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Fixed FE 1

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MBulli
2018-03-12 22:21:39 +01:00
parent c224967b19
commit 316b1d2911
11 changed files with 76 additions and 72 deletions
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10
tex/chapters/mvg.tex
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@@ -4,7 +4,7 @@
% Gauss Blur Filter
% Repetitive Box filter to approx Gauss
% Simple multipass, n/m approach, extended box filter
Digital filters are implemented by convolving the input signal with a filter kernel, i.e. the digital filter's impulse response.
Digital filters are implemented by convolving the input signal with a filter kernel, \ie{} the digital filter's impulse response.
Consequently, the filter kernel of a Gaussian filter is a Gaussian with finite support \cite{dspGuide1997}.
Assuming a finite-support Gaussian filter kernel of size $M$ and an input signal $x$, discrete convolution produces the smoothed output signal
\begin{equation}
@@ -14,8 +14,8 @@ Assuming a finite-support Gaussian filter kernel of size $M$ and an input signal
where $\sigma$ is a smoothing parameter called standard deviation.
Note that \eqref{eq:bkdeGaus} has the same structure as \eqref{eq:gausFilt}, except the varying notational symbol of the smoothing parameter and the different factor in front of the sum.
While in both equations the constant factor of the Gaussian is removed of the inner sum, \eqref{eq:bkdeGaus} has an additional normalization factor $W^{-1}$.
This factor is necessary to ensure that the estimate is a valid density function, i.e. that it integrates to one.
While in both equations the constant factor of the Gaussian is removed from the inner sum, \eqref{eq:bkdeGaus} has an additional normalization factor $W^{-1}$.
This factor is necessary to ensure that the estimate is a valid density function, \ie{} that it integrates to one.
Such a restriction is superfluous in the context of digital filters, so the normalization factor is omitted.
Computation of a digital filter using the naive implementation of the discrete convolution algorithm yields $\landau{NM}$, where $N$ is again the input size given by the length of the input signal and $M$ is the size of the filter kernel.
@@ -77,7 +77,7 @@ The overall algorithm to efficiently compute \eqref{eq:boxFilt} is listed in Alg
\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 $\sigma$.
As seen in \eqref{eq:gausFilt}, the sole parameter of the Gaussian kernel is the standard deviation $\sigma$.
In contrast, the box function \eqref{eq:boxFx} is parametrized by its width $L$.
Therefore, in order to approximate the Gaussian filter of a given $\sigma$, a corresponding value of $L$ must be found.
Given $n$ iterations of box filters with identical sizes the ideal size $\Lideal$, as suggested by Wells~\cite{wells1986efficient}, is
@@ -112,7 +112,7 @@ The approximated $\sigma$ as a function of the integer width has a staircase sha
By reducing the rounding error, the step size of the function is reduced.
However, the overall shape will not change.
\etal{Gwosdek}~\cite{gwosdek2011theoretical} proposed an approach which allows to approximate any real-valued value of $\sigma$.
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.
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.