Scala-view: Animate streams of images

Introduction

In the previous post I discussed how comonads can be useful for structuring certain kinds of scientific and statistical computations. Two of the examples I gave were concerned with the time-evolution of 2-d images. In that post I used Breeze to animate the sequence of computed images. In this post I want to describe an alternative that is better suited to animating an image sequence.

Scala-view is a small Scala library for animating a Stream of Images on-screen in a separate window managed by your window manager. It works with both ScalaFX Images (recommended) and Scala Swing/AWT BufferedImages (legacy). The stream of images is animated in a window with some simple controls to start and stop the animation, and to turn on and off the saving of image frames to disk (typically for the purpose of turning the image sequence into a movie). An example of what a window might look like is given below.

Ising window

More comprehensive documentation is available from the scala-view github repo, but here I give a quick introduction to the library to outline its capabilities.

A Scala-view tutorial

This brief tutorial gives a quick introduction to using the Scala-view library for viewing a ScalaFX Image Stream. It assumes only that you have SBT installed, and that you run SBT from an empty directory.

An SBT REPL

Start by running SBT from an empty or temporary directory to get an SBT prompt:

$ sbt
>

Now we need to configure SBT to use the Scala-view library, and start a console. From the SBT prompt:

set libraryDependencies += "com.github.darrenjw" %% "scala-view" % "0.5"
set scalaVersion := "2.12.4"
console

The should result in a scala> REPL prompt. We can now use Scala and the Scala-view library interactively.

An example REPL session

You should be able to paste the code snippets below directly into the REPL. You may find :paste mode helpful.

We will replicate the heat equation example from the examples-sfx directory, which is loosely based on the example from my blog post on comonads. We will start by defining a simple parallel Image and corresponding comonadic pointed image PImage type. If you aren’t familiar with comonads, you may find it helpful to read through that post.

import scala.collection.parallel.immutable.ParVector
case class Image[T](w: Int, h: Int, data: ParVector[T]) {
  def apply(x: Int, y: Int): T = data(x * h + y)
  def map[S](f: T => S): Image[S] = Image(w, h, data map f)
  def updated(x: Int, y: Int, value: T): Image[T] =
    Image(w, h, data.updated(x * h + y, value))
}

case class PImage[T](x: Int, y: Int, image: Image[T]) {
  def extract: T = image(x, y)
  def map[S](f: T => S): PImage[S] = PImage(x, y, image map f)
  def coflatMap[S](f: PImage[T] => S): PImage[S] = PImage(
    x, y, Image(image.w, image.h,
      (0 until (image.w * image.h)).toVector.par.map(i => {
        val xx = i / image.h
        val yy = i % image.h
        f(PImage(xx, yy, image))
      })))
  def up: PImage[T] = {
    val py = y - 1
    val ny = if (py >= 0) py else (py + image.h)
    PImage(x, ny, image)
  }
  def down: PImage[T] = {
    val py = y + 1
    val ny = if (py < image.h) py else (py - image.h)
    PImage(x, ny, image)
  }
  def left: PImage[T] = {
    val px = x - 1
    val nx = if (px >= 0) px else (px + image.w)
    PImage(nx, y, image)
  }
  def right: PImage[T] = {
    val px = x + 1
    val nx = if (px < image.w) px else (px - image.w)
    PImage(nx, y, image)
  }
}

We will need a function to convert this image into a ScalaFX WritableImage.

import scalafx.scene.image.WritableImage
import scalafx.scene.paint._
def toSfxI(im: Image[Double]): WritableImage = {
    val wi = new WritableImage(im.w, im.h)
    val pw = wi.pixelWriter
    (0 until im.w) foreach (i =>
      (0 until im.h) foreach (j =>
        pw.setColor(i, j, Color.gray(im(i,j)))
      ))
    wi
  }

We will need a starting image representing the initial condition for the heat equation.

val w = 600
val h = 500
val pim0 = PImage(0, 0, Image(w, h,
  ((0 until w*h).toVector map {i: Int => {
  val x = i / h
  val y = i % h
  0.1*math.cos(0.1*math.sqrt((x*x+y*y))) + 0.1 + 0.8*math.random
  }}).par
))

We can define a kernel associated with the update of a single image pixel based on a single time step of a finite difference solution of the heat equation.

def kernel(pi: PImage[Double]): Double = (2*pi.extract+
  pi.up.extract+pi.down.extract+pi.left.extract+pi.right.extract)/6.0

We can now create a Stream of PImage with

def pims = Stream.iterate(pim0)(_.coflatMap(kernel))

We can turn this into a Stream[WritableImage] with

def sfxis = pims map (im => toSfxI(im.image))

Note that we are essentially finished at this point, but so far everything we have done has been purely functional with no side effects. We haven’t even computed our solution to the heat equation. All we have constructed are lazy infinite streams representing the solution of the heat equation.

