Commit f30a3228 authored by Christoph Ruegge's avatar Christoph Ruegge
Browse files

move inverse to lecture/ and solutions/, created exercise

parent beb640fe
%% Cell type:markdown id: tags:
# Image deconvolution
## Task
Given a blurred image $y$, find the original image $x$, where
$$T x = y,$$
and $T$ is a blurring operator, e.g. an operator assigns to each pixel an
average of all pixels in some neighbourhood.
Depending on the blurring operator, it is not clear in general whether this
problem has a solution at all. A frequent approach is to replace the problem by
$$\text{Minimize } {\lVert T x - y \rVert}^2 \text{ with respect to } x,$$
or equivalently
$$T^* T x = T^* y.$$
This equation can e.g. be solved using CG, but it turns out to be numerically
unstable: if the blurred image $y$ is distorted with some noise, the
reconstruction $x$ has significantly amplified noise.
## Regularization
A simple way to stablilize the problem is Tikhonov regularization, which
replaces the problem by
$$\text{Minimize } {\lVert T x - y \rVert}^2 + \alpha {\lVert x \rVert}^2 \text{
with respect to } x,$$
for some *regularization parameter* $\alpha > 0$ or equivalently
$$(T^* T + \alpha \mathbb{1}) x = T^* y.$$
$\alpha$ has to be chosen large enough to avoid the instabilities of the
problem, but small enough in order not to change the problem too much.
There are several ways to choose the parameter (semi-)automatically depending on
the blurred and distorted image $y$ and the noise level. Mathematically, one is
interested in the question under what circumstances and how fast the regularized
reconstruction converge to the true solution.
For this example, we will simply choose a regularization parameter manually.
## Imports and basics
%% Cell type:code id: tags:
``` haskell
{-# OPTIONS_GHC -O3 #-}
{-# LANGUAGE BangPatterns #-}
{-# LANGUAGE ScopedTypeVariables #-}
import Control.Monad.Identity
import Data.Array.Repa as R hiding ((++))
import Data.Array.Repa.Eval
import Data.Array.Repa.Stencil
import Data.Array.Repa.Stencil.Dim2
import Data.Array.Repa.Algorithms.Pixel (doubleLuminanceOfRGB8, rgb8OfGreyDouble)
import Data.Array.Repa.IO.BMP (readImageFromBMP, writeImageToBMP)
import Data.Random.Normal (normalsIO')
```
%% Cell type:markdown id: tags:
Alias, for convenience.
%% Cell type:code id: tags:
``` haskell
type Image r = Array r DIM2 Double
```
%% Cell type:markdown id: tags:
Loading and saving of images -- rather uninteresting.
%% Cell type:code id: tags:
``` haskell
loadImage :: FilePath -> IO (Image U)
loadImage file = do
loaded <- readImageFromBMP file
case loaded of
Left err -> error $ show err
Right img -> computeP $ R.map doubleLuminanceOfRGB8 img
saveImage :: FilePath -> Image U -> IO ()
saveImage file img = do
!m <- foldAllP max 0 img
!img' <- computeP $ R.map (rgb8OfGreyDouble . (/ m)) img
writeImageToBMP file img'
```
%% Cell type:markdown id: tags:
## Operator type class
We need functions for both the blurring operator and its adjoint. This can be
done nicely using a typeclass.
%% Cell type:code id: tags:
``` haskell
class Operator a where
evalOp :: a -> Image U -> Image D
adjointOp :: a -> Image U -> Image D
```
%% Cell type:markdown id: tags:
## CG solver
The Repa CG solver for the regularized normal equation
$$(T^* T + \alpha \mathbb{1}) x = T^* y.$$
The code tries to use as few operator evaluations as possible.
