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r r r using barcode encoder for visual .net control to generate, create code 3 of 9 image in visual .net applications. Microsoft Office Official Website All signals shoul .NET barcode 3 of 9 d be de ned on the same domain (no pyramids). Increasing values of the scale parameter should produce coarser representations.

A signal at a coarser level should contain less structure than a signal at a ner level. If one considers the number of local extrema as a measure of smoothness, then the number of extrema should not increase as we go to coarser scale. This property is called scale space causality.

All representations should be generated by applications of a convolution kernel to the original image.. The last property is certainly debatable, since convolution formally requires a linear, space-invariant operator. One interesting approach to scale space which violates this requirement is to produce a scale space by using gray scale morphological. Linear operators and kernels smoothing (we wil l discuss this later) with larger and larger structuring elements [5.16]. You could use scale space concepts to represent texture [4.

16] or even a probability density function (in which case, your scale space representation becomes a clustering algorithm [5.24]) as well as brightness. We will see applications of scale representations as we proceed through the course.

One of the most interesting aspects of scale space representations is the behavior of our old friend, the Gaussian. The second derivative of the Gaussian (in two dimensions, the Laplacian of Gaussian: LOG) has been shown [5.27] to have some very nice properties when used as a kernel.

In particular, the zero crossings of the LOG are good indicators of the location of an edge. One might be inclined to ask, Is the Gaussian the best smoothing operator to use to develop a kernel like this Said another way: We want a kernel whose second derivative never generates a new zero crossing as we move to larger scale. In fact, we could state this desire in the following more general form.

Let our concept of a feature be a point where some operator has an extreme, either maximum or minimum. The concept of scale space causality says that as scale increases, as images become more blurred, new features are never created. The Gaussian is the ONLY kernel (linear operator) with this property [5.

1, 5.2]. Studies of nonlinear operators have been done to see under what conditions these operators are scale space causal [5.

22]. This idea of scale space causality is illustrated in the following example. Fig.

5.13 illustrates the brightness pro le along a single line from an image, and the scale space created by blurring that single line with one-dimensional Gaussians of increasing variance. In Fig.

5.14, we see the Laplacian of the Gaussian, and the points where the Laplacian changes sign. The features in this example, the zero crossings (which are good candidates for edges) are indicated in the right image.

Observe that as scale increases, feature points (in this case, zero-crossings) are never created as scale increases. As we go from top (low scale) to bottom (high scale), some features disappear, but no new ones are created. One obvious application of this idea is to identify the important edges in the image rst.

We can do that by going up in scale, nding those few edges, and then tracking them down to lower scale.. 5.10 Quantifying the accuracy of an edge detector Since there are m any options in the design of an edge detection algorithm, we need some objective ways to say that one edge detector works better than another. Pratt [5.33] has suggested a simple formula to address this question.

The formula is just. 5.10 Quantifying the accuracy of an edge detector increasing variance Gaussian blur 200 250 50 100 150 200 250. 20 40. 80 100 120 140 160 180 200. Fig. 5.13.

(a) Br ightness pro le of a scanline through an image. (b) Scale space representation of that scanline. Scale increases toward the bottom, so no new features should be created as one goes from top to bottom.

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