Mathias Brandewinder on .NET, F#, VSTO and Excel development, and quantitative analysis / machine learning.
by Mathias 31. July 2014 14:15

It is the summer, a time to cool off and enjoy vacations – so let’s keep it light, and hopefully fun, today! A couple of days ago, during his recent San Francisco visit, @tomaspetricek brought up an idea that I found intriguing. What if you had two images, and wanted to recreate an image similar to the first one, using only the pixels from the second?

To make this real, let’s take two images - a portrait by Velasquez, and one by Picasso, which I have conveniently cropped to be of identical size. What we are trying to do is to re-arrange the pixels from the Picasso painting, and recombine them to get something close to the Velasquez:

picasso velasquez

You can find the original images here:

My thinking on the problem was as follows: we are trying to arrange a set of pixels into an image as close as possible to an existing image. That’s not entirely trivial. Being somewhat lazy, rather than work hard, I reverted to my patented strategy “what is the simplest thing that could possibly work (TM)”.

Two images are identical if each of their matching pixels are equal; the greater the difference between pixels, the less similar they are. In that frame, one possible angle is to try and match each pixel and limit the differences.

So how could we do that? If I had two equal groups of people, and I were trying to pair them by skill level, here is what I would do: rank each group by skill, and match the lowest person from the first group with his counterpart in the second group, and so on and so forth, until everyone is paired up. It’s not perfect, but it is easy.

Problem here is that there is no obvious order over pixels. Not a problem – we’ll create a sorting function, and replace it with something else if we don’t like the result. For instance, we could sort by “maximum intensity”; the value of a pixel will be the greater of its Red, Green and Blue value.

At that point, we have an algorithm. Time to crank out F# and try it out with a script:

open System.IO
open System.Drawing

let combine (target:string) ((source1,source2):string*string) =
    // open the 2 images to combine
    let img1 = new Bitmap(source1)
    let img2 = new Bitmap(source2)
    // create the combined image
    let combo = new Bitmap(img1)
    // extract pixels from an image
    let pixelize (img:Bitmap) = [
        for x in 0 .. img.Width - 1 do
            for y in 0 .. img.Height - 1 do
                yield (x,y,img.GetPixel(x,y)) ]
    // extract pixels from the 2 images
    let pix1 = pixelize img1
    let pix2 = pixelize img2
    // sort by most intense color
    let sorter (_,_,c:Color) = [c.R;c.G;c.B] |> Seq.max
    // sort, combine and write pixels
    (pix1 |> List.sortBy sorter,
     pix2 |> List.sortBy sorter)
    |> List.iter (fun ((x1,y1,_),(_,_,c2)) -> 
    // ... and save, we're done

… and we are done. Assuming you downloaded the two images in the same place as

let root = __SOURCE_DIRECTORY__

let velasquez = Path.Combine(root,"velasquez.bmp")
let picasso = Path.Combine(root,"picasso.bmp")

let picasquez = Path.Combine(root,"picasquez.bmp")
let velasso = Path.Combine(root,"velasso.bmp")

(velasquez,picasso) |> combine velasso
(picasso,velasquez) |> combine picasquez

… which should create two images like these:

picasquez velasso

Not bad for 20 lines of code. Now you might argue that this isn’t the nicest, most functional code ever, and you would be right. There are a lot of things that could be done to improve that code; for instance, handling pictures of different sizes, or injecting an arbitrary Color sorting function – feel free to have fun with it!

Also, you might wonder why I picked that specific, and somewhat odd, sorting function. Truth be told, it happened by accident. In my first attempt, I simply summed the 3 colors, and the results were pretty bad. The reason for it is, Red, Green and Blue are encoded as bytes, and summing up 3 bytes doesn’t necessarily do what you would expect. Rather than, say, convert everything to int, I went the lazy route again…


Original images:

by Mathias 16. June 2014 22:15

Like many a good man, I too got caught into the 2048 trap, which explains in part why I have been rather quiet on this blog lately (there are a couple other reasons, too).

In case you don't know what 2048 is yet, first, consider yourself lucky - and, fair warning, you might want to back away now, while you still have a chance. 2048 is a very simple and fun game, and one of the greatest time sinks since Tetris. You can play it here, and the source code is here on GitHub.

I managed to dodge the bullet for a while, until @PrestonGuillot, a good friend of mine, decided to write a 2048 bot as a fun weekend project to sharpen his F# skills, and dragged me down with him in the process. This has been a ton of fun, and this post is a moderately organized collection of notes from my diary as a recovering 2048 addict.

Let's begin with the end result. The video below shows a F# bot, written by my friend @Blaise_V, masterfully playing the game. I recorded it a couple of weeks ago, accelerating time "for dramatic purposes":

One of the problems Preston and I ran into early was how to handle interactions with the game. A recent post by @shanselman was praising Canopy as a great library for web UI testing, which gave me the idea to try it for that purpose. In spite of my deep incompetence of things web related, I found the Canopy F# DSL super easy to pick up, and got something crude working in a jiffy. With a bit of extra help from the awesome @lefthandedgoat, the creator of Canopy (thanks Chris!), it went from crude to pretty OK, and I was ready to focus on the interesting bits, the game AI.