%% Cell type:code id: tags:
``` haskell
data CGState = CGState { cgx :: Image U
, cgp :: Image U
, cgr :: Image U
, cgr2 :: Double
}
cgreg :: Operator a => a -> Double -> Image U -> Image U -> [CGState]
cgreg op reg rhs initial =
runIdentity $ do
(res :: Image U) <- computeP $ rhs -^ evalOp op initial
rInit <- computeP $ adjointOp op res
r2Init <- normSquaredP rInit
return $ iterate cgStep (CGState initial rInit rInit r2Init)
where normSquaredP = sumAllP . R.map (^(2::Int))
scale a = R.map (* a)
cgStep :: CGState -> CGState
cgStep (CGState x p r r2) =
runIdentity $ do
!(q :: Image U) <- computeP $ evalOp op p
!p2 <- normSquaredP p
!q2 <- normSquaredP q
let alpha = r2 / (q2 + reg*p2)
!x' <- computeP $ x +^ scale alpha p
!(s :: Image U) <- computeP $ adjointOp op q
!r' <- computeP $ r -^ scale alpha (s +^ scale reg p)
!r2' <- normSquaredP r'
let beta = r2' / r2
!p' <- computeP $ r' +^ scale beta p
return $ CGState x' p' r' r2'
```
%% Cell type:markdown id: tags:
`cgreg` returns a lazy list of all iterates. `takeUntil` is uses to implement
the stopping rule; it is similar to `takeWhile`, but also returns the final
iterate.
%% Cell type:code id: tags:
``` haskell
takeUntil :: (a -> Bool) -> [a] -> [a]
takeUntil _ [] = []
takeUntil predicate (x:xs)
| predicate x = [x]
| otherwise = x : takeUntil predicate xs
```
%% Cell type:markdown id: tags:
We want to be able to output some information about the iterates, i.e. run an IO
action over the list of iterates, before returning the final one. One option
would be something like
```haskell
fmap last . forM iterates $ \it -> do [...]
```
However, this would perform the `fmap last` only *after* the IO actions are
done, therfore retaining the entire list of iterates in memory.
We want all but the final iterate to be garbage collected directly after
printing information. This can be done using a monadic fold, where we do not
actually accumulate a result but only return the last value.
%% Cell type:code id: tags:
``` haskell
process :: Monad m => [a] -> (a -> m ()) -> m a
process xs f = foldM (\_ x -> f x >> return x) undefined xs
```
%% Cell type:markdown id: tags:
`runCG` wrapper that calls `cg`, print information and implements the
stopping rule based on relative residuals.
%% Cell type:code id: tags:
``` haskell
runCG :: Operator a => Double -> a -> Double -> Image U -> IO (Image U)
runCG tol op reg rhs = do
let initial = computeS $ fromFunction (extent rhs) (const 0)
let steps' = cgreg op reg rhs initial
let r20 = cgr2 $ head steps'
let steps = takeUntil (\x -> sqrt (cgr2 x / r20) < tol) steps'
result <- process (zip [(1::Int)..] steps) $ \(n, cgs) ->
putStrLn $ show n ++ " " ++ show (sqrt $ cgr2 cgs / r20)
return $ cgx . snd $ result
```
%% Cell type:markdown id: tags:
## Blurring operator
The implementation of the averaging operator uses repa stencils.
%% Cell type:code id: tags:
``` haskell
data StencilOp = StencilOp { getStencil :: Stencil DIM2 Double }
instance Operator StencilOp where
evalOp = (delay .) . mapStencil2 (BoundConst 0) . getStencil
adjointOp = evalOp
mkKernel :: Int -> StencilOp
mkKernel n = StencilOp $ makeStencil2 n n $ \(Z:.i:.j) ->
if max (abs i) (abs j) <= n then Just x else Nothing
where x = 1 / fromIntegral (n*n)
```
%% Cell type:markdown id: tags:
## Main
The main function loads an image, blurs and distorts it, and performs
regularized and unregularized reconstructions.
%% Cell type:code id: tags:
``` haskell
main :: IO ()
main = do
img <- loadImage "grumpy.bmp"
saveImage "exact.bmp" img
let sh = extent img
-- Repa only supports stencils of size up to 7. Efficient implementations of
-- more general blurring (or other convolution) operators can be done using e.g.
-- FFTs.
let kernel = mkKernel 7
blurry <- computeP $ evalOp kernel img
saveImage "blurry.bmp" blurry
putStrLn "blurry reconstructed"
runCG 1e-3 kernel 0 blurry >>= saveImage "blurry_reconstructed.bmp"