04-10-2012, 05:31 PM
WAVELET-BASED IMAGE PROCESSING
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Abstract
The 1990's witnessed an explosion of wavelet-based methods in the eld
of image processing. This paper will focus primarily on wavelet-based
image compression. We shall describe the connection between wavelets
and vision and how wavelet techniques provide image compression algorithms
that are clearly superior to the present jpeg standard. In
particular the wavelet-based algorithms known as spiht, aswdr, and
the new standard jpeg2000, will be described and compared. Our comparison
will show that, in many respects, aswdr is the best algorithm.
Applications to denoising will also be brie
y referenced and pointers
supplied to other references on wavelet-based image processing.
Introduction
The eld of image processing is a huge one. It encompasses, at the
very least, the following areas: 1. Image Compression; 2. Image De-
noising; 3. Image Enhancement; 4. Image Recognition; 5. Feature De-
tection, 6. Texture Classication. Wavelet-based techniques apply to
all of these topics. One reason that wavelet analysis provides such an
all-encompassing tool for image processing is that a similar type of anal-
ysis occurs in the human visual system. To be more precise, the human
visual system performs hierarchical edge detection at multiple levels of
resolution|and wavelet transforms perform a similar analysis (more on
this below).
Transform-based compression and wavelets
Wavelet-based compression is one type of transform-based compres-
sion. In general, transform-based compression is done according to the
scheme shown in Fig. 1. For wavelet-based compression, a wavelet trans-
form and its inverse are used for the transform and inverse transform,
respectively. Other transforms can be used as well. For example, the
jpeg algorithm uses a block-dct and its inverse (see subsection 1.3 be-
low).
The types of images that we shall consider are digital. The image is
a matrix of integers ranging from 0 to 255. These values specify shades
of grey|with 0 being pure black and 255 pure white. These integers
can be specied using 8 bits (1 byte) for each pixel (matrix element).
It is typical in image compression to treat grey scale images only. That
is because the human visual system responds much more sensitively to
intensity (corresponding to a grey scale) than to color attributes. See
[33, Chap. 9] for more details, including interesting photos in Plate 7.
Karhunen-Loeve Transform
The optimal linear transform to use (when mean square error6 is used)
is the Karhunen-Loeve transform. The Karhunen-Loeve transform is the
best (in terms of minimizing mse) linear transform for \decorrelating"
(removing redundancy) from images (see [13] and [6]).
Unfortunately, the Karhunen-Loeve transform is very expensive to
compute; it runs very slowly on a computer, far too slowly for most
applications. Low complexity, and its consequence of rapid compres-
sion and decompression is just one of the desired features of a compres-
sion/decompression method (codec).
Desired features of an image codec
There are several desired features of an image codec. Here is a list
of some of the main features and where they prove useful: 1. Targeted
Compression Ratio [image archiving, Internet transmission]; 2. Progres-
sive/Embedded [web pages; database browsing]; 3. Low complexity/Low
memory [narrow bandwidth applications]; 4. Region of Interest (roi)
[reconnaissance; medical diagnosis]; 5. Operations on compressed data
[reconnaissance; denoising]. By Targeted Compression Ratio, we mean
the ability to precisely encode to any desired compression ratio. Pro-
gressive/Embedded refers to the ability, at any point in the transmission
of the compressed le, to reconstruct an approximate image. Such a
feature is useful for webpages and database browsing. The roi feature
refers to the ability of the compressor to allocate more bits to describing
regions of interest, a property which is of obvious importance in recon-
naissance. See Fig. 2 for an example of the roi property. Table 1 shows
how each of the compression algorithms to be compared here fare with
regard to these desiderata. We now turn to a discussion of each of these
image codecs.
Essentials of wavelet-based compression
Before we look at some particular wavelet-based image compression
algorithms, it is helpful to look at a specic image and examine a wavelet
transform of it. In Fig. 5(a) we show an image commonly used in image
processing, known as Lena. Fig. 5(b) shows a histogram for the intensi-
ties of the pixels of Lena. Notice that the histogram is widely dispersed
over the range from 0 to 255. A measure of this dispersion is entropy,9
which for this image is 7:45. This shows that there is almost no redun-
dancy in the 8-bit image values in Lena. A zip compression of Lena, for
example, yields very little compression (about 15% savings in le size).
After transforming, however, and setting all values below a threshold of
10 to zero, we obtain Fig. 5©. Fig. 5© has an entropy 1:35, which by
the principles of information theory (see [1] or [32]) can be compressed
by a factor of about 8=1:35, a savings of about 83%. When the decom-
pressed le, corresponding to Fig. 5©, is inverted the resulting image is
exactly the same as the original. Considerably more compression|with
some loss of exactness, but still perceptually the same as the original|is
obtained if a higher threshold is used.
JPEG2000 codec
The jpeg2000 codec, which is the new ISO standard for photographic
image compression, is described in great detail in [25]. Here we shall only
brie
y summarize its main features. jpeg2000 is a block-based method
like jpeg, but instead of dividing the image into blocks (subimages),
jpeg2000 divides the wavelet transform into blocks. See Fig. 7. Because
the top-left corner block inverts to a low-resolution version of the original
image, jpeg2000 avoids the blocking artifacts of jpeg. jpeg2000 has all
of the desired properties summarized in Table 1. Because jpeg2000 is
described in great detail in [25], we now turn to a comparison of the
compression performance of the codecs we have described above.