120 lines
6.7 KiB
TeX
120 lines
6.7 KiB
TeX
\section{Moving Average Filter}
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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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The Gaussian filter is a widely used smoothing filter.
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It is defined as the convolution of an input signal and the Gaussian function
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\begin{equation}
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\label{eq:gausfx}
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g(x) = \frac{1}{\sigma \sqrt{2\pi}} \expp{-\frac{x^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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In the discrete case the Gaussian filter is easily computed with the sliding window algorithm in time domain.
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It is easily extended to multi-dimensional signals, as convolution is separable if the filter kernel is separable, i.e. multidimensional convolution can be computed as individual one-dimensional convolutions with a one-dimensional kernel.
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Because of $\operatorname{e}^{x^2+y^2} = \operatorname{e}^{x^2}\cdot\operatorname{e}^{y^2}$ the Gaussian filter is separable and can be easily applied to multi-dimensional signals.
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% TODO ähnlichkeit Gauss und KDE -> schneller Gaus = schnelle KDE
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Computation of a filter using the a naive implementation of the sliding window algorithm yields $\landau{NK}$, where $N$ is the length of the input signal and $K$ is the size of the filter kernel.
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Note that in the case of the Gaussian filter $K$ depends on $\sigma$.
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In order to capture all significant values of the Gaussian function the kernel size $K$ must be adopted to the standard deviation of the Gaussian.
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A popular approach to efficiently compute a filter result is the FFT-convoultion algorithm which is $\landau{N\log(N)}$.
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For large values of $\sigma$ the computation time of the Gaussian filter might be reduced by applying the filter in frequency domain.
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In order to do so, both signals are transformed into frequency domain using the FFT.
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The convoluted time signal is equal to the point-wise multiplication of the signals in frequency domain.
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In case of the Gaussian filter the computation of the Fourier transform of the kernel can be saved, as the Gaussian is a eigenfunction for the Fourier transform \cite{?}.
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While the FFT-convolution algorithm poses an efficient algorithm for large signals, it adds an noticeable overhead for small signals.
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While the above mentions algorithms poses efficient computations schemes to compute an exact filter result, approximative algorithms can further speed up the computation.
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A well-known rapid approximation of the Guassian filter is given by the moving average filter.
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\subsection{Moving Average Filter}
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The moving average filter is a simplistic filter which takes an input function $x$ and produces a second function $y$.
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A single output value is computed by taking the average of a number of values symmetrical around a single point in the input.
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The number of values in the average can also be seen as the width $w=2r+1$, where $r$ is the \qq{radius} of the filter.
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The computation of an single output value using a moving average filter of radius $r$ is defined as
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\begin{equation}
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\label{eq:symMovAvg}
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y[i]=\frac{1}{2r+1} \sum_{j=-r}^{r}x[i+j] \text{.}
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\end{equation}
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Such a filter requires clearly $\landau{Nw}$ operations, where $N$ is the signal length.
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It is well-known that a moving average filter can approximate a Gaussian filter by repetitive recursive computations.
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\eqref{eq:symMovAvg} is equal to convolution of the input signal and the rectangular function.
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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 moving average filter in several passes approximately converges to a Gaussian filter.
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In practice three or 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 the moving average 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 moving average 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+r] - x[i-r-1]
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\end{equation}
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This recursive calculation scheme further reduces the time complexity of the moving average 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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Given a fast approximation scheme, it is necessary to construct a box filter analogous to a given Gaussian filter.
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As given in \eqref{eq:gausfx} the solely parameter of the Gaussian kernel is the standard deviation $\sigma$.
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In contrast, the moving average filter is parametrized by its width $w$.
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Therefore, in order to approximate the Gaussian filter of a given $\sigma$ a corresponding value of $w$ must be found.
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Given $n$ iterations of moving average filters with identical widths the ideal width $\wideal$, as suggested by Wells~\cite{wells1986efficient}, is
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\begin{equation}
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\label{eq:boxidealwidth}
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\wideal = \sqrt{\frac{12\sigma^2}{n}+1} \text{.}
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\end{equation}
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In general $\wideal$ can be any real number, but the moving average filter in \eqref{eq:symMovAvg} 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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w_l &=
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\begin{cases}
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\floor{w_{\text{ideal}}} - 1 & \text{if } \floor{w_{\text{ideal}}} \text{ is odd} \\
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\floor{w_{\text{ideal}}} & \text{else }
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\end{cases} \\
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w_u &= w_l + 2 \text{.}
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\end{split}
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\end{align}
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Given $w_l$ and $w_u$ the approximation is done by computing $m$ box filters of width $w_l$ followed by $(n-m)$ box filters of size $w_u$, 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-nw_l^2-4nw_l-3n}{-4w_l-4} \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 $[-r; r]$, 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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