added todos to all chapters

tex/chapters/mvg.tex

@@ -5,6 +5,7 @@
 % Repetitive Box filter to approx Gauss
 % Simple multipass, n/m approach, extended box filter
 
+\todo{normalisierungsfaktor, sigma vs. h beschreiben}
 The Gaussian filter is a widely used smoothing filter.
 It is defined as the convolution of an input signal and the Gaussian function
 \begin{equation}
@@ -13,26 +14,27 @@ g(x) = \frac{1}{\sigma \sqrt{2\pi}} \expp{-\frac{x^2}{2\sigma^2}} \text{,}
 \end{equation}
 where $\sigma$ is a smoothing parameter called standard deviation.
 
-In the discrete case the Gaussian filter is easily computed with the sliding window algorithm in time domain.
-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.
-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.
+%In the discrete case the Gaussian filter is easily computed with the sliding window algorithm in time domain.
+If the filter kernel is separable, the convolution is also separable i.e. multi-dimensional convolution can be computed as individual one-dimensional convolutions with a one-dimensional kernel.
+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. \todo{quelle}
 
 % TODO ähnlichkeit Gauss und KDE -> schneller Gaus = schnelle KDE
 
-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.
+Computation of a filter using the a naive implementation of the discrete convolution algorithm yields $\landau{NK}$, where $N$ is the length of the input signal and $K$ is the size of the filter kernel.
 Note that in the case of the Gaussian filter $K$ depends on $\sigma$.
 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.
 
 
-A popular approach to efficiently compute a filter result is the FFT-convoultion algorithm which is $\landau{N\log(N)}$.
-For large values of $\sigma$ the computation time of the Gaussian filter might be reduced by applying the filter in frequency domain.
-In order to do so, both signals are transformed into frequency domain using the FFT.
-The convoluted time signal is equal to the point-wise multiplication of the signals in frequency domain.
-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{?}.
+%A popular approach to efficiently compute a filter result is the FFT-convoultion algorithm which is $\landau{N\log(N)}$.
+%For large values of $\sigma$ the computation time of the Gaussian filter might be reduced by applying the filter in frequency domain.
+%In order to do so, both signals are transformed into frequency domain using the FFT.
+%The convoluted time signal is equal to the point-wise multiplication of the signals in frequency domain.
+%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{?}.
 
-While the FFT-convolution algorithm poses an efficient algorithm for large signals, it adds an noticeable overhead for small signals.
+%While the FFT-convolution algorithm poses an efficient algorithm for large signals, it adds an noticeable overhead for small signals.
 
-While the above mentions algorithms poses efficient computations schemes to compute an exact filter result, approximative algorithms can further speed up the computation.
+%While the above mentions algorithms poses efficient computations schemes to compute an exact filter result, approximative algorithms can further speed up the computation.
+\todo{o(nk) ist scheiße und wir wollen o(n) haben, deshalb box filter boy}
 A well-known rapid approximation of the Guassian filter is given by the moving average filter.
 
 \subsection{Moving Average Filter}