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+%% \lstset{
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Therefore, to fully optimize global runtimes, it is important to pay attention to how memory transfers are done.
This leads us to propose, in the following section, an overall code structure to be used with all our kernel examples.
-Obviously, our code originally accepts various image dimensions and can process color images.
+Obviously, our code originally accepts various image dimensions and can process color images when an extrapolated definition of the median filter is choosen.
However, so as to propose concise and more readable code, we will assume the following limitations:
-8 or 16~bit-coded gray-level input images whose dimensions $H\times W$ are multiples of 512 pixels.
+16~bit-coded gray-level input images whose dimensions $H\times W$ are multiples of 512 pixels.
\begin{algorithm}
\SetNlSty{}{}{:}
\section{Data transfers, memory management.}
This section deals with the following issues:
\begin{enumerate}
-\item Data transfer from CPU memory to GPU global memory: several GPU memory areas are available as destination memory but the 2-D caching mechanism of texture memory, specifically designed for fetching neighboring pixels, is currently the fastest way to fetch gray-level pixel values inside a kernel computation. This has lead us to choose \textbf{texture memory} as primary GPU memory area for images.
-\item Data fetching from GPU global memory to kernel local memory: as said above, we use texture memory. Depending on which process is run, texture data is used either by direct fetching in kernel local memory or through a prefetching in shared memory.
+\item Data transfer from CPU memory to GPU global memory: several GPU memory areas are available as destination memory but the 2-D caching mechanism of texture memory \index{memory~hierarchy!texture~memory}, specifically designed for fetching neighboring pixels, is currently the fastest way to fetch gray-level pixel values inside a kernel computation. This has led us to choose \textbf{texture memory} as primary GPU memory area for images.
+\item Data fetching from GPU global memory to kernel local memory: as said above, we use texture memory \index{memory~hierarchy!texture~memory}. Depending on which process is run, texture data is used either by direct fetching in kernel local memory or through a prefetching \index{prefetching} in shared memory \index{memory~hierarchy!shared~memory}.
\item Data outputting from kernels to GPU memory: there is actually no alternative to global memory, as kernels can not directly write into texture memory and as copying from texture to CPU memory would not be faster than from simple global memory.
-\item Data transfer from GPU global memory to CPU memory: it can be drastically accelerated by use of \textbf{pinned memory}, keeping in mind it has to be used sparingly.
+\item Data transfer from GPU global memory to CPU memory: it can be drastically accelerated by use of \textbf{pinned memory} \index{memory~hierarchy!pinned~memory}, keeping in mind it has to be used sparingly.
\end{enumerate}
-Algorithm \ref{algo:memcopy} summarizes all the above considerations and describe how data are handled in our examples. For more information on how to handle the different types of GPU memory, we suggest to refer to CUDA programmer's guide.
+Algorithm \ref{algo:memcopy} summarizes all the above considerations and describes how data are handled in our examples. For more information on how to handle the different types of GPU memory, we suggest to refer to the CUDA programmer's guide.
-At debug stage, for simplicity's sake, we use the \textbf{cutil} library supplied by the NVidia developpement kit (SDK). Thus, in order to easily implement our examples, we suggest readers download and install the latest NVidia-SDK (ours is SDK4.0), create a new directory \textit{SDK-root-dir/C/src/fast\_kernels} and adapt the generic \textit{Makefile} that can be found in each sub-directory of \textit{SDK-root-dir/C/src/}. Then, only two more files will be enough to have a fully operational environnement: \textit{main.cu} and \textit{fast\_kernels.cu}.
+At debug stage, for simplicity's sake, we use the \textbf{cutil} \index{Cutil library} library supplied by the NVidia development kit (SDK). Thus, in order to easily implement our examples, we suggest readers to download and to install the latest NVidia-SDK (ours is SDK4.0), create a new directory \textit{SDK-root-dir/C/src/fast\_kernels} and adapt the generic \textit{Makefile} that can be found in each sub-directory of \textit{SDK-root-dir/C/src/}. Then, only two more files will be enough to have a fully operational environnement: \textit{main.cu} and \textit{fast\_kernels.cu}.
