In the previous part of this post I introduced deconvolution and built up the core of a total-variation deconvolution algorithm based on ADMM.

def deconvolve_ADMM( K, B, num_iters=40, rho=1.0, gamma=0.1 ):
# store the image size
dim = B.shape
# pad out the kernel if its the wrong shape
if K.shape != dim:
raise ValueError('B and K must have same shape')
# define the two derivative operators
Dx = Kernel2D( [ 0, 0], [-1, 0], [-1.0, 1.0] )
Dy = Kernel2D( [-1, 0], [ 0, 0], [-1.0, 1.0] )
# define an initial solution estimate
I = numpy.zeros( dim )
# define the two splitting variables and lagrangr multipliers
Zx = Dx.mul( I )
Zy = Dy.mul( I )
Ux = numpy.zeros( Zx.shape )
Uy = numpy.zeros( Zy.shape )
# cache the necessary terms for the I update, need to circularly
# shift the kernel so it's DC spot lies at the corner
fK = numpy.fft.fft2( numpy.roll( K/numpy.sum(K), (dim[0]//2,dim[1]//2), axis=(0,1) ) )
fB = numpy.fft.fft2( B )
fDx = Dx.spectrum( dim )
fDy = Dy.spectrum( dim )
# build the numerator and denominator
num_init = numpy.conj( fK )*fB
den = numpy.conj( fK )*fK + rho*( numpy.conj(fDx)*fDx + numpy.conj(fDy)*fDy )
# define the L1 soft-thresholding operator
soft_threshold = lambda q: numpy.sign(q)*numpy.maximum( numpy.abs( q ) - gamma/rho, 0.0 )
# main solver loop
for iter in range( num_iters ):
print('ADMM iteration [%d/%d]'%(iter,num_iters))
# I-update
V = rho*( Dx.mul( Zx - Ux, trans=True) + Dy.mul( Zy - Uy, trans=True ) )
I = numpy.real( numpy.fft.ifft2( (num_init + numpy.fft.fft2(V))/den ) )
# Z-updates, cache the gradient filter results
tmp_x = Dx.mul( I )
tmp_y = Dy.mul( I )
Zx = soft_threshold( tmp_x + Ux )
Zy = soft_threshold( tmp_y + Uy )
# multiplier update
Ux += tmp_x - Zx
Uy += tmp_y - Zy
# return reconstructed result
return I

That is it.

Deconvolution is the process of removing blur from a signal. For images, there are two main types:

**Blind deconvolution** removes blur without specific knowledge of what the exact blur is. It generally requires estimating the blur kernel.
**Non-blind deconvolution** removes blur but is provided the blur kernel directly. It is an easier problem than blind deconvolution. Non-blind deconvolution is often used as a component of blind deconvolution algorithms.

Here I am focusing on non-blind deconvolution.

## Optimizing for \({\bf I}\)

Making the typical assumption of \(\epsilon\) being Gaussian noise, a solution for the sharp image \({\bf I}\) can be found by looking for an underlying sharp image that best explains, in a least-squares sense, the measured data \({\bf B}\).

\begin{equation*}
{\bf I} = \mbox{argmin}_{\bf I} \frac{1}{2} \| {\bf K} \otimes {\bf I} - {\bf B} \|_2^2
\end{equation*}

Denoting \(\mathcal{F}(\bf K)\) and \(\mathcal{F}^{-1}(\bf K)\) as the forward and inverse Fourier transform of \({\bf K}\) and \(*\) as complex conjugation, this equation can be solved very quickly in the Fourier domain:

\begin{equation*}
{\bf I} = \mathcal{F}^{-1}\left[ \frac{\mathcal{F}({\bf K})^*\mathcal{F}(\bf B)}{\mathcal{F}({\bf K})^2} \right]
\end{equation*}

Unfortunately the results are garbage. Zero or very low magnitude values of the Fourier transform of \({\bf K}\) cause divide-by-zero issues and amplify noise. These tiny values are due to convolution by \({\bf K}\) attenuating certain frequency bands to nothing. It makes sense intuitively since blurry images lose the sharp edges and fine details of focused images so the Fourier modes representing these are lost forever. Trying to undo the process is poorly defined and so the solve above just finds *anything* that minimizes the objective, whether it *looks* like an image or not.

It's possible to stabilize the above a bit by adding fudge factors here and there but the results are not very compelling. A better approach is to add regularizers to the optimization that favor specific types of underlying sharp images. One very effective set is the total variation which is incorporated below:

\begin{equation*}
{\bf I} = \mbox{argmin}_{\bf I} \frac{1}{2} \| {\bf K} \otimes {\bf I} - {\bf B} \|_2^2 + \gamma \| \frac{\partial}{\partial x} {\bf I} \|_1 + \gamma \| \frac{\partial}{\partial y} {\bf I} \|_1
\end{equation*}

The two extra terms penalize the gradients of the image and reach a zero value only for images that are constant. Using a 1-norm makes them less sensitive to large jumps in value than a 2-norm would be which makes these priors favor piecewise constant reconstructions. This is qualitatively similar to natural images that feature piecewise smooth features.

