a novel scheme for the accelerated analysis of quad-trees in the discrete wavelet spectrum of a digital image is proposed. During the pre-scanning step, the proposed scheme generates objective and specially structured binary codes for the whole set of quad-tree roots (wavelet coefficients) and thereby accumulates facts on the significance of respective descendants (wavelet coefficients comprising quad-trees on the view). The developed scheme can be successfully applied to any zero-tree based image coding procedure, such as the embedded zero-tree wavelet (EZW) algorithm of Shapiro and set partitioning in hierarchical trees (SPIHT) by Said and Pearlman. Exceptionally high performance of the proposed quad-tree analysis scheme, in the sense of image encoding times, is demonstrated using the EZW algorithm and the discrete Le Gall wavelet transform.

Keywords—Wavelets, Discrete wavelet transforms, Quad-trees, Zero-tree based image coding, Le Gall wavelets, EZW.

I. INTRODUCTION

Most modern and up-to-date digital image compression (coding) techniques employ a discrete wavelet transform (DWT), usually followed by quantization and entropy coding, [1-3]. Especially useful are image coders that allow progressive encoding with an embedded bit stream, such as the embedded zero-tree wavelet (EZW) image coder, suggested by Shapiro, [4]. With embedded bit streams, the wavelet coefficients are encoded in bit planes, with the most significant bit planes being transmitted first. In that way, the decoder can cease decoding at any point in the bit stream, and it will reconstruct an image with required level of accuracy.

Different variants of zero-tree based progressive image coders have been developed since Shapiro introduced his algorithm in 1993. The SPIHT (Set Partitioning in Hierarchical Trees) algorithm, proposed by Said and Pearlman, shows excellent results in this class of coders, [5]. Some other interesting ideas and innovative proposals in the area are presented in [6-10].

Bit plane encoding is more efficient if one reorders the wavelet coefficient data in such a way that coefficients with small absolute values tend to get clustered together, increasing the lengths of the zero run in the bit planes. Data structures such as insignificant quad-trees (zero-trees) are very efficient in achieving such clustering of zeros. They are used in EZW, SPIHT, and other wavelet-based image coders.

Though there is a number of wavelet-based image coding schemes available, the need for improved performance and wide commercial usage demand newer and better techniques to be developed.

Modest attempts to improve image compression time in zero-tree based image coding procedures were made by Kunal Mukherjee et al., [11]. Unfortunately, their RMF (Recursive Merge Filter) based EZW algorithm is bound up with Haar wavelets, and is absolutely inapplicable to higher order wavelets (Le Gall, Daubechies, etc.).

In this paper, we propose a novel idea (scheme) for accelerated analysis of quad-trees in the discrete wavelet spectrum of the image under processing. The proposed scheme generates specially structured binary codes for all roots (wavelet coefficients) of available quad-trees, estimates the current threshold value and makes a decision over the significance of wavelet coefficients (descendants) comprising quad-trees on the view. The developed scheme, being applied to EZW encoder, impressively speeds up the image encoding phase ( (41 53)%− , for lossless compression, and (39 95)%− , for lossy compression; Section III) and, naturally, the overall performance of the encoder. In parallels, impact of the image smoothness level on the efficiency of EZW image encoders is touched on.

II. THE EMBEDDED ZERO-TREE WAVELET (EZW) CODING

The embedded zero-tree wavelet (EZW) algorithm is one of the most efficient image (signal) coding schemes, which are developed for wavelets. The EZW encoder transmits the large (significant) wavelet coefficients before transmitting the smaller coefficients. This is done by realizing multiple passes (iterations, scanning) over the discrete wavelet spectrum of the image, lowering the threshold by a factor of two each time, [4].

On each iteration the EZW encoder refines the value of each coefficient determined to be significant during previous passes and then searches through the coefficients previously considered to be insignificant. A wavelet coefficient is said to be insignificant with respect to a given threshold T if it is less (by absolute value) than T . The zero-tree is based on the hypothesis that if a particular wavelet coefficient (the parent) at a high scale is insignificant with respect to a given threshold T , then all wavelet coefficients (the descendants) of the same orientation in the same spatial location at lower scales are probably to be insignificant with respect to T (Fig. 1, a). If this is true, we say that this coefficient is the root of the zero-tree, [4].

For better understanding of the proposed ideas, we here present both the encoding and decoding phases of the current EZW algorithm, [6]. Let RL (Refinement list) be the list of wavelet coefficients which have been previously found to be significant and let IL (Index list) be the list of indices of all wavelet coefficients which have not yet been found to be significant, ordered so that parents are listed before their children (say, in accordance with Morton’s trajectory; Fig. 1, b).