Listings \ref{lst:main1}, \ref{lst:fkern1} and \ref{lst:mkfile} implement all the above considerations minimally, while remaining functional.
The main file of Listing \ref{lst:main1} is a simplified version of our actual main file.
-It has to be noticed that cutil functions \texttt{cutLoadPGMi} and \texttt{cutSavePGMi} only operate on unsigned integer data. As data is coded in short integer format for performance reasons, the use of these functions involves casting data after loading and before saving. This may be overcome by use of a different library. Actually, our choice was to modify the above mentioned cutil functions.
+It has to be noticed that cutil functions \texttt{cutLoadPGMi} \index{Cutil library!cutLoadPGMi} and \texttt{cutSavePGMi} \index{Cutil library!cutSavePGMi} only operate on unsigned integer data. As data is coded in short integer format for performance reasons, the use of these functions involves casting data after loading and before saving. This may be overcome by use of a different library. Actually, our choice was to modify the above mentioned cutil functions.
Listing \ref{lst:fkern1} gives a minimal kernel skeleton that will serve as the basis for all other kernels. Lines 5 and 6 determine the coordinates $(i, j)$ of the pixel to be processed, each pixel being associated to one thread.
The instruction in line 8 combines writing the output gray-level value into global memory and fetching the input gray-level value from 2-D texture memory.
\section{Performance measurements}
-As our goal is to design very fast implementations of basic image processing algorithms, we need to make quite accurate time-measurements, within the order of magnitude of $0.01~ms$. Again, the easiest way of doing so is to use the helper functions of the cutil library. As usual, as the durations we are measuring are short and possibly suject to non neglectable variations, a good practice is to measure multiple executions and issue the mean runtime. All time results given in this chapter have been obtained through 1000 calls to each kernel.
+As our goal is to design very fast implementations of basic image processing algorithms, we need to make quite accurate time-measurements, within the order of magnitude of $0.01~ms$. Again, the easiest way of doing so is to use the helper functions of the cutil library. As usual, as the durations we are measuring are short and possibly subject to non neglectable variations, a good practice is to measure multiple executions and issue the mean runtime. All time results given in this chapter have been obtained through 1000 calls to each kernel.
-Listing \ref{lst:chronos} shows how to use the dedicated cutil functions. Timer declaration and creation only need to be performed once while reset, start and stop can be used as often as necessary. Synchronization is mandatory before stopping the timer (Line 7), to avoid runtime measurement being biased.
+Listing \ref{lst:chronos} shows how to use the dedicated cutil functions \index{Cutil library!Timer usage}. Timer declaration and creation only need to be performed once while reset, start and stop can be used as often as necessary. Synchronization is mandatory before stopping the timer (Line 7), to avoid runtime measurement being biased.
\lstinputlisting[label={lst:chronos},caption=Time measurement technique using cutil functions]{Chapters/chapter3/code/exChronos.cu}
In an attempt to provide relevant speedup values, we either implemented CPU versions of the algorithms studied, or used the values found in existing literature. Still, the large number and diversity of hardware platforms and GPU cards makes it impossible to benchmark every possible combination and significant differences may occur between the speedups we announce and those obtained with different devices. As a reference, our developing platform details as follows:
In order to estimate the potential for improvement of each kernel, a reference throughput measurement, involving identity kernel of Listing \ref{lst:fkern1}, was performed. As this kernel only fetches input values from texture memory and outputs them to global memory without doing any computation, it represents the smallest, thus fastest, possible process and is taken as the reference throughput value (100\%). The same measurement was performed on CPU, with a maximum effective pixel throughput of 130~Mpixel per second. On GPU, depending on grid parameters it amounts to 800~MPixels/s on GTX280 and 1300~Mpixels/s on C2070.
-
\chapter{Implementing a fast median filter}
+\chapterauthor{Gilles Perrot}{FEMTO-ST Institute}
\section{Introduction}
Median filtering is a well-known method used in a wide range of application frameworks as well as a standalone filter especially for \textit{salt and pepper} denoising. It is able to highly reduce power of noise without blurring edges too much.
