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Total Variation Preservation
Although the importance of total variation preservation for separators cannot be doubted, it is even more so for hierarchical decompositions like the Discrete Pulse Transform, due to the fact that they involve iterative applications of separators. Since the operators Ln, Un, n = 1, 2, …, and all their compositions, are total variation preserving, it is easy to obtain the statement of the following theorem, which shows that, irrespective of the length of the vector in (3.1) or the number of terms in the sum (3.21), no additional total variation, or noise, is created via the decomposition.
THE DISCRETE PULSE TRANSFORM
The proof of Theorem 27 can be found in [8, 52]. We should remark that representing a function as a sum of pulses can be done in many different ways. However, in general, such decompositions increase the total variation, that is, we might have strict inequality in (3.25) instead of equality. Based on Theorem 27 we can construct the total variation distribution of images. More precisely, this is the distribution of the total variation of an image among the different layers of the DPT. That is, essentially the plot of TV (Dn(f)) vs. n. In Figure 3.9 we present the total variation distributions of some images, where one can observe how the total variation of each image as given in Table 2.6 is distributed over the pulse size. A log scale is used on the vertical axis and the pulse size values are grouped to form a histogram. The different character of the images naturally manifests through different kinds of total variation distributions.
THE DISCRETE PULSE TRANSFORM
The question of measuring the smoothing ability of the DPT arises. The aim of a smoother primarily to remove the noise element present. The noise can be due to a number of factors, for example, acquisition, processing, compression, storage, transmission and reproduction of the image [235]. The easiest method of evaluation is purely subjective – namely, human visual investigation.
In order for evaluation to be objective, quantitative methods need to be used instead. Quantitative methods can be divided into three categories [235]. First, full-reference, where the complete reference (undistorted) image is known with certainty, secondly, no-reference, where this reference image is not known at all, and third, reduced-reference, where only part of the original reference image is known, for example, a set of extracted features. We measure the similarity of the smoothed images Pn(f) to the original unsmoothed image f with the structural similarity index [235], for two corresponding sets of pixels, x and y, in each image, where μi, i = x, y is the mean of the pixel values in i; σ2 i , i = x, y is the variance of the pixel values in i; σxy the covariance between x and y; cj = (kjL)2 , j = 1, 2 constants to stabilize the division by the weak denominator; L = 255 for greyscale images and kj ≪ 1 constants (we used kj = 0.05). This measure is a fullreference measure which provides a useful framework since we are comparing a smoothed version of the original distorted image with the original distorted image. The most widely used such measures are the mean-squareerror (MSE) and the peak signal-noise-ratio (PSNR), but these measures do not compare well with the perceived visual quality of the human visual system [235]. Wang et al [235] introduce the SSIM measure in order to penalize errors based on their visibility, that is, to simulate the HVS as much as possible. This measure is applied to 8 × 8 windows in the image for each pixel and a final mean structural similarity index is calculated as the average of these SSIM values, called the MSSIM. An MSSIM value closer to 1 indicates stronger similarity. Wang et al provide MATLAB code for the implementation of the SSIM at www.cns.nyu.edu/∼lcv/ssim which was made use of. Figure 3.10 provides MSSIM values for various images as the LULU smoothing progresses through the DPT from scale n = 1 up to n = N. Notice how, based on the content of the images, the reduction in the MSSIM values as the DPT progresses varies from image to image. The graphs provide a mechanism to determine where visual structure is in the image, that is, when the HVS would pick out structures of significance. Figure 3.11 plots the differences between the smoothed images as the DPT progresses. The graphs also give an indication of how the information is removed at each step. Figures 3.12 and 3.13 provide an indication of the structure found at big jumps in Figures 3.10 and 3.11.
1 Introduction
2 LULU Theory Background
2.1 Setting
2.2 Separators
2.3 One Dimensional LULU
2.4 One Dimensions to n Dimensions
2.5 n Dimensional LULU
2.6 Properties
2.7 Conclusion
3 The Discrete Pulse Transform
3.1 The Discrete Pulse Transform
3.2 Connectivity
3.3 LULU Implementation
3.4 Properties of the Discrete Pulse Transform
3.5 Conclusion
4 Multi-scale Analysis
4.1 Introduction
4.2 Background Theory
4.3 Scale-Space History
4.4 The Gaussian Scale-Space
4.5 Other Scale-Spaces Developed
4.6 Scale-Space: A Formal Definition
4.7 The LULU Scale-Space
4.8 Scale-Space Applications
4.9 Conclusion
5 Improving Image Quality
5.1 Introduction
5.2 Sharpening
5.3 Best Approximation
5.4 Noise Removal
5.5 Conclusion
6 Conclusion