The image encoding phase is as follows.

1. Compute max 2 1 2log max{| ( , ) |}r r Y k k= = , where 1 2( , )Y k k

is the 1 2( , )k k -th wavelet coefficient of the digital image 1 2[ ( , )]X m m

under processing; 1 2 1 2, , , {0,1, , 1}k k m m N∈ −K , 2nN = , N.n∈ Set

RL = ∅ .

Image Processing Theory, Tools and Applications

978-1-4244-7249-9/10/$26.00 ©2010 IEEE

2. For each 1 2( , )Y k k RL∈ , output the r -th most significant bit

of 1 2( , )Y k k .

3. Set the list IL to contain indices of all the wavelet coefficients, except of those in RL, ordered so that parents are listed before their children. For each index 1 2( , )k k IL∈ , do: if 1 2( , ) ( 2 )rr rY k k T T≥ =

then output poz and add 1 2( , )Y k k to RL, else if 1 2( , ) rY k k T≤ − then

output neg and add 1 2( , )Y k k to RL, else if no descendant (not

contained in RL) of 1 2( , )Y k k is significant with respect to rT then output zrt and remove all the indices corresponding to the descendants of 1 2( , )Y k k from IL, else output iz .

4. If 0r > then decrease the value of r by one and go to the step 2.

5. The end.

Since symbols poz , neg , zrt , iz do not occur with the same probability Huffman coding is usually applied to the EZW output to reduce data amount (bit stream).

(a)

(b)

Fig. 1. Wavelet coefficients are coded in a zero-tree structure and scanned in a left-to-right order: (a) parent-offspring dependencies in spatial orientation tree; (b) Morton’s scanning (the trajectory is such

that ancestors are always traversed before descendants).

The image decoding phase requires some data to start with,

namely: the initial threshold max

max2 ,r

r rT T= = the image size, the

sub-band decomposition scale and the encoded bit stream. The EZW scheme decodes encoded files (bit stream) into a symbol sequence, creates all the proper size sub-bands needed and proceeds image decompression process as follows.

1. Input maxr r= . Set RL = ∅ and set all the wavelet coefficients to zeros.

2. For each 1 2( , )Y k k RL∈ , input the r -th most significant bit of

1 2| ( , ) |Y k k .

3. Set the list IL to contain indices of all the coefficients, except of those in RL, ordered so that parents are listed before their children. For each index 1 2( , )k k IL∈ , do: input a current symbol σ ; if

pozσ = then set 1 2( , ) ( 2 )rr rY k k T T= = and add 1 2( , )Y k k to RL,

else if negσ = then set 1 2( , ) rY k k T= − and add 1 2( , )Y k k to RL, else if zerσ = then remove all indices corresponding to the descendants of 1 2( , )Y k k from IL, else if izσ = then take no action.

4. If 0r > then decrease the value of r by one and go to the step 2.

5. The end.

The other wavelet-based image coding procedure (algorithm), called SPIHT, is an improved version of EZW which achieves better compression and performance than EZW. For more detailed description of the algorithm, we refer the reader to [5].

III. ACCELERATED QUAD-TREE ANALYSIS SCHEME IN THE EZW ALGORITHM

The most time-consuming part of the EZW algorithm is the iterative image encoding phase (Section II), because at this phase all quad-trees in the discrete wavelet spectrum 1 2[ ( , )]Y k k of the image

1 2[ ( , )]X m m are analysed repeatedly for significant nodes (wavelet coefficients) with respect to decreasing from iteration to iteration threshold values. When an insignificant wavelet coefficient (parent)

1 2( , )Y k k is found, and a scan of its children reveals that they are insignificant too, then it is possible to encode that wavelet coefficient

1 2( , )Y k k and its children, a zero-tree, by one symbol ,zrt thus achieving compression. Wavelet coefficients (roots of quad-trees) found to be insignificant in the encoding phase but with significant children are coded as isolated zeros .iz

To avoid repeated scanning and verification of wavelet coefficients, comprising volumes of quad-trees in the DWT spectrum

1 2[ ( , )]Y k k , for significance with respect to changing threshold values, we have developed an original quad-tree analysis scheme, which guarantees exceptionally high performance of the encoding phase in EZW.

A. The scheme

Let 1 2[ ( , )]X m m ( 1 2, {0,1, , 1}m m N∈ −K , 2 , NnN n= ∈ ) be a

given digital image and 1 2[ ( , )]Y k k ( 1 2, {0,1, , 1}k k N∈ −K ) be its discrete wavelet (Haar, Le Gall, Daubechies, etc.) spectrum. Also, let

max 2 1 2 1 2log max{ | ( , ) | , {0,1, , 1}}r Y k k k k N = ∈ − K .