Finally, we can render our Stream of Images on screen with

scalaview.SfxImageViewer(sfxis,1e7.toInt)

which has a delay of 1e7 nanoseconds (10 milliseconds) between frames.

This should pop up a window on your display containing the initial image. Click on the Start button to animate the solution of the heat equation. See the API docs for SfxImageViewer for additional options. The ScalaFX API docs may also be useful, especially the docs for Image and WritableImage.

Advertisements

Comonads for scientific and statistical computing in Scala

Introduction

In a previous post I’ve given a brief introduction to monads in Scala, aimed at people interested in scientific and statistical computing. Monads are a concept from category theory which turn out to be exceptionally useful for solving many problems in functional programming. But most categorical concepts have a dual, usually prefixed with “co”, so the dual of a monad is the comonad. Comonads turn out to be especially useful for formulating algorithms from scientific and statistical computing in an elegant way. In this post I’ll illustrate their use in signal processing, image processing, numerical integration of PDEs, and Gibbs sampling (of an Ising model). Comonads enable the extension of a local computation to a global computation, and this pattern crops up all over the place in statistical computing.

Monads and comonads

Simplifying massively, from the viewpoint of a Scala programmer, a monad is a mappable (functor) type class augmented with the methods pure and flatMap:

trait Monad[M[_]] extends Functor[M] {
  def pure[T](v: T): M[T]
  def flatMap[T,S](v: M[T])(f: T => M[S]): M[S]
}

In category theory, the dual of a concept is typically obtained by “reversing the arrows”. Here that means reversing the direction of the methods pure and flatMap to get extract and coflatMap, respectively.

trait Comonad[W[_]] extends Functor[W] {
  def extract[T](v: W[T]): T
  def coflatMap[T,S](v: W[T])(f: W[T] => S): W[S]
}

So, while pure allows you to wrap plain values in a monad, extract allows you to get a value out of a comonad. So you can always get a value out of a comonad (unlike a monad). Similarly, while flatMap allows you to transform a monad using a function returning a monad, coflatMap allows you to transform a comonad using a function which collapses a comonad to a single value. It is coflatMap (sometimes called extend) which can extend a local computation (producing a single value) to the entire comonad. We’ll look at how that works in the context of some familiar examples.

Applying a linear filter to a data stream

One of the simplest examples of a comonad is an infinite stream of data. I’ve discussed streams in a previous post. By focusing on infinite streams we know the stream will never be empty, so there will always be a value that we can extract. Which value does extract give? For a Stream encoded as some kind of lazy list, the only value we actually know is the value at the head of the stream, with subsequent values to be lazily computed as required. So the head of the list is the only reasonable value for extract to return.

Understanding coflatMap is a bit more tricky, but it is coflatMap that provides us with the power to apply a non-trivial statistical computation to the stream. The input is a function which transforms a stream into a value. In our example, that will be a function which computes a weighted average of the first few values and returns that weighted average as the result. But the return type of coflatMap must be a stream of such computations. Following the types, a few minutes thought reveals that the only reasonable thing to do is to return the stream formed by applying the weighted average function to all sub-streams, recursively. So, for a Stream s (of type Stream[T]) and an input function f: W[T] => S, we form a stream whose head is f(s) and whose tail is coflatMap(f) applied to s.tail. Again, since we are working with an infinite stream, we don’t have to worry about whether or not the tail is empty. This gives us our comonadic Stream, and it is exactly what we need for applying a linear filter to the data stream.

In Scala, Cats is a library providing type classes from Category theory, and instances of those type classes for parametrised types in the standard library. In particular, it provides us with comonadic functionality for the standard Scala Stream. Let’s start by defining a stream corresponding to the logistic map.

import cats._
import cats.implicits._

val lam = 3.7
def s = Stream.iterate(0.5)(x => lam*x*(1-x))
s.take(10).toList
// res0: List[Double] = List(0.5, 0.925, 0.25668749999999985,
//  0.7059564011718747, 0.7680532550204203, 0.6591455741499428, ...

