Support width

If we take the particular case = 0 (all the others are just translations of it) we get 

The basis function can be expressed as the sum of scaled copies of itself, and the scaling factors are just the entries in the mask. The cascade algorithm can then be viewed as making the basis function from tinier and tinier copies of itself as iteration proceeds. 

An even more important theoretical result is that if we take some other function (with some small print) bo(t) and apply the cascade algorithm by 

We now look at the question again, of how much of the curve is influenced by a single control point. 

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This gives a more precise meaning to the term support width. It is the width in abscissa units of the closure15 of the abscissa region over which the basis function is non-zero (the support region). 

For example, the binary scheme whose mask is [1,3,3, l]/4 (the binary quadratic B-spline) has a support width of (4 — l)/2 = 3/2 spans on each side. 

We have to consider two cases such a point may be either at the ends of two different control points supports, which will be the case if the support width is integral, or at the end of only one, which will be the case if the support width is fractional. 

Consider first the case of integer support width. If the support width is even, the end-point will have an integer label and correspond to an original control point. If the support width is odd, the end-point will have a halfinteger label and correspond to a midedge of the original polygon. 

Looking at the support width, we see that at each refinement just half of the remaining amount of parameter line is filled with straight line pieces. Thus as refinement steps proceed, the amount remaining is halved at each step the limit curve consists entirely of pieces of straight line. 

If the maximum support width acceptable is w, then the expression for the generic scheme satisfying this constraint becomes... 

It is worthwhile to mention a few other wavelets which have gone unmentioned in the discussion thus far. These are the morlet. mexican hat and Meyer wavelets (see Fig. 13). The Meyer wavelet is one which does not have compact support, stated another way, the support width of the wavelet is infinite. The Meyer wavelet is orthogonal and symmetrical as well. 

The reasonable support width and caving step distance should be integer times of L in order to achieve the maximum limit of top-coal caving. So, the width of new end-support is chosen to 1.75 m. 

Design Data Abutment Support Width Design Abutment... 

In practice, the minimum abutment support width may be calculated as shown in Equation 6.4 ... 

The support width will be Nj, = 23.6 in. Add 3 in required temperature movement, the total required support width equals to 26.5 in. The required minimum support width for seismic case equals to the sum of bridge seismic displacement, bridge temperature displacement, and the reserved edge displacement (usually 4 in). In this example, this requirement equals to 14 in, not in control. Based on the 26.5 in minimum requirement, the design uses 30 in, OK. A preliminary abutment configuration is shown in Figure 6.14 based on the given information and calculated support width. 

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