First introduced by Tukey in \cite{tukey77}, it has been widely studied since then, and many researchers have proposed efficient implementations of it, adapted to various hypothesis, architectures and processors.
-Originally, its main drawbacks were its compute complexity, its non linearity and its data-dependent runtime. Several researchers have adress these issues and designed, for example, efficient histogram-based median filter with predictible runtime \cite{Huang:1981:TDS:539567, Weiss:2006:FMB:1179352.1141918}.
+Originally, its main drawbacks were its compute complexity, its non linearity and its data-dependent runtime. Several researchers have addressed these issues and designed, for example, efficient histogram-based median filter with predictible runtimes \cite{Huang:1981:TDS:539567, Weiss:2006:FMB:1179352.1141918}.
-More recently, the advent of GPUs opened new perspectives in terms of image processing performance, and some researchers managed to take advantage of the new graphic capabilities: in that respect, we can cite the Branchless Vectorized Median filter (BVM) \cite{5402362, chen09} which allows very interesting runtimes on CUDA-enabled devices but, as far as we know, the fastest implementation to date is the histogram-based CCDS median filter \cite{6288187}.
+More recently, the advent of GPUs opened new perspectives in terms of image processing performance, and some researchers managed to take advantage of the new graphic capabilities: in that respect, we can cite the Branchless Vectorized Median filter (BVM) \cite{5402362, chen09} which allows very interesting runtimes on CUDA-enabled devices but, as far as we know, the fastest implementation to date is the histogram-based PCMF median filter \cite{Sanchez-2-2012}.
Some of the following implementations feature very fast runtimes; They are targeted on Nvidia Tesla GPU (Fermi architecture, compute capability 2.x) but may easily be adapted to other models e.g. those of compute capability 1.3.
As mentioned earlier, the selection of the median value can be performed by more than one technique, using either histogram-based or sorting methods, each of them having its own benefits and drawbacks as will be discussed further down.
\subsection{A naive implementation}
-As a reference, Listing \ref{lst:medianGeneric} gives a simple, not to say simplistic implementation of a CUDA kernel (\texttt{kernel\_medianR}) achieving generic $n\times n$ histogram-based median filtering. Its runtime has a very low data dependency, but this implementation does not suit very well GPU architecture. Each pixel loads the whole of its $n\times n$ neighborhood meaning that one pixel is loaded multiple times inside one single thread block, and above all, the use of a local vector (histogram[]) considerably downgrades performance, as the compiler automatically stores such vectors in local memory (slow).
+As a reference, Listing \ref{lst:medianGeneric} gives a simple, not to say simplistic implementation of a CUDA kernel (\texttt{kernel\_medianR}) achieving generic $n\times n$ histogram-based median filtering. Its runtime has a very low data dependency, but this implementation does not suit very well GPU architecture. Each pixel loads the whole of its $n\times n$ neighborhood meaning that one pixel is loaded multiple times inside one single thread block, and above all, the use of a local vector (histogram[]) considerably downgrades performance, as the compiler automatically stores such vectors in local memory (slow) \index{memory~hierarchy!local~memory}.
-Table \ref{tab:medianHisto1} displays measured runtimes of \texttt{kernel\_medianR} and pixel throughputs for each GPU version and for both CPU and GPU implementations. Usual window sizes of $3\times 3$, $5\times 5$ and $7\times 7$ are shown. Though some specific applications require larger window sizes and dedicated algorithms , such small square window sizes are most widely used in general purpose image processing. GPU runtimes have been obtained with a grid of 64-thread blocks. This block size, is a good compromise in this case.
+Table \ref{tab:medianHisto1} displays measured runtimes of \texttt{kernel\_medianR} and pixel throughputs for each GPU version and for both CPU and GPU implementations. Usual window sizes of $3\times 3$, $5\times 5$ and $7\times 7$ are shown. Though some specific applications require larger window sizes and dedicated algorithms, such small square window sizes are most widely used in general purpose image processing. GPU runtimes have been obtained with a grid of 64-thread blocks. This block size, is a good compromise in this case.