Let us now suppose that we want to apply a linear filter to this stream, in order to smooth the values. The idea behind using comonads is that you figure out how to generate one desired value, and let coflatMap take care of applying the same logic to the rest of the structure. So here, we need a function to generate the first filtered value (since extract is focused on the head of the stream). A simple first attempt a function to do this might look like the following.

  def linearFilterS(weights: Stream[Double])(s: Stream[Double]): Double =
    (weights, s).parMapN(_*_).sum

This aligns each weight in parallel with a corresponding value from the stream, and combines them using multiplication. The resulting (hopefully finite length) stream is then summed (with addition). We can test this with

linearFilterS(Stream(0.25,0.5,0.25))(s)
// res1: Double = 0.651671875

and let coflatMap extend this computation to the rest of the stream with something like:

s.coflatMap(linearFilterS(Stream(0.25,0.5,0.25))).take(5).toList
// res2: List[Double] = List(0.651671875, 0.5360828502929686, ...

This is all completely fine, but our linearFilterS function is specific to the Stream comonad, despite the fact that all we’ve used about it in the function is that it is a parallelly composable and foldable. We can make this much more generic as follows:

  def linearFilter[F[_]: Foldable, G[_]](
    weights: F[Double], s: F[Double]
  )(implicit ev: NonEmptyParallel[F, G]): Double =
    (weights, s).parMapN(_*_).fold

This uses some fairly advanced Scala concepts which I don’t want to get into right now (I should also acknowledge that I had trouble getting the syntax right for this, and got help from Fabio Labella (@SystemFw) on the Cats gitter channel). But this version is more generic, and can be used to linearly filter other data structures than Stream. We can use this for regular Streams as follows:

s.coflatMap(s => linearFilter(Stream(0.25,0.5,0.25),s))
// res3: scala.collection.immutable.Stream[Double] = Stream(0.651671875, ?)

But we can apply this new filter to other collections. This could be other, more sophisticated, streams such as provided by FS2, Monix or Akka streams. But it could also be a non-stream collection, such as List:

val sl = s.take(10).toList
sl.coflatMap(sl => linearFilter(List(0.25,0.5,0.25),sl))
// res4: List[Double] = List(0.651671875, 0.5360828502929686, ...

Assuming that we have the Breeze scientific library available, we can plot the raw and smoothed trajectories.

def myFilter(s: Stream[Double]): Double =
  linearFilter(Stream(0.25, 0.5, 0.25),s)
val n = 500
import breeze.plot._
import breeze.linalg._
val fig = Figure(s"The (smoothed) logistic map (lambda=$lam)")
val p0 = fig.subplot(3,1,0)
p0 += plot(linspace(1,n,n),s.take(n))
p0.ylim = (0.0,1.0)
p0.title = s"The logistic map (lambda=$lam)"
val p1 = fig.subplot(3,1,1)
p1 += plot(linspace(1,n,n),s.coflatMap(myFilter).take(n))
p1.ylim = (0.0,1.0)
p1.title = "Smoothed by a simple linear filter"
val p2 = fig.subplot(3,1,2)
p2 += plot(linspace(1,n,n),s.coflatMap(myFilter).coflatMap(myFilter).coflatMap(myFilter).coflatMap(myFilter).coflatMap(myFilter).take(n))
p2.ylim = (0.0,1.0)
p2.title = "Smoothed with 5 applications of the linear filter"
fig.refresh

Image processing and the heat equation

Streaming data is in no way the only context in which a comonadic approach facilitates an elegant approach to scientific and statistical computing. Comonads crop up anywhere where we want to extend a computation that is local to a small part of a data structure to the full data structure. Another commonly cited area of application of comonadic approaches is image processing (I should acknowledge that this section of the post is very much influenced by a blog post on comonadic image processing in Haskell). However, the kinds of operations used in image processing are in many cases very similar to the operations used in finite difference approaches to numerical integration of partial differential equations (PDEs) such as the heat equation, so in this section I will blur (sic) the distinction between the two, and numerically integrate the 2D heat equation in order to Gaussian blur a noisy image.

