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PaintDotNet.Quantization

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Quantization code extracted from Paint.NET 4.2.16 w/ many fixes, improvements and optimizations to the standard Octree/Quantization code from MSDN

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# PaintDotNet.Quantization
**NOTE:** This is still pre-first draft! I'll update this to say final draft when it's all finished 🙂

This repository has the image quantization code from Paint.NET, along with the supporting utility and helper classes that are used along the way. Since images are always stored in-memory at 32-bit color depth (BGRA), quantization is necessary in order to permit saving images at 8-bit (or less) color depths. You can also use the Quantize effect (added in 4.2.16) to do this in-place (the image is still 32-bit BGRA, albeit using only up to 256 unique colors).

Please note that this is NOT intended to be used as a library or "nuget package" that you drop into your app and use as-is. You *can* do that (as per the license), but the code in this repository is meant for *integration*, not *linking.* As such, don't expect the public API surface to be beautiful or amazing, don't expect lots of attention given to Issues or especially Pull Requests. Things like `IBitmapSource` and `IImagingFactory`, in particular, are mere shadows of their counterparts in the full Paint.NET source code.

## Background

Paint.NET's quantization code is based on an old MSDN article from 2003, _Optimizing Color Quantization for ASP.NET Images_ (https://docs.microsoft.com/en-us/previous-versions/dotnet/articles/aa479306(v=msdn.10)). The code in the article has made its way into a number of projects besides Paint.NET, including ImageSharp (https://github.com/SixLabors/ImageSharp).

Unfortunately, it has a few bugs and quirks that this repository has fixes for. I recently dove in and completely gutted this code, making fixes and improvements, and I thought it would be useful to share this with everyone else. I've also optimized performance quite a lot, especially the code that maps from colors to palette entries (see section 7 below).

If you're new to quantization, here is some recommended reading:

- **Optimizing Color Quantization for ASP.NET Images**, Microsoft -- https://docs.microsoft.com/en-us/previous-versions/dotnet/articles/aa479306(v=msdn.10)
- **Color quantization**, Wikipedia -- https://en.wikipedia.org/wiki/Color_quantization
- **Octree** (data structure employed by this code), Wikipedia -- https://en.wikipedia.org/wiki/Octree
- **Median Cut** (another algorithm often used for palette generation), Wikipedia -- https://en.wikipedia.org/wiki/Median_cut
- **ImageSharp's Quantization code**, GitHub -- https://github.com/SixLabors/ImageSharp/tree/master/src/ImageSharp/Processing/Processors/Quantization

## Discussion

So, here are the changes I've made to ye ol' MSDN quantization code, in order from simple to crazy:

### 1) Right Shift Bug

The first fix is an easy one. The MSDN code has a `GetColorIndex` method whose job is to provide the child node index for a color while navigating down the octree. However, it has what is essentially a precision bug that causes colors to land in the wrong leaf nodes:

```
private static int GetColorIndex(ref Rgba32 color, int level)
{
    DebugGuard.MustBeLessThan(level, Mask.Length, nameof(level));

int shift = 7 - level;
    ref byte maskRef = ref MemoryMarshal.GetReference(Mask);
    byte mask = Unsafe.Add(ref maskRef, level);
    
    return ((color.R & mask) >> shift)
            | ((color.G & mask) >> (shift - 1))
            | ((color.B & mask) >> (shift - 2));
}
```

*(I'm using the code from ImageSharp, btw, not the MSDN article.)*

The problem is that when `level` is large, `shift` will be small, and `(shift - 1)` and `(shift - 2)` will be negative. The intention is to do a shift-left operation, and maybe that's how it works in C, but definitely not in C#. The result is zero.

The fix is to only shift-right by `shift`, and then do a shift-left by 1 or 2:

```
private static int GetColorIndex(ref Rgba32 color, int level)
{
    DebugGuard.MustBeLessThan(level, Mask.Length, nameof(level));

int shift = 7 - level;
    ref byte maskRef = ref MemoryMarshal.GetReference(Mask);
    byte mask = Unsafe.Add(ref maskRef, level);
    
    return ((color.R & mask) >> shift)
            | (((color.G & mask) >> shift) << 1)
            | (((color.B & mask) >> shift) << 2);
}
```

This code will now produce the correct output. Be warned, however, that memory usage will be higher since there will be substantially more leaf nodes. Reducing the octree and generating the palette will also be slower.

This is not just a theoretical quality improvement, by the way. I saw images being reduced to 64 colors instead of 256! I believe it was an image of a black-to-red gradient that caused this to happen, which should have fit nicely into 256 colors.

