121 lines
6.8 KiB
TeX
121 lines
6.8 KiB
TeX
\section{Gaussian Filter Approximation}
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% Basic box filter formula
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% Recursive form
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% Gauss Blur Filter
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% Repetitive Box filter to approx Gauss
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% Simple multipass, n/m approach, extended box filter
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Digital filters are implemented by convolving the input signal with a filter kernel, i.e. the digital filter's impulse response.
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Consequently, the filter kernel of a Gaussian filter is a Gaussian with finite support \cite{dspGuide1997}.
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Assuming a finite-support Gaussian filter kernel of size $M$ and an input signal $x$, discrete convolution produces the smoothed output signal
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\begin{equation}
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\label{eq:gausFilt}
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(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{,}
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\end{equation}
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where $\sigma$ is a smoothing parameter called standard deviation.
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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.
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While in both equations the constant factor of the Gaussian is removed of the inner sum, \eqref{eq:bkdeGaus} has an additional normalization factor $W^{-1}$.
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This factor is necessary to ensure that the estimate is a valid density function, i.e. that it integrates to one.
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Such a restriction is superfluous in the context of digital filters, so the normalization factor is omitted.
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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.
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In order to capture all significant values of the Gaussian function the kernel size $M$ must be adopted to the standard deviation of the Gaussian.
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A faster $\landau{N}$ algorithm is given by the well-known approximation scheme based on iterated box filters.
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While reducing the algorithmic complexity this approximation also reduces computational time significantly due to the simplistic computation scheme of the box filter.
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Following that, an implementation of the iterated box filter is one of the fastest ways to obtain an approximative Gaussian filtered signal.
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% TODO mit einem geringen Fehler
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% TODO und somit kann der mvg filter auch die BKDE approxen?
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%\subsection{Box Filter}
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The box filter is a simplistic filter defined as convolution of the input signal and the box (or rectangular) function.
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A discrete box filter of radius $l\in \N_0$ is given as
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\begin{equation}
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\label{eq:boxFilt}
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(B_L*x)(i) = \sum_{k=0}^{L-1} x(k) B_L(i-k)
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\end{equation}
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where $L=2l+1$ is the width of the filter and
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\begin{equation}
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\label{eq:boxFx}
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B_L(i)=
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\begin{cases}
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L^{-1} & \text{if } -l \le i \le l \\
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0 & \text{else }
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\end{cases}
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\end{equation}
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is the discrete box kernel.
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Such a filter clearly requires $\landau{NL}$ operations, where $N$ is the input signal length.
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It is well-known that a box filter can approximate a Gaussian filter by repetitive recursive computations.
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Given by the central limit theorem of probability, repetitive convolution of a rectangular function with itself eventually yields a Gaussian in the limit.
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Likewise, filtering a signal with the box filter several times approximately converges to a Gaussian filter.
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In practice three to five iterations are most likely enough to obtain a reasonable close Gaussian approximation \cite{kovesi2010fast}.
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This allows rapid approximation of the Gaussian filter using iterated box filter, which only requires a few addition and multiplication operations.
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Opposed to the Gaussian filter, where several evaluations of the exponential function are necessary.
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The most efficient way to compute a single output value of the box filter is:
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\begin{equation}
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\label{eq:recMovAvg}
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y(i) = y(i-1) + x(i+l) - x(i-l-1)
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\end{equation}
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This recursive calculation scheme further reduces the time complexity of the box filter to $\landau{N}$, by reusing already calculated values.
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Furthermore, only one addition and subtraction is required to calculate a single output value.
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The overall algorithm to efficiently compute \eqref{eq:boxFilt} is listed in Algorithm~\ref{alg:naiveboxalgo}.
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\begin{algorithm}[t]
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\caption{Recursive 1D box filter}
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\label{alg:naiveboxalgo}
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\begin{algorithmic}[1]
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\Statex \textbf{Input:} $f$ defined on $[1,N]$ and box radius $l$
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\Statex \textbf{Output:} $u \gets B_L*f$
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\State $a := \sum_{i=-l}^{l} f(i) $
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\State $u(1) := a/(2l+1)$
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\For{$i=2 \textbf{ to } N$}
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\State $a:= a + f(i+l) - f(i-l-1) $ \Comment{\eqref{eq:recMovAvg}}
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\State $u(i) := a/(2l+1)$
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\EndFor
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\end{algorithmic}
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\end{algorithm}
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Given a fast approximation scheme, it is necessary to construct a box filter analogous to a given Gaussian filter.
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As seen in \eqref{eq:gausFilt}, the solely parameter of the Gaussian kernel is the standard deviation $\sigma$.
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In contrast, the box function \eqref{eq:boxFx} is parametrized by its width $L$.
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Therefore, in order to approximate the Gaussian filter of a given $\sigma$, a corresponding value of $L$ must be found.
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Given $n$ iterations of box filters with identical sizes the ideal size $\Lideal$, as suggested by Wells~\cite{wells1986efficient}, is
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\begin{equation}
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\label{eq:boxidealwidth}
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\Lideal = \sqrt{\frac{12\sigma^2}{n}+1} \quad \text{.}
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\end{equation}
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In general $\Lideal$ can be any real number, but $B_L$ in \eqref{eq:boxFx} is restricted to odd integer values.
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Hence, the set of possible approximated standard deviations is limited, because the ideal width must be rounded to the next valid value.
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In order to reduce the rounding error Kovesi~\cite{kovesi2010fast} proposes to perform two box filters with different widths
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\begin{align}
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\label{eq:boxwidthtwo}
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\begin{split}
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L_1 &=
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\begin{cases}
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\floor{\Lideal} - 1 & \text{if } \floor{\Lideal} \text{ is odd} \\
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\floor{\Lideal} & \text{else }
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\end{cases} \\
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L_2 &= L_1 + 2 \quad \text{.}
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\end{split}
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\end{align}
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Given $L_1$ and $L_2$ the approximation is done by computing $m$ box filters of width $L_1$ followed by $(n-m)$ box filters of size $L_2$, where $0\le m\le n$.
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As all other values are known, $m$ can be computed with
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\begin{equation}
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\label{eq:boxrepeatm}
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m=\frac{12\sigma^2-nL_1^2-4nL_1-3n}{-4L_1-4} \quad \text{.}
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\end{equation}
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The approximated $\sigma$ as a function of the integer width has a staircase shaped graph.
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By reducing the rounding error, the step size of the function is reduced.
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However, the overall shape will not change.
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\etal{Gwosdek}~\cite{gwosdek2011theoretical} proposed an approach which allows to approximate any real-valued value of $\sigma$.
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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.
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This extension introduces only marginal computational overhead over conventional box filtering.
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