First we need a simple image type which can have pixels of arbitrary type T (this is very important – all functors must be fully type polymorphic).

  import scala.collection.parallel.immutable.ParVector
  case class Image[T](w: Int, h: Int, data: ParVector[T]) {
    def apply(x: Int, y: Int): T = data(x*h+y)
    def map[S](f: T => S): Image[S] = Image(w, h, data map f)
    def updated(x: Int, y: Int, value: T): Image[T] =
      Image(w,h,data.updated(x*h+y,value))
  }

Here I’ve chosen to back the image with a parallel immutable vector. This wasn’t necessary, but since this type has a map operation which automatically parallelises over multiple cores, any map operations applied to the image will be automatically parallelised. This will ultimately lead to all of our statistical computations being automatically parallelised without us having to think about it.

As it stands, this image isn’t comonadic, since it doesn’t implement extract or coflatMap. Unlike the case of Stream, there isn’t really a uniquely privileged pixel, so it’s not clear what extract should return. For many data structures of this type, we make them comonadic by adding a “cursor” pointing to a “current” element of interest, and use this as the focus for computations applied with coflatMap. This is simplest to explain by example. We can define our “pointed” image type as follows:

  case class PImage[T](x: Int, y: Int, image: Image[T]) {
    def extract: T = image(x, y)
    def map[S](f: T => S): PImage[S] = PImage(x, y, image map f)
    def coflatMap[S](f: PImage[T] => S): PImage[S] = PImage(
      x, y, Image(image.w, image.h,
      (0 until (image.w * image.h)).toVector.par.map(i => {
        val xx = i / image.h
        val yy = i % image.h
        f(PImage(xx, yy, image))
      })))

There is missing a closing brace, as I’m not quite finished. Here x and y represent the location of our cursor, so extract returns the value of the pixel indexed by our cursor. Similarly, coflatMap forms an image where the value of the image at each location is the result of applying the function f to the image which had the cursor set to that location. Clearly f should use the cursor in some way, otherwise the image will have the same value at every pixel location. Note that map and coflatMap operations will be automatically parallelised. The intuitive idea behind coflatMap is that it extends local computations. For the stream example, the local computation was a linear combination of nearby values. Similarly, in image analysis problems, we often want to apply a linear filter to nearby pixels. We can get at the pixel at the cursor location using extract, but we probably also want to be able to move the cursor around to nearby locations. We can do that by adding some appropriate methods to complete the class definition.

    def up: PImage[T] = {
      val py = y-1
      val ny = if (py >= 0) py else (py + image.h)
      PImage(x,ny,image)
    }
    def down: PImage[T] = {
      val py = y+1
      val ny = if (py < image.h) py else (py - image.h)
      PImage(x,ny,image)
    }
    def left: PImage[T] = {
      val px = x-1
      val nx = if (px >= 0) px else (px + image.w)
      PImage(nx,y,image)
    }
    def right: PImage[T] = {
      val px = x+1
      val nx = if (px < image.w) px else (px - image.w)
      PImage(nx,y,image)
    }
  }

Here each method returns a new pointed image with the cursor shifted by one pixel in the appropriate direction. Note that I’ve used periodic boundary conditions here, which often makes sense for numerical integration of PDEs, but makes less sense for real image analysis problems. Note that we have embedded all “indexing” issues inside the definition of our classes. Now that we have it, none of the statistical algorithms that we develop will involve any explicit indexing. This makes it much less likely to develop algorithms containing bugs corresponding to “off-by-one” or flipped axis errors.

This class is now fine for our requirements. But if we wanted Cats to understand that this structure is really a comonad (perhaps because we wanted to use derived methods, such as coflatten), we would need to provide evidence for this. The details aren’t especially important for this post, but we can do it simply as follows:

  implicit val pimageComonad = new Comonad[PImage] {
    def extract[A](wa: PImage[A]) = wa.extract
    def coflatMap[A,B](wa: PImage[A])(f: PImage[A] => B): PImage[B] =
      wa.coflatMap(f)
    def map[A,B](wa: PImage[A])(f: A => B): PImage[B] = wa.map(f)
  }

It’s handy to have some functions for converting Breeze dense matrices back and forth with our image class.

  import breeze.linalg.{Vector => BVec, _}
  def BDM2I[T](m: DenseMatrix[T]): Image[T] =
    Image(m.cols, m.rows, m.data.toVector.par)
  def I2BDM(im: Image[Double]): DenseMatrix[Double] =
    new DenseMatrix(im.h,im.w,im.data.toArray)

Now we are ready to see how to use this in practice. Let’s start by defining a very simple linear filter.