The first observation to make when analysing results of Table \ref{tab:medianHisto1} is that, on CPU, window size has almost no influence on the effective pixel throughput.
Since inner loops that fill the histogram vector contain very few fetching instructions (from 9 to 49, depending on the window size), it is not surprising to note their neglectable impact compared to outer loops that fetch image pixels (from 256k to 16M instructions).
\section{NVidia GPU tuning recipes}
When designing GPU code, besides thinking of the actual data computing process, one must choose the memory type into which to store temporary data. Three types of GPU memory are available:
\begin{enumerate}
-\item \textbf{Global memory, the most versatile:}\\Offers the largest storing space and global scope but is slowest (400 cycles latency). \textbf{Texture memory} is physically included into it, but allows access through an efficient 2-D caching mechanism.
-\item \textbf{Registers, the fastest:}\\Allow access wihtout latency, but only 63 registers are available per thread (thread scope), with a maximum of 32K per Symetric Multiprocessor (SM).
-\item \textbf{Shared memory, a complex compromise:}\\All threads in one block can access 48~KBytes of shared memory, which is faster than global memory (20 cycles latency) but slower than registers.
+\item \textbf{Global memory, the most versatile:} \index{memory~hierarchy!global~memory}\\Offers the largest storing space and global scope but is slowest (400 cycles latency). \textbf{Texture memory} is physically included into it, but allows access through an efficient 2-D caching mechanism.
+\item \textbf{Registers, the fastest:} \index{memory~hierarchy!registers}\\Allow access wihtout latency, but only 63 registers are available per thread (thread scope), with a maximum of 32K per Symetric Multiprocessor (SM) \index{register count}w.
+\item \textbf{Shared memory, a complex compromise:} \index{memory~hierarchy!shared~memory}\\All threads in one block can access 48~KBytes of shared memory, which is faster than global memory (20 cycles latency) but slower than registers.
However, bank conflicts can occur if two threads of a warp try to access data stored in one single memory bank. In such cases, the parallel process is re-serialized which may cause significant performance decrease. One easy way to avoid it is to ensure that two consecutive threads in one block always access 32-bit data at two consecutive adresses.
\end{enumerate}
\noindent As observed earlier, designing a median filter GPU implementation using only global memory is fairly straightforward, but its performance remains quite low even if it is faster than CPU.
-To overcome this, the most frequent choice made in efficient implementations found in literature is to use shared memory. Such option implies prefetching data prior to doing the actual computations, a relevant choice, as each pixel of an image belongs to n$\times$n different neighborhoods. Thus, it can be expected that fetching each gray-level value from global memory only once should be more efficient than doing it each time it is required. One of the most efficient implementations using shared memory is presented in \cite{5402362}. In the case of the generic kernel of Listing \ref{lst:medianGeneric}, using shared memory without further optimization would not bring valuable speedup because that would just move redundancy from texture to shared memory fetching and would generate bank conflicts. For information, we wrote such a version of the generic median kernel and our measurements showed a speedup of around 3\% (as an example: 32ms for 5$\times$5 median on a 1024$^2$ pixel image, i.e. 33~Mpixel/s ).
+To overcome this, the most frequent choice made in efficient implementations found in literature is to use shared memory. Such option implies prefetching \index{prefetching}data prior to doing the actual computations, a relevant choice, as each pixel of an image belongs to n$\times$n different neighborhoods. Thus, it can be expected that fetching each gray-level value from global memory only once should be more efficient than doing it each time it is required. One of the most efficient implementations using shared memory is presented in \cite{5402362}. In the case of the generic kernel of Listing \ref{lst:medianGeneric}, using shared memory without further optimization would not bring valuable speedup because that would just move redundancy from texture to shared memory fetching and would generate bank conflicts. For information, we wrote such a version of the generic median kernel and our measurements showed a speedup of around 3\% (as an example: 32ms for 5$\times$5 median on a 1024$^2$ pixel image, i.e. 33~Mpixel/s ).