I had so much fun in the process, I figured others might too, and turned this into another Community for F# Dojo, which you can find here.


by Mathias 12. April 2014 11:05

A lightweight post this week. One of my favorite F# type providers is the World Bank type provider, which enables ridiculously easy access to a boatload of socio-economic data for every country in the world. However, numbers are cold – wouldn’t it be nice to visualize them using a map? Turns out it’s pretty easy to do, using another of my favorites, the R type provider. The rworldmap R package, as its name suggests, is all about world maps, and is a perfect fit with the World Bank data.

The video below shows you the results in action; I also added the code below, for good measure. The only caveat relates to the integration between the Deedle data frame library and R. I had to manually copy the Deedle.dll and Deedle.RProvider.Plugin.dll into packages\RProvider.1.0.5\lib for the R Provider to properly convert Deedle data frames into R data frames. Enjoy!

Here is the script I used:

#I @"..\packages\"
#r @"R.NET.1.5.5\lib\net40\RDotNet.dll"
#r @"RProvider.1.0.5\lib\RProvider.dll"
#r @"FSharp.Data.2.0.5\lib\net40\FSharp.Data.dll"
#r @"Deedle.0.9.12\lib\net40\Deedle.dll"
#r @"Deedle.RPlugin.0.9.12\lib\net40\Deedle.RProvider.Plugin.dll"

open FSharp.Data
open RProvider
open RProvider.``base``
open Deedle
open Deedle.RPlugin
open RProviderConverters

let wb = WorldBankData.GetDataContext()
wb.Countries.France.Indicators.``Population (Total)``.[2000]

let countries = wb.Countries

let pop2000 = series [ for c in countries -> c.Code => c.Indicators.``Population (Total)``.[2000]]
let pop2010 = series [ for c in countries -> c.Code => c.Indicators.``Population (Total)``.[2010]]
let surface = series [ for c in countries -> c.Code => c.Indicators.``Surface area (sq. km)``.[2010]]

let df = frame [ "Pop2000" => pop2000; "Pop2010" => pop2010; "Surface" => surface ]
df?Codes <- df.RowKeys

open RProvider.rworldmap

let map = R.joinCountryData2Map(df,"ISO3","Codes")

df?Density <- df?Pop2010 / df?Surface
df?Growth <- (df?Pop2010 - df?Pop2000) / df?Pop2000

let map2 = R.joinCountryData2Map(df,"ISO3","Codes")

Have a great week-end, everybody! And big thanks to Tomas for helping me figure out a couple of things about Deedle.

by Mathias 22. March 2014 13:11

A couple of days ago, I got into the following Twitter exchange:


So why do I think FsCheck + XUnit = The Bomb?

I have a long history with Test-Driven Development; to this day, I consider Kent Beck’s “Test-Driven Development by Example” one of the biggest influences in the way I write code (any terrible code I might have written is, of course, to be blamed entirely on me, and not on the book).

In classic TDD style, you typically proceed by writing incremental test cases which match your requirements, and progressively write the code that will satisfy the requirements. Let’s illustrate on an example, a password strength validator. Suppose that my requirements are “a password must be at least 8 characters long to be valid”. Using XUnit, I would probably write something along these lines:

namespace FSharpTests

open Xunit
open CSharpCode

module ``Password validator tests`` =

    let ``length above 8 should be valid`` () =
        let password = "12345678"
        let validator = Validator ()

… and in the CSharpCode project, I would then write the dumbest minimal implementation that could passes that requirement, that is:

public class Validator
    public bool IsValid(string password)
        return true;

Next, I would write a second test, to verify the obvious negative:

namespace FSharpTests

open Xunit
open CSharpCode

module ``Password validator tests`` =

    let ``length above 8 should be valid`` () =
        let password = "12345678"
        let validator = Validator ()

    let ``length under 8 should not be valid`` () =
        let password = "1234567"
        let validator = Validator ()

This fails, producing the following output in Visual Studio:


… which forces me to fix my implementation, for instance like this:

public class Validator
    public bool IsValid(string password)
        if (password.Length < 8)
            return false;

        return true;

Let’s pause here for a couple of remarks. First, note that while my tests are written in F#, the code base I am testing against is in C#. Mixing the two languages in one solution is a non-issue. Then, after years of writing C# test cases with names like Length_Above_8 _Should_Be_Valid, and arguing whether this was better or worse than LengthAbove8 ShouldBeValid, I find that having the ability to simply write “length above 8 should be valid”, in plain old English (and seeing my tests show that way in the test runner as well), is pleasantly refreshing. For that reason alone, I would encourage F#-curious C# developers to try out writing tests in F#; it’s a nice way to get your toes in the water, and has neat advantages.

