Image compression using the distance transform on curved space (DTOCS) and Delaunay triangulation
Introduction
In recent years, there has been a dramatically increasing demand for handling images in digital form. Owing to performance improvements and significant reductions in the cost of image scanners, photographs, printed text and other media can now be easily converted into digital form. Direct acquisition of digital images is also becoming more common as sensors and associated electronics improve. The use of satellite imaging, e.g. LANDSAT, in remote sensing of the earth and the advent of electronic still-cameras in the consumer market are good examples. In addition, many different imaging modalities in medicine, such as computed tomography (CT) or magnetic resonance imaging (MRI), generate images directly in digital form. Computer-generated images (synthetic images) are also becoming a major source of digital data. However, although much progress has been made, there is one potential problem with digital images, the large number of bits required to represent them. Fortunately, digital images generally contain a significant amount of redundancy.
Image compression techniques are commonly divided into first and second generation methods (Kunt et al., 1987). The second generation methods can further be divided into two groups. The first group is characterized by the use of local operators. Pyramidal coding (Burt and Adelson, 1983) and anisotropic nonstationary predictive coding (Wilson et al., 1983) are the main examples of this group of methods. The second group methods attempt to describe an image in terms of contour and texture. Directional decomposition-based coding (Ikonomopoulos and Kunt, 1985) and segmentation-based coding (Kocher and Kunt, 1983) are two major examples of this second group of methods.
A recently introduced image compression method is based on the linear wavelet theory. With this method it is possible to obtain both time and space resolution at the same time giving better compression ratios than classical methods (DeVore et al., 1992; Sriram and Marcellin, 1993).
In (Wang et al., 1981) the min-max-Medial Axis Transform is used for image compression giving 1.0 and 2.5 bits per pixel for noisy chromosome images. Such rates are comparable with those typically obtained in interpolative and transform coding schemes.
Commonly distance transforms are used in feature extraction in pattern matching and learning. Their use in image compression is very rare. This paper presents a new image compression method in which the encoding process is based on the distance transform on curved space (DTOCS) (Vepsäläinen and Toivanen, 1991; Toivanen, 1996) and the decoding is based on Delaunay triangulation (Rajan, 1994; De Floriani, 1989). The algorithm presented in this paper could best be used in applications where an image is made more accurate iteratively. First, a modest number of control points, i.e. points which have been taken out from the original image and hold corresponding gray values, is sent just to make the subject of the image visible to the viewer, and more control points are sent as soon as possible so that the decompressed image is enhanced iteratively.
Section snippets
Definition of the DTOCS
This paper utilizes a new distance transform in which the distance values are weighted by the gray value differences of the original image (Toivanen, 1996). Definition 1 Let X⊂Z2. Let B⊂Z2 be the structuring element. Let the external boundary of X be denoted by ∂X and be defined by ∂X=(X⊕B)⧹X. ∂X⊂XC (Giardina and Dougherty, 1988).
In the definition below we will use the following notation. x∈X and y∈∂X. Let ΨX(x,y) denote the set of digital 8-paths in X∪∂X linking x and y. Let γ∈ΨX(x,y) and let γ have n
The DTOCS algorithm
A sequential two-pass algorithm to calculate the distance transform on curved space (DTOCS) is presented in (Vepsäläinen and Toivanen, 1991) and with some extensions in (Toivanen, 1996). The DTOCS algorithm requires two images: the original gray-level image and a binary image which determines the region(s) X in which the transform is performed and the background area XC. It should be noted that the region X in which the following transform is performed may consist of several disjoint
The compression algorithm
The basic idea behind the proposed compression method is to transform the original gray-level image to an image which contains points, which can be considered fundamental for the reconstruction of the image. The problem is to find as optimal positions for those points as possible. In one dimension, it is clear that a point should be put every time there is a change in the gray value, and that point should be placed at the coordinate where the first derivative starts to grow. In two dimensions,
Time complexity of the compression algorithm
Let NG denote the number of pixels in the gray-level image G. The number of control points is denoted by N. b,d,e,f,m and n denote constant scalars. Proposition 1 The time complexity of the compression algorithm is O(NG). Proof Let a be a repetition counter.
Step 1. O(1).
Step 2. O(max(N,1))=O(N).
Step 3. O(NG−N).
Step 4. O(max(aNG,aNG,aNG))=O(NG).
Step 5–6. O(max(3a,4am2n))=O(max(a,amn).
Step 8. Huffman could be O.
Huffman is left out of the following calculation because there are several ways to implement
Compression results
Fig. 14 shows an image which has been compressed with an algorithm presented in (Toivanen, 1996). It was decompressed using Delaunay triangulation. Fig. 15, Fig. 16, Fig. 17, Fig. 18 depict images which have been compressed by the method presented in this paper. Fig. 15 shows an image which has been compressed using the raster pattern of Fig. 1. The distance between the center pixel and the four other pixels was 6 pixels giving a compression ratio of 1:21. In Fig. 16 the center points of the
Conclusion
In this paper, a new image compression method based on the DTOCS is presented and the results are compared to JPEG DCT images. The overall performance of the proposed method is of the same category as the DCT method with a 4×4 normalization matrix. In the decompression, the Delaunay triangulation-based algorithm gives visually better images than the 8 kernels' method presented in (Toivanen, 1993, 1991). The obtained gain in SNR is about 2.0 dB and is visually significant. In low frequency areas
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