Objective

2

As a powerful image compression method

set partition coding (SPC) method effectively uses the correlation between the wavelet coefficients to obtain a higher data compression ratio

and it has been widely used for all types of image compression. The SPC method uses the idea of successive quantization approximation of the wavelet coefficient set

such as set partition coding system (SPACS)

which partitions the coefficient set step-by-step to find significant coefficients and code them. A significance map was used to decide whether a set is significant

based on which set will be partitioned. If a set is significant

then SPC will output a location bit "1" and the set will be partitioned into 4 subsets; otherwise

the set is prepared for the next set coding operation. If set partition operations are conformable to the distribution of the insignificant coefficients

then location bits and unnecessary bits will be decreased. When the coefficient set is sparse

the SPC method can use fewer bits to encode the image. However

with the decrease of the bit plane

the sparseness of the coefficient set decreases

and the SPC method will waste many location and unnecessary bits

especially for lossless compression. To increase the lossless encoding performance of the set SPC method

we construct a set partition coding method that embeds a generalized tree classifier named SPACS_C.

Method

2

The previous SPC methods process all coordinate sets with "test before partition

" thereby increasing the number of location and unnecessary bits when the correlation between data decreases. SPACS_C calculates the bit costs of two different coding ways called "test before partition" and "partition before test" for each coordinate set. Then

it chooses the one with lower bit cost to process the set. The process method used in SPACS_C takes advantage of the data characteristics in which the sparseness of bottom bit-planes decrease rapidly in wavelet transformed images. SPACS_C performs the coding process in the domain of the wavelet transform of the image. Daubechies (4

4) integer wavelet transform is used in this study. The level of wavelet transform is determined by the image size. For instance

a level of 5 is recommended for an image with a spatial size of 512×512. Similar to SPACS

a general tree (GT) is used to simultaneously represent the tree and square sets in the wavelet domain. The processing flow of SPACS_C is as follows:1)Initialization. Let the threshold n be most significant digit of the maximum of the wavelet coefficient and the list of significant points (LSP) be an empty list. Add all GTs into the list of insignificant points (LIP)

where all GTs that have descendants are added into the list of insignificant sets (LIS). 2)Sorting pass. Perform the significant test for every entry (

$$i

j$$

) in LIP. If the result is positive

bit=1

add GT (

$$i

j$$

; 1) into LSP to output the sign of the root node coefficient

and then remove (

$$i

j$$

) from LIP. For every entry in LIS

conduct prediction to calculate the bit costs of the two different coding ways called "test before partition" and "partition before test". The bit costs are the mathematical expectation of the bits used in coding using the two methods. If "test before partition" uses less bits than "partition before test"

then do "test before partition". Otherwise do "partition before test". If the result of the "test before partition" is positive

then the out location bit "1" moves to the corresponding GT from LIP to LSP and outputs the sign of the root node coefficient. For "partition before test"

the executed partition is removed from LIS. Let List1 be the partition result of a GT in LIS. Perform a significant test for every entry in List1. If the result is positive

partition the list to obtain a new set List2

and then place all entries at the end of LIS. Otherwise

place the entries at LIS. 3)Refinement pass. For every entry in LSP

if type=0

output the

$$n$$

th most significant bit

whereas if type=1

update type=0.4)Threshold update. Let

$$n$$

=

$$n$$

-1 and return to sorting pass until n is equal to 0 to achieve a lossless compression. In SPACS_C

the significant test is performed by a significant test function

where for any element

$$c$$

in a set

$$\mathit{\boldsymbol{c}}$$

if

$$c$$

≥

$${2^n}$$

and

$$c$$

<

$${2^{n + 1}}$$

then

$$c$$

is significantly relative to the threshold value

$${2^n}$$

.

Result

2

Various visible and infrared images with different sizes

statistical properties

and bit depths were used to evaluate SPACS_C

and JPEG2000 and JPEG-LS were used for comparison. For SPACS_C

a 5-level Daubechies (4

4) integer wavelet transform was used for decomposition. In addition

the wavelet coefficients were encoded. For infrared images with bit depths of 16 and 8 bits

the lossless encoding performance of SPACS_C was improved and became superior to that of JPEG 2000; an average of 3.1% less bits were used by the former. Notably

JPEG-LS can only be used for 8-bit image compression. For visible images with 8-bit depth

SPACS_C was superior to that of JPEG2000 and comparable with JPEG-LS. Unlike JPEG-LS

SPACS_C can provide a quality progressive code flow

which means that SPACS_C can also be used in loss compression and can stop coding when a limited bit rate is satisfied. SPACS_C can use part of the code stream to reconstruct the entire image

whereas JPEG-LS can only reconstruct part of the image.

Conclusion

2

The proposed method enhances the coding performance of SPC for lossless image compression by decreasing the output of location bit "1". Extensive experiment results show that the lossless compression performance of SPACS_C is better than that of JPEG2000 and comparable with JPEG-LS. The process mode used in SPACS_C suits the low sparseness in the bottom bit planes of wavelet-transformed images. Moreover

SPACS_C can progressively compress images such as JPEG2000. Unlike JPEG-LS

which can only compress images with 8-bit depth

SPACS_C can be used for images with any bit depth.