Windowed sinc function

There is a lot of perhaps unnecessary lore about choice of a window function, and practically every function that rises from zero to a peak and then falls again has been named after someone. in Numerical Recipes in C [49] So far we some popular reconstruction filters which try to approximate the sinc filter were considered. Why cannot the sinc filter itself be used? Because, as already mentioned at the beginning of this chapter, it has infinite spatial extend and therefore is impracticable. One possibility to make the sinc function finite is to simply truncate it. However, we can again investigate the effects of such a truncation in frequency domain. Truncating a function means multiplying it with a box function. This means that in frequency domain the box function (the frequency response of the sinc function) gets convolved with a sinc (the frequency response of the box function). It would be a better idea to multiply the sinc with some other function which is also zero outside some extend but has better properties in frequency domain.

A lot of such functions, usually called windows, can be found in the literature. Definitions of some windows are:

• The rectangular window or box function (as defined by Eq. 4.1) which is tantamount to a pure truncation.
• The Bartlett window, which is actually just a tent function (as defined by Eq. 4.6).
• The Welch window
  

  $$\textrm{Welch}(x,\tau)=\left \{ \begin{array}{cl} 1-\left(... ...rt x\vert < \tau$}\\ 0 & \textrm{else} \end{array} \right.$$ (4.10)

  
  

• The Parzen window
  

  $$\textrm{Parzen}(x)=\frac{1}{4}\left \{ \begin{array}{cl} 4... ...\vert x\vert < 2$}\\ 0 & \textrm{else} \end{array} \right.$$ (4.11)

  
  
  is a piece-wise cubic approximation of the Gaussian window of extend two. Although its width is not directly adjustable it can, of course, be scaled to every desired extend.
• The Hann (due to Julius van Hann, often wrongly referred to as Hanning window [64], sometimes just cosine bell window) and Hamming window, which are quite similar, they only differ in the choice of one parameter $\alpha$:
  

  $$\textrm{H}(x,\tau,\alpha)=\left \{ \begin{array}{cl} \alph... ...rt x\vert < \tau$}\\ 0 & \textrm{else} \end{array} \right.$$ (4.12)

  
  
  with $\alpha=\frac{1}{2}$ being the Hann window and $\alpha=0.54$ the Hamming Window. The Hamming window has the disadvantage of being discontinuous at the edges, which leads to clearly visible artifacts in the images.
• The Blackman window, it has one additional cosine term to the Hann and Hamming window to further reduce the ripple ratio.
  

  $$\textrm{Blackman}(x,\tau)=\left \{ \begin{array}{cl} 0.42+... ...rt x\vert < \tau$}\\ 0 & \textrm{else} \end{array} \right.$$ (4.13)

  
  

• The Lanczos window, which is the central lobe of a sinc function scaled to a certain extend.
  

  $$\textrm{Lanczos}(x,\tau)=\left \{ \begin{array}{cl} \frac{... ...rt x\vert < \tau$}\\ 0 & \textrm{else} \end{array} \right.$$ (4.14)

  
  

• The Kaiser window [23], which has an adjustable parameter $\alpha$ which controls how steeply it approaches zero at the edges. It is defined by
  

  $$\textrm{Kaiser}(x,\tau,\alpha)=\left \{ \begin{array}{cl} ... ...t x\vert\leq \tau \\ 0 & \textrm{else} \end{array} \right.$$ (4.15)

  
  
  where I₀(x) is the zeroth order modified Bessel function (a definition, and a more detailed discussion of the Bessel functions can be found, for instance, in the Numerical Recipes in C [49]). The higher $\alpha$ gets the narrower gets the Kaiser window.
• and we can also use a Gaussian function as window, although it actually never reaches zero, by truncating it. It is in its general form defined by
  

  $$\textrm{Gauss}(x,\tau,\sigma)=\left \{ \begin{array}{cl} 2... ...rt x\vert < \tau$}\\ 0 & \textrm{else} \end{array} \right.$$ (4.16)

  
  
  with $\sigma$ being the standard deviation. The higher $\sigma$ gets, the wider gets the Gaussian window and, on the other hand, the more severe gets the truncation.

The earthier bits of dialogue, I explained, were the result of Gentry's years with the hairy-knuckled, hard drinking engineers and mathematicians of JPL's Astrodynamics Division, where the Pasadena cops often have to be called in to settle bare-fisted fights over Bessel functions and non-linear partial differential equations.Arthur C. Clarke, prologue of ``Rama II'' Are these windows worth the effort? A first clue can be gained from looking at the frequency responses. The frequency responses of the sinc filter windowed with all these windows and the windows themselves are depicted in Appendix B.

Obviously, some windows show a really bad behavior in frequency domain, e.g, the rectangular or the Bartlett window, but also Kaiser and Gaussian window for certain values for $\alpha$ and $\sigma$ respectively. This indicates that the advantage of Kaiser and Gaussian window of being adjustable can easily turn into a drawback of having to find the right value.

Figure 4.7: Frequency responses of windowed sinc windowed with a Hamming window of different widths.

One advantage, however, that all windows have in common is that if we want a better result we can simply increase the width. Of course, this increases also the computational effort. The plots in Appendix B were generated for a half-width of two, i.e., a contribution of four samples which is the same as for the cubic splines. So how does the window width affect the frequency response? To get an impression for this in Fig. 4.7 are several windowed sinc filters depicted which were windowed with a Hamming window of different widths. As we can see the approximation gets closer and closer the wider the window gets until it almost reaches it for quite wide windows. Although, admittedly, a filter width of one thousand is not really reasonable.

Symmetric piecewise n-th order polynomial filters (which of course include cubic ones) and windowed sinc functions for function reconstruction have recently been subject of extended research [35,36,38]. In the case of piecewise polynomial kernels it is stated that in all cases cardinal splines are superior to all other reconstruction filters of equal width. Windowed sinc functions are reported to perform quite well, based on a purely quantitative comparison to linear interpolation, with Welch, Lanzcos and Kaiser windows, the truncated sinc kernel is reported to be one of the worst performing kernels. In Chapter 5 we will concentrate on the quality of derivative reconstruction of the windows presented in this chapter.