Rochitha.S.M1, Savitri Raju2
1,2Dept of Electronics and communication Engineering, SDM College of Engineering and Technology Dharwad-580 002,Karnataka, India
1[email protected], 2[email protected]
Abstract: For low bit-rate compression applications, segmentation-based coding methods provide, in general, high compression ratios when compared with traditional (e.g., transform and subband) coding approaches. In this paper, we present a segmentation based image coding method that divides the desired image using binary space partitioning (BSP). Geometric wavelet is a recent development in the field of multivariate nonlinear piecewise polynomials approximation. The present study improves the geometric wavelet (GW) image coding method by using the slope intercept representation of the straight line in the binary space partition scheme.
Keywords: segmentation based coding; convex domain; BSP; Geometric Wavelet; sparse geometric wavelet representation
1. INTRODUCTION
The Binary Space Partitioning Tree (BSP Tree) is a binary tree used to represent an organization of continuous space by recursive subdivision. The BSP tree was initially introduced to organize a set of polygons so that visible surface renderings could be produced from an arbitrary viewing position.
The notion of segmentation-based image coding was introduced during the early 1980’s [1], [2]. A contour texture coding method was used to partition the image into rather complex geometric regions. The image signals within these regions were represented using low-order polynomials.
Quadtree-based image compression, which recursively divides the image signal into simple geometric regions (squares or rectangles), has been one of the most popular segmentation based coding schemes investigated by researchers [3], [4];
Segmentation-based compression methods usually describe the desired image as a set of regions. In general, the description of each region requires two types of information: 1) the geometry of the region boundaries and 2) the attributes of the image signal within the region. In order to achieve high compression ratio and good image quality, one needs to segment the image into a minimum number of regions such that the geometric
description of the region boundaries is simple and the image signal within each region is continuous (or smooth).
Therefore, the most challenging aspect of a segmentation based coding approach is to balance between a small number of geometrically simple regions and the smoothness (or continuity) of the image signal within these regions.
Previous segmentation-based methods have focused on one of these two conflicting requirements at the expense of the other. For example, due to the complexity of the regions used in the contour-texture coding method, the geometric information uses 75% of the total bit-rate cost even with a very small number of regions. On the other hand, using rigid geometric descriptions, such as the quadtree representation method, a very large number of regions is needed to generate a piecewise smooth image signal [3], [4]. From the mid 80s there have been many attempts to design second generation image coding techniques that exploit the geometry of the edge singularities of an image. Recently, many image compression algorithms such as the Bandelets, the Prune tree the Prune-Join tree and the GW image coding method
[7] based on the sparse geometric representation have been introduced.
2. BINARY SPACE PARTITIONING TECHNIQUE
BSP scheme achieves the above balance by using a simple, yet flexible description of the images. It has wide applications in the field of image processing and computer graphics. This technique subdivides an initial convex domain into two subdomains by intersecting it with a hyperplane. In image processing applications, the convex domain is the plane on which a straight line acts as a hyperplane.
The subdivision process is performed to minimize the given cost functional. The BSP approach partitions the desired image recursively by straight lines in a hierarchical manner.
A binary tree image representation method that divides the image domain using binary space partitioning (BSP) was introduced by the authors in 1990 and further developed in a series of papers.
The BSP approach partitions the desired image recursively by straight lines in a hierarchical manner. First, a line (arbitrarily oriented) is selected, based on an appropriate criterion, to partition the whole image into two subimages.
Using the same (or another) criterion, two lines are selected to split the two subimages resulting from the first partitioning. This procedure is repeated until a terminating criterion is reached. The outcome of this recursive partitioning is a set of (unpartitioned) convex regions (polygons) that are referred to as the cells of the segmented image (see Fig. 1), and a binary tree that is referred to as the binary space partitioning tree representation of that image. The nonleaf nodes of the BSP tree are associated with the partitioning lines, and the leaves represent the cells (unpartitioned regions) of the image. Each region of the image's BSP tree representation may have one or more attributes that describe the characteristics of the image signal. An example of such attributes is a zero-order (i.e., the mean value) or higher order polynomial model of the pixel intensities within the region.
Figure 1: Tilling of cameraman image of size 256x256.
The most critical aspect of the BSP tree construction process is the method used for selecting the partitioning lines. At each stage of the recursive partitioning (including the first stage when partitioning the whole image), the BSP line- selection method consists of two major steps, explained in the following:
1) Since the number of lines that pass through (i.e., partition) the polygon under consideration is infinite, in practice, and especially for image coding applications, one has to quantize the space of all possible lines that partition the polygon. This line-quantization process generates a finite set of lines that one needs to consider. Furthermore, because lines in the image (two-dimensional) domain can be represented using two parameters (e.g., the slope-intercept or normal representation), a line- quantization process implies the quantization of two parameters. In general, and independent of the particular line representation used, one of the two line parameters is referred to as the line orientation. The quantization of the line orientation is an important concept and will be discussed further below.
2) From the finite set of (quantized) lines, one has to select the partitioning line of the polygon under consideration. For this step, one needs to define a line-selection criterion that can be used for selecting the partitioning line.
Regarding the line-criterion used for selecting the partitioning lines we can employ either boundary or least-square-error-based criteria. Under the boundary-based approach, the selected lines pass through the boundaries (edges) of the objects in the image. For the least-square-error case, a selected
line minimizes a square error function defined over the region under consideration.
