---What is Waifu2x?---

https://github.com/nagadomi/waifu2x
Waifu2x is a neural network based image upscaler for anime images. It is primarily designed to run on CUDA.
The quality increase is practically unparalleled.
Demonstration Image: https://raw.githubusercontent.com/nagadomi/waifu2x/master/images/slide.png
You can test it on images yourself at http://waifu2x.udp.jp/
Most crucially, it's been ported to run on .avi videos, hence it can upscale anime to quality previously unavalible.

---What is this process for?---

This process takes a .avi video and runs waifu2x on it to generate an unparalleled high quality upscale.
Uncompressed .avi files have monsterously impractical file sizes (200gb per episode would be typical)
Hence this process carries out a convulted system of encoding to generate very small file sizes for the quality of video.
The audio gets split from the video by the process and then needs to be remerged.

---What is this process worth using on?---

Anime where the best release avalible is DVD quality or equivalent or worse will see the most benefit from this process.
Anime that already recieved a BD or HD quality release will recieve less benefit from this process (and take longer)
Maybe there's someone out there with a fetish for UHD anime?

---Requirements---

A .avi video
An Nvidia GPU that can utilize CUDA
A lot of time to set up all the programs you need.
A lot of time (dependant upon GPU model) to process the video using waifu2x-w2xc (0.25fps on a GTX770 on DVD quality)
Can be done without CUDA but expect ~0.02fps

---Step 1: Getting AviUtl---

1. Go to http://www.htasoft.com/aviutl/aviutl.htm for information
2. Download AviUtl here: http://spring-fragrance.mints.ne.jp/aviutl/
3. Download http://www.htasoft.com/aviutl/aviutlengsettings.zip and unzip it all into the AviUtl folder, replacing files
4. You now have an English-translated AviUtl

---Step 2: Getting waifu2x-w2xc plugin---

1. Go to https://github.com/YSRKEN/aviutl-waifu2x-w2xc and google translate it to read up on it
2. Ensure you have Visual C++ redistributable package 2013
3. Install by downloading the plugins folder and extracting it
4. The folder titled "plugins" needs to be within the AviUtl directory and contain both models_rgb folders and the .auf

---Step 3: Getting x264vfw---

1. This doesn't need an explanation: http://sourceforge.net/projects/x264vfw/

---Step 4: Getting Avi-Mux GUI---

1. This doesn't need an explanation: http://www.alexander-noe.com/video/amg/

---Step 5: Getting AviUtl-waifu2x-w2xc to happen---

1. Load up the target .avi
2. Under Filter select waifu2x-w2xc
3. Under settings select waifu2x-w2xc

Explanation of next dialogue:

Noise 0 is no noise reduction. Noise 1 is Normal noise reduction. Noise 2 is High noise reduction
High noise reduction is most suited to JPEG, not to video. You may want to test all the settings to determine the best
Scale 0 is no increase in resolution. Scale 1 is a 2x increase in resolution (what we want)
Block is related to how CUDA works in some ridiculously complicated way
Have Use GPU ticked or you will die of old age before the video finishes processing
Leave photography unticked

4. Set Block to the vertical resolution of the input footage.
To adjust the slider quickly instead of reprocessing every tick, you may want to close the file then reopen when done
5. Close the dialogue

If at this point the program crashes or hangs, set block back to 128 and use that in future.

You will get an error with different text depending upon whether you are using Japanese AppLocal
Japanese: サイズが大きすぎます！
English: ƒTƒCƒY‚ª‘å‚«‚·‚¬‚Ü‚·I

This translates to "Size is too big".
This means waifu2x-w2xc is increasing the resolution of the file beyond what AviUtl settings allow
6. File - General settings - system configuration
7. Set MAX picture size X and Y to double the resolution of the source AVI. Hit OK
8. Reboot AviUtl and reload the Avi. You are now good to go.

---Step 6: Exporting from AviUtl---

1. File -> Save Avi
2. Click "Video Codec"
3. From the dropdown select x264vfw - H.264/MPEG-4 AVC codec
4. Click "Configure"
5. Under Rate Control, set the Rate Factor to about 18.
6. Under Output, set output mode to File and select an output file location/name along the lines of AnimeEP01-Video.avi
7. Hit ok and ok again
8. Back in the main window for saving, use same location with file name along the lines of AnimeEp01-Audio.avi
9. MAKE SURE EVERYTHING YOU WANT IS SET UP CORRECTLY BEFORE CONTINUING BECAUSE YOU'RE IN FOR THE LONG HAUL NOW.
10. Click save and wait anywhere between several hours and several weeks (years if not using CUDA)
11. You now have a .avi file which has the audio but a blank video, and a .avi file with the video but no audio.

---Step 7: Remerging the audio and video---

1. Load up AVI-Mux GUI
2. Open both the audio and video files via drag and drop
3. Select each of the files in the program and click "Generate data source from files" (you need to do this 1 at a time)
4. In the lower GUI find the line that says something like "FourCC: x264 (unknown bitrate. 0.00kByte, 12:34:56.789
5. Untick the tick box next to that line
6. Click start to save the file, giving it whatever name you want
7. Save. This process is very quick compared to the previous step
8. You are done. Enjoy your extremely high quality anime

---Extra 1: waifu2x-w2xc in multiple stages---

Leaving waifu2x-w2xc doing it's stuff in AviUtl for several days straight might not be the best idea.
Any interuption to the process and you would have to start over
Hence, it may be important to do the process in multiple stages.

1. Install AviDemux http://www.videohelp.com/software/AviDemux
2. Do the above process as normal until you reach Step 6
3. Get the total number of frames at the top (it looks like [1/34567] where the second number is total)
4. Split the number of frames into equalish sized segments (like 1 - 17283 and 17284-34567)
5. FOR EACH SEGMENT...
Edit -> select processing range
Enter the start and end frame of that segment and hit ok
Do step 6 as per normal, giving each segment a different name
6. Carry out step 7 on every segment to create the merged segments
7. Boot up AviDemux
8. Open the FIRST merged segment
9. Use File -> Append to append the SECOND through LAST merged segment
10. File -> Save
11. You are done. Enjoy your extremely high quality anime

---Extra 2: Speeding shit up---

This document should already have made it very clear how slow this process is
Doing it on an entire anime series with typical hardware would take almost as long as it does for each episode to come out
Hence you would need a dedicated CUDA / Neural Network rig to do serious, frequent releases at this quality
https://timdettmers.wordpress.com/2014/08/14/which-gpu-for-deep-learning/ gives some advice on making that rig
In short, you probably want the Titan X. I don't know if having multiple Titan X will speed up waifu2x further.