### 2) Unnecessary lookup table
There's also some silliness with regard to the use of a lookup table for calculating the mask in `GetColorIndex`:

```
/// <summary>
/// Gets the mask used when getting the appropriate pixels for a given node.
/// </summary>
private static ReadOnlySpan<byte> Mask => new byte[]
{
    0b10000000,
    0b1000000,
    0b100000,
    0b10000,
    0b1000,
    0b100,
    0b10,
    0b1
};
```
I'm guessing that, like most people, when I integrated this code into my project I thought it was a bunch of complicated wizardry that I didn't have the patience to read through and fully comprehend. However, upon finally doing this thorough inspection, it's obvious that this is a lookup table for `(1 << (7-i))`, which definitely doesn't need a lookup table.

So, kill the `Mask` and just compute the value when it's needed. We can even use `shift` because it already equals `(1 - level)`:

```
private static int GetColorIndex(ref Rgba32 color, int level)
{
    int shift = 7 - level;
    byte mask = (byte)(1 << shift)
    
    return ((color.R & mask) >> shift)
            | (((color.G & mask) >> shift) << 1))
            | (((color.B & mask) >> shift) << 2));
}
```

This will save the static field access, pointer math, and memory dereference.

### 3) Octree.AddColor() is slow

`Octree.AddColor()` is called once for every pixel in the image. The problem is, it's slow. It's not poorly written, it just requires a lot of jumping around to get its job done. When you have millions or even billions of pixels in an image, it gets *really* bad.

I found it was much faster to pre-process the image and create a color histogram, essentially a `[color, count]` list, and amend `AddColor()` to take the count value. Then, each color is only sent down the octree once and performance is much better. Generating the histogram is also easily parallelizable, greatly improving performance on higher core count systems.

Be sure to multiply the `red`, `green`, and `blue` values by the `count` if you take this approach. `red = color.R * count`, in other words, and set `pixelCount` to `count`.

In this repo, look for the `ColorHistogram` class for my implementation. I support full 64-bit counts, as Paint.NET is intended to work correctly with very large images, so storing the histogram efficiently is important: since there are 2^24 maximum RGB colors, and a `[colorBGR, long]` tuple would take 11 bytes, that would mean about 176 MiB for an image that uses all colors. Instead, I store 3 separate lists: one for colors that only show up once (no need to store count), another list that uses 32-bit `uint`s, and a final list that uses 64-bit `long`s. Enumeration for the client is still homogenous, it's just `IEnumerable<Entry>`, where `Entry` is a struct with a `ColorBgr24` and a `long` for the count.

More memory is required for this approach (building and using a histogram), but it's temporary, and is still quite a bit less than the `OctreeNode`s (see section 6). It is probably possible to improve this, maybe by destroying/trimming the histogram while enumerating it, but I didn't think it was worth pursuing.

### 4) Exact palette size is not always achieved

A problem with using the standard Octree quantization algorithm is that the `Reduce()` method, on a per-node basis, is all or nothing. If you have reduced the octree down to 260 leaf nodes and you're now reducing a node that has 8 children, the color count will be reduced by 7. You'll end up with 253 colors instead of 256. This is *very* common in practice.

I found a CodeProject article by someone who figured out a way to fix this: https://www.codeproject.com/Articles/109133/Octree-Color-Palette. Search for "Merging for Exact Colors Count."

The code in this repo doesn't use this approach because the next section fixes both this and another problem. If you're looking to ease into all of these fixes one-at-a-time, you don't need to skip this one before tackling the next section (which is much more complicated to implement). I also include this in the discussion because it was a critical point in my learning and discovery process for working with the quantization code.

### 5) Colors are unevenly reduced

The MSDN code makes use of a `reducibleNodes` array that is built while populating the Octree. It's essentially a 2D jagged list of all the nodes in the octree: the first index is the level of the octree (zero being just the root node), with the second index being an unordered (in principal anyway) list of the nodes at that level. *(Note that it is not actually a 2D jagged list! It's an array of references to the first node in the list, and then the nodes themselves form a linked list by way of their `NextReducible` property. Yuck!)*

In practice, however, the list's ordering is important to the overall quality of the palette. The second level of that list is filled in as you call `AddColor()`, with new nodes being inserted at the head of the linked list. This means that the final stages of the reduction process will prefer to merge colors that were added last; those that first appeared toward the bottom of the image, in other words, assuming the image is processed top-to-bottom.

To fix this, I changed the reduction algorithm somewhat. See `OctreeQuantizer::Octree::Reduce()` for the implementation.

I removed the `reducibleNodes` list and `NextReducible` property, opting instead to traverse the octree when I need to count or gather nodes (the alternative was extra bookkeeping fields and it got very complicated/buggy). The octree is, as usual, reduced from bottom to top. However, after each level is completely reduced (and therefore removed), I look at the number of nodes in the last and last-1 levels. Once the last level has more nodes than the target color count, but the last-1 level has less than or equal, I break out of the loop and switch to the final stage of reduction.

In the final stage of reduction, all of the leaf nodes are gathered into a list and then sorted by their weight (`pixelCount`). The order of this list then establishes the priority ordering for node reduction. Nodes with the lowest weight, which store information about colors that were seen less often in the image, are reduced first. Because many nodes could have the same weight, tie-breaking for the sort order is done by comparing the hash code of the node's output color at high precision (32-bit floating point per component). This is done via the `ColorRgb96Float` struct in this repo's code.