-As for registers, designing a generic median filter that would only use that type of memory seems difficult, due to the above mentioned 63 register-per-thread limitation.
+As for registers, designing a generic median filter that would only use that type of memory seems difficult, due to the above mentioned 63 register-per-thread limitation \index{register count}.
Yet, nothing forbids us to design fixed-size filters, each of them specific to one of the most popular window sizes. It might be worth the effort as dramatic increase in performance could be expected.
-Another track to follow in order to improve performance of GPU implementations consists in hiding latencies generated by arithmetic instruction calls and memory accesses. Both can be partially hidden by introducing Instruction-Level Parallelism (ILP) and by increasing the data count output by each thread. Though such techniques may seem to break the NVidia occupancy paradigm, they can lead to dramatically higher data throughput values.
+Another track to follow in order to improve performance of GPU implementations consists in hiding latencies generated by arithmetic instruction calls and memory accesses. Both can be partially hidden by introducing Instruction-Level Parallelism \index{Instruction-Level Parallelism}(ILP) and by increasing the data count output by each thread. Though such techniques may seem to break the NVidia occupancy paradigm, they can lead to dramatically higher data throughput values.
The following sections illustrate these ideas and detail the design of the fastest CUDA median filter known to date.
\section{A 3$\times$3 median filter: using registers }
Designing a median filter dedicated to the smallest possible square window size is a good challenge to start using registers.
-One first issue is that the exclusive use of registers forbids us to implement a naive histogram-based method. In a \textit{8-bit gray level pixel per thread} rule, each histogram requires one 256-element vector to store its values, i.e. four times the maximum register count allowed per thread (63). Considering that a 3$\times$3 median filter involves only 9 pixel values per thread, it seem obvious they can be sorted within the 63-register limit.
+One first issue is that the exclusive use of registers forbids us to implement a naive histogram-based method. In a \textit{8-bit gray level pixel per thread} rule, each histogram requires one 256-element vector to store its values, i.e. four times the maximum register count allowed per thread (63)\index{register count}. Considering that a 3$\times$3 median filter involves only 9 pixel values per thread, it seem obvious they can be sorted within the 63-register limit.
\subsection{The simplest way}
-In the case of a 3$\times$3 median filter, the simplest solution consists in associating one register to each gray-level value, then sorting those 9 values and selecting the fifth one, i.e. the median value. For such a small amount of data to sort, a simple selection method is well indicated. As shown in Listing \ref{lst:kernelMedian3RegTri9} (\texttt{kernelMedian3RegTri9()}), the constraint of only using registers leads to adopt an unusual manner of coding. However, results are persuasive: runtimes are divided by around 120 on GTX280 and 80 on C2070, while only reduced by a 3.5 factor on CPU.
-The diagram of Figure \ref{fig:compMedians1} summarizes these first results. Only C2070 throughputs are shown and compared to CPU results. We included the maximum effective pixel throughput in order to see the improvement potential of the different implementations. We also introduced throughputd achieved by \textit{libJacket}, a commercial implementation, as it was the fastest known implementation of a 3$\times$3 median filter to date, as illustrated in \cite{chen09}. One of the authors of libJacket kindly posted the CUDA code of its 3$\times$3 median filter, that we inserted into our own coding structure. The algorithm itself is quite similar to ours, but running it in our own environement produced higher throughput values than those published in \cite{chen09}, not due to different hardware capabilities between our GTX280 and the GTX260 used in the paper, but to the way we perform memory transfers and to our register-only method of storing temporary data.
+In the case of a 3$\times$3 median filter, the simplest solution consists in associating one register to each gray-level value, then sorting those 9 values and selecting the fifth one, i.e. the median value. For such a small amount of data to sort, a simple selection method is well indicated. As shown in Listing \ref{lst:kernelMedian3RegTri9} (\texttt{kernelMedian3RegSort9()}), the constraint of only using registers leads to adopt an unusual manner of coding. However, results are persuasive: runtimes are divided by around 120 on GTX280 and 80 on C2070, while only reduced by a 3.5 factor on CPU.