Consider a wavelet coefficient 1 2( , )Y k k ( 2n iss sk j−= + ,

{0, 1, ..., 2 1}, 1,2n issj s−∈ − = ) which is the root (parent) of the

quad-tree (provided 1 1i > and 2 1i > ), comprising the set of wavelet

coefficients (descendants) 1 2 1 2{ ( , ) | ( , ) }Y k k k k∗ ∗ ∗ ∗ ∈ℑ , where

1 2

min{ 1, 1}1 21

{ ( )x ( )}i i

k ktt t

− −

=ℑ = ℑ ℑU ,

( ) {2 , 2 1, ..., 2 ( 1) 1}s

t t tk s s st k k kℑ = ⋅ ⋅ + + − , 1,2s = .

Low scale

High scale

High scale

Low scale

Let us associate 1 2( , )Y k k with two binary codes (one for the

offspring of 1 2( , )Y k k , another for the descendants of 1 2( , )Y k k , except offspring), namely:

max1 2 1 2 1 1 2 0 1 2( , ) ( , ) ( , ) ( , )rO k k u k k u k k u k k= ⟨ ⟩K ,

max1 2 1 2 1 1 2 0 1 2( , ) ( , ) ( , ) ( , )rD k k v k k v k k v k k= ⟨ ⟩K .

The above codes are generated by the onetime scanning of the discrete wavelet spectrum 1 2[ ( , )]Y k k as shown below.

1. 1 2( , ) 1ru k k = , if at least one of coefficients (taken by absolute

value) 1 22 , 2| |k ky ,

1 22 1, 2| |k ky + , 1 22 , 2 1| |k ky + or

1 22 1, 2 1| |k ky + + falls into the

interval 1[2 ,2 )r r+ , max{0,1, , }r r∈ K , and 1 2( , ) 0ru k k = , otherwise;

2.

1 2 1 2

1 2 1 2

1 2

1 2 1 2 1 21 2

1 2 1 2

1

(2 ,2 ) (2 1,2 )

(2 ,2 1) (2 1,2 1),

for 8 max{ , } 4 1,

(2 ,2 ) (2 ,2 ) (2 ,2 1)( , )

(2 ,2 1) (2 1,2 )

(2 1

r r

r r

r r rr

r r

r

u k k u k k

u k k u k k

N k k N

u k k v k k u k kv k k

v k k u k k

v k

∨ + ∨

∨ + ∨ + +

≤ ≤ −

∨ ∨ + ∨=

∨ + ∨ + ∨

∨ + 2 1 2

1 2

1 2

,2 ) (2 1,2 1)

(2 1,2 1),

for 1 max{ , } 8 1,

r

r

k u k k

v k k

k k N

∨ + + ∨

∨ + +

≤ ≤ −

for all max0,1, ,r r= K .

Thus, to state that the wavelet coefficient 1 2( , )Y k k

( 1 2, {0,1, , 2 1}k k N∈ −K ) is the root of the zero-tree, comprising the

set of wavelet coefficients (descendants) 1 2 1 2{ ( , ) | ( , ) }Y k k k k∗ ∗ ∗ ∗ ∈ℑ ,

with respect to the threshold 2rrT T= = ( max{0,1, , }r r∈ K ), it

suffices to ascertain that 1 2( , ) 0ru k k = and 1 2( , ) 0rv k k = (the key moment of the proposed idea).

We here emphasize that in the first instance one must compute binary codes 1 2( , )D k k with indices 1 2( , )k k satisfying the inequality

1 28 max{ , } 4 1N k k N≤ ≤ − , then codes 1 2( , )D k k with indices

1 2( , )k k satisfying the inequality 1 216 max{ , } 8 1N k k N≤ ≤ − , and,

finally, codes 1 2( , )D k k with 1 2( , )k k such that 1 2max{ , } 1k k = .

Also, we observe that generation of binary codes 1 2( , )O k k , for all

possible values of 1k and 2k , requires ( 2 4N − ) verifications of

falling into one or another half-open interval, while that for 1 2( , )D k k

requires ( 2 4 7N − ) logical additions on binary codes of length

max 1r + .

Evidently, the described quad-tree analysis scheme can be used with other coding algorithms similar to the EZW, as the SPIHT algorithm (in processing lists of insignificant pixels and those of insignificant sets, [5]), and for other data as images.

B. Experimental results To implement both the current (Section II) and the modified

(supplemented with the proposed quad-tree analysis scheme; Section III) versions of the EZW algorithm, the discrete Le Gall (wavelet) transform (DLGT) was employed. The latter transform possesses a tolerable “energy compaction” property, has a fast performing technique and facilitates lossless compression of digital images. Incidentally, the default reversible transform in JPEG 2000 is implemented exactly by means of DLGT, [3].