def fil(pi: PImage[Double]): Double = (2*pi.extract+
  pi.up.extract+pi.down.extract+pi.left.extract+pi.right.extract)/6.0

This simple filter can be used to “smooth” or “blur” an image. However, from a more sophisticated viewpoint, exactly this type of filter can be used to represent one time step of a numerical method for time integration of the 2D heat equation. Now we can simulate a noisy image and apply our filter to it using coflatMap:

import breeze.stats.distributions.Gaussian
val bdm = DenseMatrix.tabulate(200,250){case (i,j) => math.cos(
  0.1*math.sqrt((i*i+j*j))) + Gaussian(0.0,2.0).draw}
val pim0 = PImage(0,0,BDM2I(bdm))
def pims = Stream.iterate(pim0)(_.coflatMap(fil))

Note that here, rather than just applying the filter once, I’ve generated an infinite stream of pointed images, each one representing an additional application of the linear filter. Thus the sequence represents the time solution of the heat equation with initial condition corresponding to our simulated noisy image.

We can render the first few frames to check that it seems to be working.

import breeze.plot._
val fig = Figure("Diffusing a noisy image")
pims.take(25).zipWithIndex.foreach{case (pim,i) => {
  val p = fig.subplot(5,5,i)
  p += image(I2BDM(pim.image))
}}

Note that the numerical integration is carried out in parallel on all available cores automatically. Other image filters can be applied, and other (parabolic) PDEs can be numerically integrated in an essentially similar way.

Gibbs sampling the Ising model

Another place where the concept of extending a local computation to a global computation crops up is in the context of Gibbs sampling a high-dimensional probability distribution by cycling through the sampling of each variable in turn from its full-conditional distribution. I’ll illustrate this here using the Ising model, so that I can reuse the pointed image class from above, but the principles apply to any Gibbs sampling problem. In particular, the Ising model that we consider has a conditional independence structure corresponding to a graph of a square lattice. As above, we will use the comonadic structure of the square lattice to construct a Gibbs sampler. However, we can construct a Gibbs sampler for arbitrary graphical models in an essentially identical way by using a graph comonad.

Let’s begin by simulating a random image containing +/-1s:

import breeze.stats.distributions.{Binomial,Bernoulli}
val beta = 0.4
val bdm = DenseMatrix.tabulate(500,600){
  case (i,j) => (new Binomial(1,0.2)).draw
}.map(_*2 - 1) // random matrix of +/-1s
val pim0 = PImage(0,0,BDM2I(bdm))

We can use this to initialise our Gibbs sampler. We now need a Gibbs kernel representing the update of each pixel.

def gibbsKernel(pi: PImage[Int]): Int = {
   val sum = pi.up.extract+pi.down.extract+pi.left.extract+pi.right.extract
   val p1 = math.exp(beta*sum)
   val p2 = math.exp(-beta*sum)
   val probplus = p1/(p1+p2)
   if (new Bernoulli(probplus).draw) 1 else -1
}

So far so good, but there a couple of issues that we need to consider before we plough ahead and start coflatMapping. The first is that pure functional programmers will object to the fact that this function is not pure. It is a stochastic function which has the side-effect of mutating the random number state. I’m just going to duck that issue here, as I’ve previously discussed how to fix it using probability monads, and I don’t want it to distract us here.

However, there is a more fundamental problem here relating to parallel versus sequential application of Gibbs kernels. coflatMap is conceptually parallel (irrespective of how it is implemented) in that all computations used to build the new comonad are based solely on the information available in the starting comonad. OTOH, detailed balance of the Markov chain will only be preserved if the kernels for each pixel are applied sequentially. So if we coflatMap this kernel over the image we will break detailed balance. I should emphasise that this has nothing to do with the fact that I’ve implemented the pointed image using a parallel vector. Exactly the same issue would arise if we switched to backing the image with a regular (sequential) immutable Vector.