+The diagram of Figure \ref{fig:compMedians1} summarizes these first results, obtained with a block size of 128 threads on GTX280 and 256 on C2070. Only C2070 throughputs are shown and compared to CPU results. We included the maximum effective pixel throughput in order to see the improvement potential of the different implementations. We also introduced throughputd achieved by \textit{libJacket}, a commercial implementation, as it was the fastest known implementation of a 3$\times$3 median filter to date, as illustrated in \cite{chen09}. One of the authors of libJacket kindly posted the CUDA code of its 3$\times$3 median filter, that we inserted into our own coding structure. The algorithm itself is quite similar to ours, but running it in our own environement produced higher throughput values than those published in \cite{chen09}, not due to different hardware capabilities between our GTX280 and the GTX260 used in the paper, but to the way we perform memory transfers and to our register-only method of storing temporary data.
\lstinputlisting[label={lst:kernelMedian3RegTri9},caption= 3$\times$3 median filter kernel using one register per neighborhood pixel and bubble sort]{Chapters/chapter3/code/kernMedianRegTri9.cu}
\begin{figure}
\centering
- \includegraphics[width=11cm]{Chapters/chapter3/img/debitPlot1.pdf}
- \caption{Comparison of pixel throughputs on GPU C2070 and CPU for generic median, in 3$\times$3 median register-only and \textit{libJacket}.}
+ \includegraphics[width=15cm]{Chapters/chapter3/img/debitPlot1.pdf}
+ \caption{Comparison of pixel throughputs on GPU C2070 and CPU for generic median, 3$\times$3 median register-only and \textit{libJacket}.}
\label{fig:compMedians1}
\end{figure}
\end{itemize}
-\subsubsection{Reducing register count}
-Our current kernel (\texttt{kernelMedian3RegTri9}) uses one register per gray-level value, which amounts to 9 registers for the entire 3$\times$3 window.
+\subsubsection{Reducing register count \index{register count}}
+Our current kernel (\texttt{kernelMedian3RegSort9}) uses one register per gray-level value, which amounts to 9 registers for the entire 3$\times$3 window.
This count can be reduced by use of an iterative sorting process called \textit{forgetful selection}, where both \textit{extrema} are eliminated at each sorting stage, until only 3 elements remain. The question is to find out the minimal register count $k_{n^2}$ that allows the selection of the median amoung $n^2$ values. The answer can be evaluated considering that, when eliminating the maximum and the minimum values, one has to make sure not to eliminate the global median value. Such a situation is illustrated in Figure \ref{fig:forgetful_selection} for a 3$\times$3 median filter. For better comprehension, the 9 elements of the 3$\times$3 pixel window have been represented in a row.
\begin{figure}
\centering
$$k_{n^2}=\lceil \frac{n^2}{2}\rceil+1 $$
This rule can be applied to the first eliminating stage and remains true with the next ones as each stage suppresses exactly two values, one above and one below the median value.
In our 3$\times$3 pixel window example, the minimum register count becomes $k_9=\lceil 9/2\rceil+1 = 6$.
+This iterative process is illustrated in Figure \ref{fig:forgetful3}, where it achieves one entire $3\times 3$ median selection, beginning with $k_9=6$ elements.
+\begin{figure}
+ \centering
+ \includegraphics[width=5cm]{Chapters/chapter3/img/forgetful_selectionb.png}
+ \caption{Determination of the Median value by the forgetful selection process, applied to a $3\times 3$ neighborhood window.}
+ \label{fig:forgetful3}
+\end{figure}
+
The \textit{forgetful selection} method, used in \cite{mcguire2008median} does not imply full sorting of values, but only selecting minimum and maximum values, which, at the price of a few iteration steps ($n^2-k$), reduces arithmetic complexity.
