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Gaussian blur

One common pattern of array computation is the so-called stencil, where we change the value of an element in the array based on its neighbours. For example, we might implement image blurring by assigning each pixel the average value of all of its neighbors. Futhark does not have a special-purpose stencil language construct. Instead, stencil computations are expressed as maps on the index space, using explicit array indexing to access the stencil source array and returning the new value for the index. While this is rather verbose, at least until Futhark grows more syntactical conveniences, it works and performs well. Let’s look at how to implement a simple image blurring program.

We will represent an image as a three-dimensional array [rows][cols][3]u8. The innermost size-3 dimension encodes the three colour channels for red, green, and blue, respectively. When blurring, it is useful to operate on each colour channel separately. Furthermore, instead of the colour being a number from 0 to 255, it is more convenient to store it as a floating-point number between 0 and 1.0. Therefore, we define a function that transforms an array of type [rows][cols][3]u8 into three arrays of type [rows][cols]f32 each. The result is that we have one array for each of the three colour channels:

let split_channels [rows][cols]
                   (image: [rows][cols][3]u8): ([rows][cols]f32,
                                                [rows][cols]f32,
                                                [rows][cols]f32) =
  unzip3 (map (\row ->
                unzip3 (map (\pixel ->
                              (f32.u8 pixel[0] / 255,
                               f32.u8 pixel[1] / 255,
                               f32.u8 pixel[2] / 255))
                            row))
              image)

The [rows][cols] notation preceding the image parameter is not a normal function parameter. Rather, it is a size parameter, a way of indicating that the function split_channels is polymorphic in the sizes rows and cols. The main purpose is that we can then use these names to indicate the sizes of the parameter and return values of the function. When the function is called, size parameters need not be passed arguments explicitly, but are automatically inferred from the concrete image argument. If we did not explicitly add these size parameters, the Futhark compiler would look for variables rows and cols in scope.

The function split_channels maps across each inner [3]u8 element (pixel), turns this into a triple instead of a three-element array, then uses unzip to turn the resulting array-of-triples into a triple-of-arrays, which is then returned. For readability, we could have chosen to explicitly indicate the return and parameter types of the anonymous function, but in the interest of brevity we have left them for the compiler to infer. It is only required to explicitly indicate the types of all top-level functions.

We will also need to re-combine the colour channel arrays into a single array. That function looks like this:

let combine_channels [rows][cols]
                     (rs: [rows][cols]f32)
                     (gs: [rows][cols]f32)
                     (bs: [rows][cols]f32): [rows][cols][3]u8 =
  map3 (\rs_row gs_row bs_row ->
         map3 (\r g b ->
                [u8.f32 (r * 255),
                 u8.f32 (g * 255),
                 u8.f32 (b * 255)])
              rs_row gs_row bs_row)
       rs gs bs

Another thing we will need is the actual stencil function. That is, the function we wish to apply to every pixel in the image. For blurring, we will take the average value of the pixel itself plus each of its eight neighbors (nine values in total):

let new_value [rows][cols]
             (image: [rows][cols]f32) (row: i32) (col: i32): f32 =
  unsafe
  let sum =
    image[row-1,col-1] + image[row-1,col] + image[row-1,col+1] +
    image[row,  col-1] + image[row,  col] + image[row,  col+1] +
    image[row+1,col-1] + image[row+1,col] + image[row+1,col+1]
  in sum / 9

The function call new_value(image, row, col) computes the new value for the pixel at position (row, col) in image.

The alert reader will have noticed that new_value cannot be applied to pixels on the edge of the image - doing so would result in out-of-bounds accesses to the image array. We will take care to only call the new_value function with safe indices, but the Futhark compiler is sadly not yet smart enough to realise this - thus we are forced to use the unsafe keyword to prevent the insertion of bounds checks that would otherwise hinder parallelisation. If we did not use unsafe, the Futhark compiler would fail with an error message pointing at the problematic array access.

Now we can write the actual stencil function, which applies new_value to every inner element of a colour channel array. This uses the iota function for constructing an array of integers ranging from 0 to the provided argument (the latter not inclusive). The edges are left unchanged:

let blur [rows][cols]
         (channel: [rows][cols]f32): [rows][cols]f32 =
  map (\row ->
         map(\col ->
               if row > 0 && row < rows-1 && col > 0 && col < cols-1
               then new_value channel row col
               else channel[row,col])
             (iota cols))
      (iota rows)

You may have heard that branches are expensive on a GPU. While this is a good basic rule of thumb, what is actually expensive is branch divergence - that is, when neighboring threads take different paths through a branch. In our stencil, only the edge elements will take the false branch, and these are few in number compared to the interior.

Stencil computations usually have an outer (sequential) loop for applying the stencil several times. Our program is no different - we will apply the blurring transformation a user-defined number of times. The more iterations we run, the more blurred the image will become:

let main [rows][cols]
         (iterations: i32) (image: [rows][cols][3]u8): [rows][cols][3]u8 =
  let (rs, gs, bs) = split_channels image
  let (rs, gs, bs) = loop (rs, gs, bs) for i < iterations do
    let rs = blur rs
    let gs = blur gs
    let bs = blur bs
    in (rs, gs, bs)
  in combine_channels rs gs bs

Our main function is quite simple. We split the input image into three different channels, use a sequential loop to blur each colour channel the requested number of times, then recombine the resulting channel arrays into a single final image.

The Futhark loop construct merits an explanation: in the above function, we declare three loop variant variables, rs, gs, and bs. These take their initial values from the incidentally identically named variables in scope (but this is not in general requirement). The loop body then returns three values that become the values of the loop variant variables in the next iteration of the loop. In essence, the loop construct is just syntactical suger for a particularly simple (but common) pattern of tail-recursive function. However, the Futhark compiler is able to perform transformations involving loops that it cannot for recursive functions (although it does not perform any such for this simple program).

The three separate calls to blur may seem wasteful, but the Futhark compiler is smart enough to fuse them together into a single GPU kernel that traverses the three colour channel arrays simultaneously. This is an instance of horisontal fusion.

Our Futhark program is now done. We can make it a little more useful by writing a small Python wrapper program for reading and writing PNGs: blur-png.py. We must compile blur.fut using the PyOpenCL backend:

$ futhark pyopencl --library blur.fut

This produces a Python module blur.py which is then imported by blur-png.py. We can try it out on any PNG image, say, this illustration of the spirit of Futhark:

$ python blur-png.py gottagofast.png --output-file gottagofast-blurred.png

Which produces this slightly smushed image. We can also ask for a hundred iterations::

$ python blur-png.py gottagofast.png --output-file gottagofast-blurred.png --iterations 100

Which produces this blurry mess. Notice the edges - perhaps simply keeping them unchanged is not the best way to implement image blurring. Still, this program is a decent description of how to implement stencils in Futhark.