The trick here is to recognise that if we coloured alternate pixels black and white using a chequerboard pattern, then all of the black pixels are conditionally independent given the white pixels and vice-versa. Conditionally independent pixels can be updated by parallel application of a Gibbs kernel. So we just need separate kernels for updating odd and even pixels.

def oddKernel(pi: PImage[Int]): Int =
  if ((pi.x+pi.y) % 2 != 0) pi.extract else gibbsKernel(pi)
def evenKernel(pi: PImage[Int]): Int =
  if ((pi.x+pi.y) % 2 == 0) pi.extract else gibbsKernel(pi)

Each of these kernels can be coflatMapped over the image preserving detailed balance of the chain. So we can now construct an infinite stream of MCMC iterations as follows.

def pims = Stream.iterate(pim0)(_.coflatMap(oddKernel).
  coflatMap(evenKernel))

We can animate the first few iterations with:

import breeze.plot._
val fig = Figure("Ising model Gibbs sampler")
fig.width = 1000
fig.height = 800
pims.take(50).zipWithIndex.foreach{case (pim,i) => {
  print(s"$i ")
  fig.clear
  val p = fig.subplot(1,1,0)
  p.title = s"Ising model: frame $i"
  p += image(I2BDM(pim.image.map{_.toDouble}))
  fig.refresh
}}
println

Here I have a movie showing the first 1000 iterations. Note that youtube seems to have over-compressed it, but you should get the basic idea.

Again, note that this MCMC sampler runs in parallel on all available cores, automatically. This issue of odd/even pixel updating emphasises another issue that crops up a lot in functional programming: very often, thinking about how to express an algorithm functionally leads to an algorithm which parallelises naturally. For general graphs, figuring out which groups of nodes can be updated in parallel is essentially the graph colouring problem. I’ve discussed this previously in relation to parallel MCMC in:

Wilkinson, D. J. (2005) Parallel Bayesian Computation, Chapter 16 in E. J. Kontoghiorghes (ed.) Handbook of Parallel Computing and Statistics, Marcel Dekker/CRC Press, 481-512.

Further reading

There are quite a few blog posts discussing comonads in the context of Haskell. In particular, the post on comonads for image analysis I mentioned previously, and this one on cellular automata. Bartosz’s post on comonads gives some connection back to the mathematical origins. Runar’s Scala comonad tutorial is the best source I know for comonads in Scala.

Full runnable code corresponding to this blog post is available from my blog repo.

Gibbs sampling a Gaussian Markov random field (GMRF) using Java

Introduction

As I’ve explained previously, I’m gradually coming around to the idea of using Java for the development of MCMC codes, and I’m starting to build up a collection of simple examples for getting started. One of the advantages of Java is that it includes a standard cross-platform GUI library. This might not seem like the most important requirement for MCMC, but can actually be very handy in several contexts, particularly for monitoring convergence. One obvious context is that of image analysis, where it can be useful to monitor image reconstructions as the sampler is running. In this post I’ll show three very small simple Java classes which together provide an application for running a Gibbs sampler on a (non-stationary, unconditioned) Gaussian Markov random field.

The model is essentially that the distribution of each pixel is defined intrinsically, dependent only on its four nearest neighbours on a rectangular lattice, and here the distribution will be Gaussian with mean equal to the sample mean of the four neighbouring pixels and a fixed (unit) variance. On its own this isn’t especially useful, but it is a key component of many image analysis applications.

A simple Java implementation

We will start with the class MrfApp containing the main method for the application:

MrfApp.java

import java.io.*;
class MrfApp {
    public static void main(String[] arg)
	throws IOException
    {
	Mrf mrf;
	System.out.println("started program");
	mrf=new Mrf(800,600);
	System.out.println("created mrf object");
	mrf.update(1000);
	System.out.println("done updates");
	mrf.saveImage("mrf.png");
	System.out.println("finished program");
	mrf.frame.dispose();
	System.exit(0);
    }
}

Hopefully this code is largely self-explanatory, but relies on a class called Mrf which contains all of the logic associated with the GMRF.

Mrf.java

import java.io.*;
import java.util.*;
import java.awt.image.*;
import javax.swing.*;
import javax.imageio.ImageIO;


class Mrf 
{
    int n,m;
    double[][] cells;
    Random rng;
    BufferedImage bi;
    WritableRaster wr;
    JFrame frame;
    ImagePanel ip;
    