In a convex polygonal domain Ω with a function f Є L2 ([0, 1]2), BSP algorithm is applied to find two subdomains Ω’, Ω’’ and two bivariate linear polynomials QΩ’ and QΩ’’ by using the least square method to minimize the cost functional
'
2 2
"
fQΩ' fQΩ" (1)
The same process is applied on the resulting subdomains Ω’ and Ω’’. This procedure is repeated until some exit criterion is met [6].
Figure 2. Two partition levels using bisecting lines
3. GEOMETRIC WAVELETS
Geometric wavelet is the recent development in the field of multivariate nonlinear piecewise polynomial approximation [8], [9] and appears for the first time in [10]. Geometric wavelets have been considered in context of image compression in [7].
Geometric wavelet, ΨΩ’ associated with the subdomain Ω’ and function f is given by
Ψ Ω’ Ψ Ω’(f) 1 Ω’(Q Ω’ - Q Ω) (2)
where Ω’ is a child of Ω in a BSP tree, i.e., Ω’
Ω and Q' , Q are '
the polynomial approximations found by (1).
Like isotropic wavelets [11], ΨΩ’ is a “local difference” component that belongs to the detail space between two levels in the BSP tree, a “low resolution” level associated with Ω and a “high resolution” level associated with Ω’. Geometric wavelets also satisfy the vanishing moment property i.e., if f is locally a polynomial over Ω then minimizing of (1) gives Q' Q f , and therefore ' 0. . However, unlike classical wavelets, geometric wavelets do not satisfy the two scale relation and the biorthogonality property.
They do not form even a basis of L2 nevertheless they can be used in the adaptive approximation [8], [10].
If P is a full BSP tree, then under certain mild conditions on, we have [10]
( ), . . [0,1]2 p
f f a e in
(3)Where Ψ[0,1]2 Ψ[0,1]2(f)
≜
1[0,1]2 Q[0,1]2 (4)GW image coding algorithm [7] is based on the fact that among all the geometric wavelets only a
“few” wavelets have large norm. Once all the geometric wavelets are created, they are sorted according to their L2 norm, i.e.,
1 2 2 2 3 2
...
k k k
(5)
and then the sparse geometric representation is extracted using the greedy methodology of nonlinear approximation. Thereafter, function f is approximated using the n-term geometric wavelet sum
1 k j n
j
(6)where n is the number of wavelets used in the sparse representation.
This sum is the generalization of the classical greedy wavelet n-term approximation [11], [12].
Figure 3.Example of a greedy selection.
Figure 4.The final GW tree with the additional nodes.
4. ALGORITHM
A typical digital camera contains a sensor with many millions of individual pixels. After the picture is taken, pixel data is pulled off the chip and placed in memory. Once in memory, the image data is compressed using something like the JPEG algorithm, at which point the raw pixel data is thrown away. On reflection, this seems to be sub- optimal. We went to a lot of trouble (costing us valuable time, energy, and memory) to get all that data off the chip, only to immediately toss most of it in the trash. Why not just grab fewer pixels in the first place? That's the general idea behind compressive sensing, and it's the basis.
In this we reconstruct the actual pixel values of an
"uncompressed" sensor image. We will do this by requesting "compressed" sensor values, each of which is the sum of pixel values within the uncompressed image. To query the values, supply a mask for the region you're interested in, and in return you will receive the sum of the pixel data corresponding to that mask. There is a limit on how many compressed sensor values you can request in
order to attempt to reconstruct the uncompressed pixel values
A. Binary Space Partitioning 1. Read the Image
2. Tile the image using tiles of size 128 X 128 3. Apply a mask for the region of interest 4. Compress the pixel values in this region 5. Take the Sum in the particular interested
region
6. Assign the sums to the appropriate regions of the image estimate
7. Reconstruct a sub-sampled image from the sums over different masks.
The different masked regions can be seen in the Figure.3. The masked region contains the compressed pixel values. Considering these compressed pixels and the uncompressed pixels of the original image we can reconstruct the image with compression.
B. Sparse Geometric Wavelets Representation 1. Select the important regions applying greedy
approximation to the binary partitioned image obtained from A
2. If any child has a missing parent we include it.
Here some more compression can be achieved by applying Greedy approximation technique.
5. RESULTS
Figure 5. Images and its Binary Partitioned scheme
Figure 6. DWT of a Binary Partitioned Cameraman Image
Figure 7. IDWT of Figure 4.
Figure 8. Simulink Model for PSNR Calculation PSNR of 7.18 is obtained for an image size of 256 x 256.
Figure 9.Sparse Geometric Wavelet Representation of Binary Space Partitioned
Cameraman Image
Figure 10. Sparse Geometric Wavelet epresentation of Binary Space Partitioned Lena
Image
Figure 11.Sparse Geometric Wavelet epresentation of Binary Space Partitioned Moon
Image
6. CONCLUSION
The key idea behind this work is the fact that to consider the image as segments and compress the few of them and applying Geometric Wavelets on it to achieve the compression.
7. ACKNOWLEDGMENT
The authors wish to thank the authorities of SDMCET Dharwad, for the facility extended to carry out the present work. The authors also would like to thank the unknown reviewers for their invaluable comments which help in improving the paper.
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