An Improved EZW Hyperspectral Image Compression

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Figure 4. 2-D wavelet transform coefﬁcients are stored in a

data cube.

level decomposition on each band.

The DWT has two important properties which are

critical for image compression:

1) Energy packing: When an image is wavelet trans-

formed, the transformed image has energy compaction in

spectral dimensions, that is, the wavelet coefficients in

the higher level subbands will, on average, be larger than

those in the lower level subbands.

2) Self-similarity: A wavelet coefficient at a higher

level subband and all wavelet coefficients of the same

spatial orientation at lower level subbands have certain

predictable relationships.

4. Shapiro’s EZW Algorithm

Shapiro invented his Embedded Zerostree Wavelet

(EZW) algorithm taking advantage of the wavelet trans-

form [2]. The EZW algorithm implements a progressive,

embedded image coding method based on the zerotrees

of data structure. All currently significant bits at the same

bitplane together and recursively encodes other pixels for

the next significant bitplane until reaching the least sig-

nificant bitplane. As a result, the lower significant bits

are embedded behind the higher significant bits, so that a

decoder quickly displays a low quality image and better

quality as more bits are received.

We notethree crucial components that make Shapiro’s

EZW algorithm effectivein image compression. First,

due to energy packing, these wavelet coefficients in

higher level subbands could be scanned earlier than oth-

ers in low level subbands. In other words, larger coeffi-

cients will be encoded first. In addition, both receiver and

transmitter know what scanning order is selected such

that it does not include the scanning order in the over-

head. Scanning order used in this paper is Morton scan.

Second, the quad-tree is the fundamental idea of the

EZW algorithm to interpolate the relations among wave-

let coefficients in different subbands; therefore, it is set

up based on the self-similarity. The definition of the

quad-tree was introduced in [2].

There are two steps to complete EZW algorithm: the-

dominant pass and the subordinate pass. The dominant

pass keeps track of the search for significant coefficients

by labeling each pixel among these four labels: signific-

ance positive symbol (POS), significant negative symbol

(NEG), zerotree root (ZTR) and isolated zeros (IZ) in

Figures 5(a)-(c). The subordinate pass quantizes each

significant coefficient that has been found in the domi-

nant pass.

The definitions of four symbols are described below. If

a root coefficient, in absolute value, is larger than a thre-

shold, it is labeled as significant positive (POS) or sig-

nificant negative (NEG) in Figure 5(b). It implies that

some of the coefficients’ descendants are significant. An

isolated zero (IZ) is a root coefficient that is insignificant

but has some significant descendants in Figure 5(c). If

the coefficient is zerotree root (ZTR), it means the root

coefficient itself and its descendants are all insignificant

such that these descendants don’t have to be encoded in

the current iteration in Figure 5(a) .

5. Modified EZW Algorithm

We propose some modifications to simplify the conven-

tion EZW algorithm and improved the compression re-

sults. [10-13] have studied the asymmetrical 3D-DWT

decomposition that causes the asymmetrical statistics of

the transformed hyperspectral image; thus the asymme-

trical tree structure is more suitable for describing the

transformed hyperspectral image. In this paper, the 3D

asymmetric tree structure was designed according to the

properties of hybrid transforms. In Figure 4, a wavelet

coefficient at a higher subband is not only relative to all

wavelet coefficients of the same spatial orientation at

lower subbands at the same band but also in the neighbor

band. Therefore, if the approximation subband is

-by-

with -level decomposition at the

first band, while the spatial dimensions of the image are

m-by-n. Any root

in the approximation has

four immediate children at

,

(,/2 ,0),(/2 ,/2,0)

l ll

xymxmym+ ++

and

In addition, except ones in the approximation, any root (x,

y, 0) at the first band has four children located at the

same spatial orientations (2x, 2y, 0), (2x + 1, 2y, 0), (2x,

2y + 1, 0), and (2x + 1, 2y + 1, 0) on the first band and

one more child (x, y, 1) below it. Note that any pixel not

in the first band has only one child below it. The new tree

structure will continue to branch until no offspring can be

found. The new and simple definition of tree structure is

demonstrated in Figure 6. Each pixel in the low-pass

(approximation) section of band 1 is a tree root.

In addition to the four labels defined in Shapiro’s

EZW algorithm, in this study, we define two more labels,

called positive and negative ZTR (PZT and NZT) in

Figure 5(d). According to [14], they are degree-1 zero-

trees since every coefficient except the root coefficient is

all zeros. Roots of PZT and NZT are positive and nega-