But that’s not the main point I am interested here. While this process works, it is not without issues. From a single requirement, “a password must be at least 8 characters long to be valid”, we ended up writing 2 test cases. First, the cases we ended up are somewhat arbitrary, and don’t fully reflect what they say. I only tested two instances, one 7 characters long, one 8 characters long. This is really relying on my ability as a developer to identify “interesting cases” in a vast universe of possible passwords, hoping that I happened to cover sufficient ground.

This is where FsCheck comes in. FsCheck is a port of Haskell’s QuickCheck, a property-based testing framework. The term “property” is somewhat overloaded, so let’s clarify: what “Property” means in that context is a property of our program that should be true, in the same sense as mathematically, a property of any number x is “x * x is positive”. It should always be true, for any input x.

Install FsCheck via Nuget, as well as the FsCheck XUnit extension; you can now write tests that verify properties by marking them with the attribute [<Property>], instead of [<Fact>], and the XUnit test runner will pick them up as normal tests. For instance, taking our example from right above, we can write:

namespace FSharpTests

open Xunit
open FsCheck
open FsCheck.Xunit
open CSharpCode

module Specification =

    let ``square should be positive`` (x:float) = 
        x * x > 0.

Let’s run that – fail. If you click on the test results, here is what you’ll see:


FsCheck found a counter-example, 0.0. Ooops! Our specification is incorrect here, the square value doesn’t have to be strictly positive, and could be zero. This is an obvious mistake, let’s fix the test, and get on with our lives:

let ``square should be positive`` (x:float) = 
    x * x >= 0.


by Mathias 1. March 2014 14:32

During some recent meanderings through the confines of the internet, I ended up discovering the Winnow Algorithm. The simplicity of the approach intrigued me, so I thought it would be interesting to try and implement it in F# and see how well it worked.

The purpose of the algorithm is to train a binary classifier, based on binary features. In other words, the goal is to predict one of two states, using a collection of features which are all binary. The prediction model assigns weights to each feature; to predict the state of an observation, it checks all the features that are “active” (true), and sums up the weights assigned to these features. If the total is above a certain threshold, the result is true, otherwise it’s false. Dead simple – and so is the corresponding F# code:

type Observation = bool []
type Label = bool
type Example = Label * Observation
type Weights = float []

let predict (theta:float) (w:Weights) (obs:Observation) = 
    (obs,w) ||> 
    |> Seq.filter fst 
    |> Seq.sumBy snd 
    |> ((<) theta)

We create some type aliases for convenience, and write a predict function which takes in theta (the threshold), weights and and observation; we zip together the features and the weights, exclude the pairs where the feature is not active, sum the weights, check whether the threshold is lower that the total, and we are done.

In a nutshell, the learning process feeds examples (observations with known label), and progressively updates the weights when the model makes mistakes. If the current model predicts the output correctly, don’t change anything. If it predicts true but should predict false, it is over-shooting, so weights that were used in the prediction (i.e. the weights attached to active features) are reduced. Conversely, if the prediction is false but the correct result should be true, the active features are not used enough to reach the threshold, so they should be bumped up.

And that’s pretty much it – the algorithm starts with arbitrary initial weights of 1 for every feature, and either doubles or halves them based on the mistakes. Again, the F# implementation is completely straightforward. The weights update can be written as follows:

let update (theta:float) (alpha:float) (w:Weights) (ex:Example) =
    let real,obs = ex
    match (real,predict theta w obs) with
    | (true,false) -> w |> Array.mapi (fun i x -> if obs.[i] then alpha * x else x)
    | (false,true) -> w |> Array.mapi (fun i x -> if obs.[i] then x / alpha else x)
    | _ -> w

Let’s check that the update mechanism works:

> update 0.5 2. [|1.;1.;|] (false,[|false;true;|]);;
val it : float [] = [|1.0; 0.5|]

The threshold is 0.5, the adjustment multiplier is 2, and each feature is currently weighted at 1. The state of our example is [| false; true; |], so only the second feature is active, which means that the predicted value will be 1. (the weight of that feature). This is above the threshold 0.5, so the predicted value is true. However, because the correct value attached to that example is false, our prediction is incorrect, and the weight of the second feature is reduced, while the first one, which was not active, remains unchanged.

Let’s wrap this up in a convenience function which will learn from a sequence of examples, and give us directly a function that will classify observations:

let learn (theta:float) (alpha:float) (fs:int) (xs:Example seq) =
    let updater = update theta alpha
    let w0 = [| for f in 1 .. fs -> 1. |]    
    let w = Seq.fold (fun w x -> updater w x) w0 xs
    fun (obs:Observation) -> predict theta w obs

We pass in the number of features, fs, to initialize the weights at the correct size, and use a fold to update the weights for each example in the sequence. Finally, we create and return a function that, given an observation, will predict the label, based on the weights we just learnt.

And that’s it – in 20 lines of code, we are done, the Winnow is implemented.



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