Box & boxKDE algos + notation fixes

tex/chapters/mvg.tex

@@ -9,7 +9,7 @@ Consequently, the filter kernel of a Gaussian filter is a Gaussian with finite s
 Assuming a finite-support Gaussian filter kernel of size $M$ and a input signal $x$, discrete convolution produces the smoothed output signal 
 \begin{equation}
 \label{eq:gausFilt}
-y[n] = \frac{1}{\sigma\sqrt{2\pi}} \sum_{k=0}^{M-1} x[k]\expp{-\frac{(n-k)^2}{2\sigma^2}} \text{,}
+(G_{\sigma}*x)(i) = \frac{1}{\sigma\sqrt{2\pi}} \sum_{k=0}^{M-1} x(k) \expp{-\frac{(i-k)^2}{2\sigma^2}} \text{,}
 \end{equation}
 where $\sigma$ is a smoothing parameter called standard deviation.
 
@@ -31,14 +31,14 @@ The box filter is a simplistic filter defined as convolution of the input signal
 A discrete box filter of \qq{radius} $l\in N_0$ is given as 
 \begin{equation}
 \label{eq:boxFilt}
-y[n] = \sum_{k=0}^{L-1} x[k] B_L(n-k)
+(B_L*x)(i) = \sum_{k=0}^{L-1} x(k) B_L(i-k)
 \end{equation}
 where $L=2l+1$ is the width of the filter and 
 \begin{equation}
 \label{eq:boxFx}
-    B_L(n)=
+    B_L(i)=
     \begin{cases} 
-      L^{-1} & \text{if } -l \le n \le l  \\
+      L^{-1} & \text{if } -l \le i \le l  \\
       0      & \text{else }
     \end{cases}
 \end{equation}
@@ -55,11 +55,26 @@ Opposed to the Gaussian filter where several evaluations of the exponential func
 The most efficient way to compute a single output value of the box filter is:
 \begin{equation}
 \label{eq:recMovAvg}
-    y[i] = y[i-1] + x[i+r] - x[i-r-1]
+    y(i) = y(i-1) + x(i+l) - x(i-l-1)
 \end{equation}
 This recursive calculation scheme further reduces the time complexity of the box filter to $\landau{N}$, by reusing already calculated values.
 Furthermore, only one addition and subtraction is required to calculate a single output value.
+The overall algorithm to efficiently compute \eqref{eq:boxFilt} is listed in Algorithm~\ref{alg:naiveboxalgo}.
 
+\begin{algorithm}[ht]
+    \caption{Recursive 1D box filter}
+    \label{alg:naiveboxalgo}
+    \begin{algorithmic}[1]
+        \Statex \textbf{Input:} $f$ defined on $[1,N]$ and box radius $l$
+        \Statex \textbf{Output:} $u \gets B_L*f$ 
+        \State $a := \sum_{i=-l}^{l} f(i) $
+        \State $u(1) := a/(2l+1)$
+        \For{$i=2 \textbf{ to } N$}
+        \State $a:= a + f(i+l) - f(i-l-1) $ \Comment{\eqref{eq:recMovAvg}}
+        \State $u(i) := a/(2l+1)$
+        \EndFor
+    \end{algorithmic}
+\end{algorithm}
 
 Given a fast approximation scheme, it is necessary to construct a box filter analogous to a given Gaussian filter.
 As seen in \eqref{eq:gausFilt}, the solely parameter of the Gaussian kernel is the standard deviation $\sigma$.
@@ -100,6 +115,7 @@ However, the overall shape will not change.
 Just like the conventional box filter, the extended version has a uniform value in the range $[-l; l]$, but unlike the conventional the extended box filter has different values at its edges.
 This extension introduces only marginal computational overhead over conventional box filtering.
 
+
 \commentByToni{Warum benutzen wir den extended box filter nicht? oder tun wir das? Liest sich so, als wäre er der heilige Gral.
 
 Ansonsten: Kapitel find ich gut. Vielleicht kann man hier und da noch paar sätze fusionieren um etwas Platz zu sparen.}