    Mrf(int n_arg,int m_arg)
    {
	n=n_arg;
	m=m_arg;
	cells=new double[n][m];
	rng=new Random();
	bi=new BufferedImage(n,m,BufferedImage.TYPE_BYTE_GRAY);
	wr=bi.getRaster();
	frame=new JFrame("MRF");
	frame.setSize(n,m);
	frame.add(new ImagePanel(bi));
	frame.setVisible(true);
    }
    
    public void saveImage(String filename)
	throws IOException
    {
	ImageIO.write(bi,"PNG",new File(filename));
    }
    
    public void updateImage()
    {
	double mx=-1e+100;
	double mn=1e+100;
	for (int i=0;i<n;i++) {
	    for (int j=0;j<m;j++) {
		if (cells[i][j]>mx) { mx=cells[i][j]; }
		if (cells[i][j]<mn) { mn=cells[i][j]; }
	    }
	}
	for (int i=0;i<n;i++) {
	    for (int j=0;j<m;j++) {
		int level=(int) (255*(cells[i][j]-mn)/(mx-mn));
		wr.setSample(i,j,0,level);
	    }
	}
	frame.repaint();
    }
    
    public void update(int num)
    {
	for (int i=0;i<num;i++) {
	    updateOnce();
	}
    }
    
    private void updateOnce()
    {
	double mean;
	for (int i=0;i<n;i++) {
	    for (int j=0;j<m;j++) {
		if (i==0) {
		    if (j==0) {
			mean=0.5*(cells[0][1]+cells[1][0]);
		    } 
		    else if (j==m-1) {
			mean=0.5*(cells[0][j-1]+cells[1][j]);
		    } 
		    else {
			mean=(cells[0][j-1]+cells[0][j+1]+cells[1][j])/3.0;
		    }
		}
		else if (i==n-1) {
		    if (j==0) {
			mean=0.5*(cells[i][1]+cells[i-1][0]);
		    }
		    else if (j==m-1) {
			mean=0.5*(cells[i][j-1]+cells[i-1][j]);
		    }
		    else {
			mean=(cells[i][j-1]+cells[i][j+1]+cells[i-1][j])/3.0;
		    }
		}
		else if (j==0) {
		    mean=(cells[i-1][0]+cells[i+1][0]+cells[i][1])/3.0;
		}
		else if (j==m-1) {
		    mean=(cells[i-1][j]+cells[i+1][j]+cells[i][j-1])/3.0;
		}
		else {
		    mean=0.25*(cells[i][j-1]+cells[i][j+1]+cells[i+1][j]
			       +cells[i-1][j]);
		}
		cells[i][j]=mean+rng.nextGaussian();
	    }
	}
	updateImage();
    }
    
}

This class contains a few simple methods for creating and updating the GMRF, and also for maintaining and updating a graphical view of the GMRF as the sampler is running. The Gibbs sampler update itself is encoded in the final method, updateOnce, and most of the code is to deal with edge and corner cases (in the literal rather than metaphorical sense!). This is called repeatedly by the method update for the required number of iterations. At the end of each iteration, the method updateOnce triggers updateImage which updates the image associated GMRF. The GMRF itself is stored in a 2-dimensional array of doubles, but an image pixel typically consists of a grayscale value represented by an unsigned byte – that is, an integer from 0 to 255. So updateImage scans through the GMRF to find the maximum and minimum values and then maps the GMRF values onto the 0 to 255 scale. The image itself is set up by the constructor method, Mrf. This class relies on an additional class called ImagePanel, which is a simple GUI panel for displaying images:

ImagePanel.java

import java.awt.*;
import java.awt.image.*;
import javax.swing.*;

class ImagePanel extends JPanel {

	protected BufferedImage image;

	public ImagePanel(BufferedImage image) {
		this.image=image;
		Dimension dim=new Dimension(image.getWidth(),image.getHeight());
		setPreferredSize(dim);
		setMinimumSize(dim);
		revalidate();
		repaint();
	}

	public void paintComponent(Graphics g) {
		g.drawImage(image,0,0,this);
	}

}

This completes the application, which can be compiled and run from the command line with

javac *.java
java MrfApp

This should compile the code and run the application, which will show a GMRF updating for 1000 iterations. When the 1000 iterations are complete, the application writes the final image to a file and then quits.

Using Parallel COLT

The above classes are very convenient, as they should work with any standard Java installation. However, in more complex scenarios, it is likely that a math library such as Parallel COLT will be required. In this case it will make sense to make use of features in the COLT library, such as random number generators and 2d matrix objects. We can adapt the above application by replacing the MrfApp and Mrf classes with the following versions (the ImagePanel class remains unchanged):

MrfApp.java

import java.io.*;
import cern.jet.random.tdouble.engine.*;

class MrfApp {

    public static void main(String[] arg)
	throws IOException
    {
	Mrf mrf;
	int seed=1234;
	System.out.println("started program");
        DoubleRandomEngine rngEngine=new DoubleMersenneTwister(seed);
	mrf=new Mrf(800,600,rngEngine);
	System.out.println("created mrf object");
	mrf.update(1000);
	System.out.println("done updates");
	mrf.saveImage("mrf.png");
	System.out.println("finished program");
	mrf.frame.dispose();
	System.exit(0);
    }