With a view to estimate efficiency of the developed accelerated quad-tree analysis scheme, a number of digital images of size 256x256, characterized by different smoothness level, were processed, namely (Fig. 2): Cameraman.bmp, Nature.bmp, Cows.bmp. Computer simulation of both versions of the EZW algorithm was performed on a PC with CPU Intel(R) Core(TM) i7 2.8 GHz, RAM 8 GB, 64-bit OS Windows 7; Programming language Java. Computer’s environment – the home user’s environment with all unnecessary processes (except antivirus, services, etc.) turned off.

(a)

(b)

(c)

Fig. 2. Test images: (a) Cameraman.bmp; (b) Nature.bmp; (c) Cows.bmp.

As it can be seen from Table 1, application of the proposed accelerated quad-tree analysis scheme to lossless encoding (the threshold value 0 1T T= = ), as well as to lossy encoding ( 1rT T= > ),

of test images leads to striking image encoding time gains EZW EZWτ τ ∗−

( EZWτ signifies the time needed to encode the test image using the

current EZW algorithm, whereas EZWτ ∗ - the time needed to encode the same image using the modified version of the EZW algorithm).

Table 1. Image encoding times EZWτ and EZWτ ∗ (in seconds).

Test image

Threshold Cameraman.bmp Nature.bmp Cows.bmp

EZWτ EZWτ ∗

EZWτ EZWτ ∗

EZWτ EZWτ ∗

0 1T T= = 3.581 1.668 10.387 5.973 14.026 8.243

1 2T T= = 1.276 0.45 7.359 4.146 10.514 6.382

2 4T T= = 0.815 0.159 4.829 2.651 7.693 4.602

3 8T T= = 0.596 0.069 2.82 1.509 5.205 3.055

4 16T T= = 0.51 0.027 1.282 0.647 2.672 1.653

5 32T T= = 0.497 0.009 0.384 0.141 0.732 0.449

For instance, in the case of lossless image encoding ( 0 1T T= = ), the modified version of the EZW algorithm runs (on the average) twice faster than the current EZW algorithm. Furthermore, with increasing values of the threshold ,T the obtainable image encoding

speed gains ( ) 100EZW EZW EZWτω τ τ τ∗= − ⋅ (%) have tendency to increase (Fig. 3).

Fig. 3. Image encoding speed gains (lossless and lossy image

compression)

Similar promising results were obtained for images of other sizes. For instance, the losless encoding time speed gains for the image peppers.bmp 512x512 equal 78.09τω = (%), and that for the image

brain.bmp 128x128 equal 37.12τω = (%).

Worth emphasizing, the overall performance of both the current and the modified EZW coders depends on the smoothness class (level) of the image under processing, [12]. The higher smoothness of the image, the lower image encoding times and noticeably better performance of the modified EZW coder in comparison with the current EZW coder (Fig. 4). We here notice that the smoothness level α of test images was determined by computing the rate of “decay” of Le Gall wavelet coefficients.

IV. CONCLUSION

In the paper, a novel scheme for the accelerated analysis of quad-trees in the discrete wavelet spectrum of a digital image is proposed. The scheme can be applied to any zero-tree based image encoder, such as the embedded zero-tree wavelet (EZW) algorithm of Shapiro, the set partitioning in hierarchical trees (SPIHT) by Said and Pearlman, and others.

Numerous experimental results show that implementation of the proposed quad-tree analysis scheme in the EZW algorithm increases the overall performance of the EZW encoder somewhere (39 95)%− .

Worth emphasizing, the image encoding speed gains directly depend on the smoothness level of the image under processing. The lower smoothness of the image, the higher image encoding speed gains are obtained.

In the future, efficiency analysis of the application of the developed quad-tree analysis scheme to other wavelet-based image encoders (SPIHT, etc.) is planned.

Fig. 4. Image smoothness impact on the efficiency of EZW encoders ( 0.311α = , for Cameraman.bmp, 0.211α = , for Nature.bmp , and

0.111α = , for Cows.bmp).

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[12] T. Žumbakis and J. Valantinas. The use of image smoothness estimates in speeding up fractal image compression. In 14th Scandinavian Conference on Image Analysis (SCIA’05), proceedings, Joensuu, Finland, LNCS 3540: 1167-1176, June 2005.

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0 1 2 3 4 5

Cameraman.bmpNature.bmpCows.bmp

τω

2log T

0

5

10

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0 0,1 0,2 0,3 0,4

TEZW

TEZW*

τ

α

EZWτ ∗ EZWτ