Changeset 2174

Timestamp:

Oct 19, 2004, 4:27:32 PM (22 years ago)

Author:

Paul Price

Message:

Generalised entry on Gaussian smoothing.

File:

• trunk/doc/pslib/psLibADD.tex (modified) (2 diffs)

trunk/doc/pslib/psLibADD.tex

@@ -1 @@
-%%% $Id: psLibADD.tex,v 1.48 2004-10-09 03:29:29 price Exp $
+%%% $Id: psLibADD.tex,v 1.49 2004-10-20 02:27:32 price Exp $
 \documentclass[panstarrs]{panstarrs}
 
@@ -102 +102 @@
 Smoothing may occasionally be perfomed on data.  We present the
 algorithms for two typical versions: boxcar and Gaussian smoothing.
-In both smoothing techniques, given a series of data values $f_i$, the
-smoothed values $g_i$ are determined by calculating a linear
-combination based on the input data point and its nearest $2N$
+In both smoothing techniques, given a series of data values $f_i(x_i)$
+where $x_i$ are the values of the corresponding to the center of the
+bin, the smoothed values $g_i(x_i)$ are determined by calculating a
+linear combination based on the input data point and its nearest $2N$
 neighbors in the form:
 
 \begin{equation}
- g_i = \sum_{n=-N}^N c_n f_i
-\end{equation}
-%
-where the values of $c_n$ determine the filter type.  For boxcar
-smoothing, the values $c_n$ are constant, and must be equal to $1/(2N
-+ 1)$ to maintain the zeroth moment of the data (care must be taken at
-the ends of the data range to reduce the value of $c_n$ as fewer input
-data points may be used).  For Gaussian smoothing, the crucial
-parameter is $\sigma$, the standard deviation.  The value of $N$
-should be chosen to be large enough to sample the Gaussian, $N =
-5\sigma$, and the values of $c_n$ are then just the Gaussian curve:
-
-\begin{equation}
-c_n = \frac{e^{\frac{-n^2}{2\sigma^2}}}{\sqrt{2\pi\sigma^2}}
+g_i = \sum_{j=i_{\rm min}}^{i_{\rm max}} c_{ij} f_j
+\end{equation}
+%
+where the values of $c_{ij}$ determine the filter type.  For boxcar
+smoothing, the values $c_{ij}$ are constant and scaled to maintain the
+zeroth moment of the data (care must be taken at the ends of the data
+range to reduce the value of $c_{ij}$ as fewer input data points may
+be used).  For Gaussian smoothing, the crucial parameter is
+$\sigma_x$, the standard deviation.  The values of $i_{\rm min}$ and
+$i_{\rm max}$ are functions of the standard deviation: $i_{\rm min}$
+corresponds to the bin in which $x_i - N\sigma_x$ is found; similarly
+$i_{\rm max}$ is the bin corresponding to $x_i + N\sigma_x$.  The
+value of $N$ should be chosen to be large enough to sample the
+Gaussian, $N = 5$.  The values of $c_{ij}$ are then just the Gaussian
+curve:
+
+\begin{equation}
+c_{ij} = \frac{e^{\frac{-(x_j - x_i)^2}{2\sigma_x^2}}}{\sqrt{2\pi\sigma_x^2}}
 \end{equation}