Using the hash code achieves a pseudo-random but *deterministic* balancing to the order that nodes are reduced. I contemplated other balancing methods, but ultimately nothing was satisfactory. Using the hash code has some benefits, especially around being straightforwardly portable to other platforms, languages, compilers, and not being dependent on an opaque random number generator (i.e. if you shuffled the list instead of sorting it).

I'm not fully convinced that reducing low-weight nodes first is the best approach, nor that using the color's hash code for tie-breaking the sort order is. However, it's a simple approach that is easy to change and experiment with, and it does produce great looking results.

**Note** that there is another change you must understand for this section. Nodes in an Octree are usually classified as *interior* or *leaf*, with leaf nodes emitting colors after the octree is reduced and the palette is finally built. However, we must introduce an extra flag here: does the node have color information? Rather, can it emit a color? (normally the answer is yes for leaves, no for interior nodes) Because we are partially reducing nodes during the final stage of reduction, we will have non-leaf (interior) nodes that store color information about their children that were reduced, even if some of the children are still present. We are not reducing all-at-once, in other words. These nodes will emit a color when building the palette, as will their children (which are leaves). This is encapsulated by the `OctreeNode.HasColorInfo` property.

### 6) Leaf nodes take up a ton of memory
If you're using 32-bit integers to store color sums and counts, like the MSDN article does, your `OctreeNode` will consume 40 bytes plus object/allocation overhead. If you're using 64-bit integers for sums and counts, this goes way up to around 56 bytes. If the image being processed has 500,000 unique colors, the total here is 20MB for 32-bit and 28MB for 64-bit. For a "worst" case image, with all 2^24 RGB colors, this balloons to 671MB and 939MB.

This can be ameliorated by using inheritance and polymorphism (virtual methods/properties, i.o.w.). In my `Octree` code, `OctreeNode` is a base class that does not store these values. Instead, derived classes with specializations for different tiers of color counts are used. See the code for more details. The code quality here is not as good as I'd like it to be, but it does work and isn't too complicated. It can also be optimized further with more `OctreeNode`-derived class that specialize different cases, and I experimented with this, but the complexity grew very quickly.

(Other approaches to solving this are also possible, such as using structs, pool allocators, and other clever ways of keeping track of sums and counts.)

### 7) Mapping colors to the palette is REALLY slow
The previous sections went into detail fixing and optimizing the palette generation process. Once you have a palette, which is just an arbitrarily ordered array of `ColorBgr24`s, you need to map the original image onto it.

The algorithm for doing this is simple. For each pixel in the source image, find the color in the palette that is *closest* to it, and use that for the output image. "Closest" is usually defined as the Euclidean distance (https://en.wikipedia.org/wiki/Euclidean_distance) in three-dimensional RGB space.

> (NOTE: Quantization generally does *not* work with the alpha channel, hence why distance is calculated in RGB space and not RGBA space. There may be 1 palette slot that is fully transparent (i.e. #00000000). Not all image formats support this, but GIF and PNG do. Whether semi-transparent colors are mapped to fully transparent, or are first blended with white or something, is up to the application. In Paint.NET, the "alpha threshold" property is used to control this.)

However, finding the closest palette color is not easily optimizable, and most code out out there uses a simple linear search. Other data structures, such as the K-d Tree (https://en.wikipedia.org/wiki/K-d_tree), look very promising but end up complicated to implement and slower to execute. The latter is due to the traversal steps being complicated with lots of branch predictions and caches misses (discussion: https://github.com/SixLabors/ImageSharp/issues/1350#issuecomment-707249531).

Usually a cache is layered on top of this approach, which allows for skipping the linear search, but *only* for colors that have already been seen before. The downside of this approach is that it can use a *lot* of memory in common cases. Also worth noting is that it is not easy to "prefetch" or "pre-fill" the cache ahead of time: quantization usually involves dithering, which means that many of the colors being mapped to the palette are not found in the source image (although they are usually "nearby").

#### Introducing the Proximity Map

I've come up with an approach that has these benefits:

1. It is much faster than the traditional linear search for images with *many* unique colors (millions of them)
2. It is a bit faster than the traditional linear search for most other images
3. I've not seen any case where it's slower (although it might be for very small palette sizes!)
4. The maximum memory usage is about 1 MiB of memory. Compared to linear search + caching, this can be a savings of 10s to 100s of megabytes.
5. It's simple to implement and understand. There are no complicated data structures, it's just arrays, loops, and some basic algebra.

Here's how it works:

The 24-bit RGB color space, which is 256x256x256 in size, is divided up into cubes of size 16x16x16. There are, of course, 16x16x16 = 4,096 of these cubes.

**TODO: Finish writing this up.** For now, there's a summary (w/o pictures sadly) over on Twitter: https://twitter.com/rickbrewPDN/status/1379238853832155136

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