-Listing \ref{lst:medianForget1pix3} details this process where forgetful selection is achieved by use of simple 2-value sorting function ($s()$, lines 1 to 5) that swaps input values if necessary. Moreover, whenever possible, in order to increase the Instruction-Level Parallelism, successive calls to $s()$ are done with independant elements as arguments. This is illustrated by the macro definitions of lines 7 to 14.
+Listing \ref{lst:medianForget1pix3} details this process where forgetful selection is achieved by use of simple 2-value swapping function ($s()$, lines 1 to 5) that swaps input values if necessary, so as to achieve the first steps of a \textit{bitonic}-like sorting network (\cite{Batcher:1968:SNA:1468075.1468121}). Moreover, whenever possible, in order to increase the Instruction-Level Parallelism \index{Instruction-Level Parallelism}, successive calls to $s()$ are done with independant elements as arguments. This is illustrated by the macro definitions of lines 7 to 14 and by Figure \ref{fig:bitonic} which details the first iteration of the $5\times 5$ selection, starting with $k_{25}=14$ elements.
-\lstinputlisting[label={lst:medianForget1pix3},caption= 3$\times$3 median filter kernel using the minimum register count of 6 to find the median value by forgetful selection method]{Chapters/chapter3/code/kernMedianForget1pix3.cu}
+\lstinputlisting[label={lst:medianForget1pix3},caption= 3$\times$3 median filter kernel using the minimum register count of 6 to find the median value by forgetful selection method. The optimal thread block size is 128 on GTX280 and 256 on C2070. ]{Chapters/chapter3/code/kernMedianForget1pix3.cu}
Our such modified kernel provides significantly improved runtimes: a speedup of around 16\% is obtained, and pixel throughput reaches around 1000~MPixel/s on C2070.
\begin{figure}
\centering
- \includegraphics[width=11cm]{Chapters/chapter3/img/debitPlot2.pdf}
+ \includegraphics[width=15cm]{Chapters/chapter3/img/debitPlot2.pdf}
\caption{Comparison of pixel throughput on GPU C2070 for the different 3$\times$3 median kernels.}
\label{fig:compMedians2}
\end{figure}
The minimum register count required to apply the forgetful selection method to a 5$\times$5 median filter is $k_{25}=\lceil 25/2\rceil+1 = 14$. Moreover, two adjacent overlapping windows share 20 pixels ($n^2-one\_column$) so that, when processing 2 pixels simultaneously, a count of 7 common selection stages can be carried out from the first selection stage with 14 common values to the processing of the last common value. That allows to limit register count to 22 per thread. Figure \ref{fig:median5overlap} describes the distribution of overlapping pixels, implemented in Listing \ref{lst:medianForget2pix5}: common selection stages take place from line 25 to line 37, while the remaining separate selection stages occur between lines 45 and 62 after the separation of line 40.
\begin{figure}
\centering
- \includegraphics[width=6cm]{Chapters/chapter3/img/median5_overlap.png}
+ \includegraphics[width=6cm]{Chapters/chapter3/img/median5_overlap4.png}
\caption{Reducing register count in a 5$\times$5 register-only median kernel outputting 2 pixels simultaneously. The first 7 forgetful selection stages are common to both processed center pixels. Only the last 5 selections have to be done separately.}
\label{fig:median5overlap}
\end{figure}
+\begin{figure}
+ \centering
+ \includegraphics[width=6cm]{Chapters/chapter3/img/forgetful_selection4.png}
+ \caption{First iteration of the $5\times 5$ selection process, with $k_{25}=14$, which shows how Instruction Level Parallelism is maximized by the use of an incomplete sorting network. Arrows represent the result of the swapping function, with the lowest value at the starting point and the highest value at the end point.}
+ \label{fig:median5overlap}
+\end{figure}
+
\lstinputlisting[label={lst:medianForget2pix5},caption=kernel 5$\times$5 median filter processing 2 output pixel values per thread by a combined forgetfull selection.]{Chapters/chapter3/code/kernMedian2pix5.cu}
Timing results follow the same variations with image size as in previously presented kernels. That is why Table \ref{tab:median5comp} shows only throughput values obtained for C2070 card and 4096$\times$4096 pixel image.