}

Mrf.java

import java.io.*;
import java.util.*;
import java.awt.image.*;
import javax.swing.*;
import javax.imageio.ImageIO;
import cern.jet.random.tdouble.*;
import cern.jet.random.tdouble.engine.*;
import cern.colt.matrix.tdouble.impl.*;

class Mrf 
{
    int n,m;
    DenseDoubleMatrix2D cells;
    DoubleRandomEngine rng;
    Normal rngN;
    BufferedImage bi;
    WritableRaster wr;
    JFrame frame;
    ImagePanel ip;
    
    Mrf(int n_arg,int m_arg,DoubleRandomEngine rng)
    {
	n=n_arg;
	m=m_arg;
	cells=new DenseDoubleMatrix2D(n,m);
	this.rng=rng;
	rngN=new Normal(0.0,1.0,rng);
	bi=new BufferedImage(n,m,BufferedImage.TYPE_BYTE_GRAY);
	wr=bi.getRaster();
	frame=new JFrame("MRF");
	frame.setSize(n,m);
	frame.add(new ImagePanel(bi));
	frame.setVisible(true);
    }
    
    public void saveImage(String filename)
	throws IOException
    {
	ImageIO.write(bi,"PNG",new File(filename));
    }
    
    public void updateImage()
    {
	double mx=-1e+100;
	double mn=1e+100;
	for (int i=0;i<n;i++) {
	    for (int j=0;j<m;j++) {
		if (cells.getQuick(i,j)>mx) { mx=cells.getQuick(i,j); }
		if (cells.getQuick(i,j)<mn) { mn=cells.getQuick(i,j); }
	    }
	}
	for (int i=0;i<n;i++) {
	    for (int j=0;j<m;j++) {
		int level=(int) (255*(cells.getQuick(i,j)-mn)/(mx-mn));
		wr.setSample(i,j,0,level);
	    }
	}
	frame.repaint();
    }
    
    public void update(int num)
    {
	for (int i=0;i<num;i++) {
	    updateOnce();
	}
    }
    
    private void updateOnce()
    {
	double mean;
	for (int i=0;i<n;i++) {
	    for (int j=0;j<m;j++) {
		if (i==0) {
		    if (j==0) {
			mean=0.5*(cells.getQuick(0,1)+cells.getQuick(1,0));
		    } 
		    else if (j==m-1) {
			mean=0.5*(cells.getQuick(0,j-1)+cells.getQuick(1,j));
		    } 
		    else {
			mean=(cells.getQuick(0,j-1)+cells.getQuick(0,j+1)+cells.getQuick(1,j))/3.0;
		    }
		}
		else if (i==n-1) {
		    if (j==0) {
			mean=0.5*(cells.getQuick(i,1)+cells.getQuick(i-1,0));
		    }
		    else if (j==m-1) {
			mean=0.5*(cells.getQuick(i,j-1)+cells.getQuick(i-1,j));
		    }
		    else {
			mean=(cells.getQuick(i,j-1)+cells.getQuick(i,j+1)+cells.getQuick(i-1,j))/3.0;
		    }
		}
		else if (j==0) {
		    mean=(cells.getQuick(i-1,0)+cells.getQuick(i+1,0)+cells.getQuick(i,1))/3.0;
		}
		else if (j==m-1) {
		    mean=(cells.getQuick(i-1,j)+cells.getQuick(i+1,j)+cells.getQuick(i,j-1))/3.0;
		}
		else {
		    mean=0.25*(cells.getQuick(i,j-1)+cells.getQuick(i,j+1)+cells.getQuick(i+1,j)
			       +cells.getQuick(i-1,j));
		}
		cells.setQuick(i,j,mean+rngN.nextDouble());
	    }
	}
	updateImage();
    }
    
}

Again, the code should be reasonably self explanatory, and will compile and run in the same way provided that Parallel COLT is installed and in your classpath. This version runs approximately twice as fast as the previous version on all of the machines I’ve tried it on.

Reference

I have found the following book very useful for understanding how to work with images in Java:

Hunt, K.A. (2010) The Art of Image Processing with Java, A K Peters/CRC Press.