The only problem is that these 1-norm terms make the optimization much more difficult and costly to solve. However, the effort is worth it.

## Formulating the Optimization with ADMM

The problem with solving the optimization above is that the 1-norm terms are non-smooth. This is challenging for optimization methods. Fortunately, there are some very effective tools for solving 1-norm problems in isolation and a framework call the *Alternating Direction Method of Multipliers* can be used to transform intractible monolithic objective functions (such as the one above) into several coupled simpler problems.

By being careful about which terms go where, this can result in very efficient solves for problems involving complex priors. What follows is only one option for the above problem.

The overall goal is to split the \(L_1\) solves into separate pixel-wise solves. This is done by introducing new variables and constraints that make the problem look more complex but actually help. The data term will stay as a coupled \(L_2\) problem and will absorb the coupling terms as well (which are also \(L_2\)) and serve to stabilize the solve.

The first step is to isolate the \(L_1\) terms. The differential operators in the priors have the effect of coupling all pixels to each other, but the fast solvers for \(L_1\) problems expect terms that look like \(\|{\bf A}\|_1\) for some image \({\bf A}\). This can be accomplished by introducing splitting variables \({\bf Z}_x,{\bf Z}_y\) with appropriate constraints.

\begin{align*}
{\bf I} = \mbox{argmin} & & \frac{1}{2} \| {\bf K} \otimes {\bf I} - {\bf B} \|_2^2 + \gamma \| {\bf Z}_x \|_1 + \gamma \| {\bf Z}_y \|_1 \\
\mbox{subject to} & & \frac{\partial}{\partial x} {\bf I} - {\bf Z}_x = 0 \\
& & \frac{\partial}{\partial y} {\bf I} - {\bf Z}_y = 0
\end{align*}

It is also important to make sure that the \({\bf Z}_x, {\bf Z}_y\) terms appear without complex linear operators in the constraints. This ensures that the terms will diagonalize to a large set of pixelwise 1D optimizations rather than couple together.

With the splitting variables introduced, the constraints can be incorporated into the objective via penalty terms with (scaled) Lagrange multipliers \({\bf U}_x,{\bf U}_y\). The multipliers keep the constraints stiff. Without the multipliers, the high weights needed on the penalty terms to enforce the constraints can dominate the solves.

Applying this step transform back from a constrained problem to a (much more complicated) unconstrained problem again:

\begin{equation*}
{\bf I} = \mbox{argmin} \frac{1}{2} \| {\bf K} \otimes {\bf I} - {\bf B} \|_2^2 + \frac{\rho}{2} \| \frac{\partial}{\partial x} {\bf I} - {\bf Z}_x + {\bf U}_x \|_2^2 + \frac{\rho}{2} \| \frac{\partial}{\partial y} {\bf I} - {\bf Z}_y + {\bf U}_y \|_2^2 + \gamma \| {\bf Z}_x \|_1 + \gamma \| {\bf Z}_y \|_1
\end{equation*}

This looks like a mess, but is just the original data term, the two added \(L_1\) terms and two additional \(L_2\) coupling terms. Calling the whole thing \(F({\bf I},{\bf Z}_x, {\bf Z}_y, {\bf U}_x, {\bf U}_y)\) the ADMM algorithm solves this by the following steps:

- Updates \({\bf I}\) by solving \({\bf I} = \mbox{argmin}_{\bf I} F({\bf I},{\bf Z}_x, {\bf Z}_y, {\bf U}_x, {\bf U}_y)\)
- Updates \({\bf Z}_x\) by solving \({\bf Z}_x = \mbox{argmin}_{{\bf Z}_x} F({\bf I},{\bf Z}_x, {\bf Z}_y, {\bf U}_x, {\bf U}_y)\)
- Updates \({\bf U}_x\) by setting \({\bf U}_x = {\bf U}_x + \frac{\partial}{\partial x} {\bf I} - {\bf Z}_x\)
- Repeats the last two steps, appropriately modified, for \({\bf Z}_y\) and \({\bf U}_y\) respectively.

### The \({\bf I}\) Update

The first step of ADMM is to update \({\bf I}\). This is a straightforward \(L_2\) problem since the only terms that \({\bf I}\) appears is in terms with a 2-norm. Denoting \(\frac{\partial}{\partial x}{\bf I}\) as \({\bf D}_x \otimes {\bf I}\) and likewise for \(\frac{\partial}{\partial y}\), the solve ends up being:

\begin{equation*}
{\bf I} = \mbox{argmin} \frac{1}{2} \| {\bf K} \otimes {\bf I} - {\bf B} \|_2^2 + \frac{\rho}{2} \| {\bf D}_x \otimes {\bf I} - {\bf Z}_x + {\bf U}_x \|_2^2 + \frac{\rho}{2} \| {\bf D}_y \otimes {\bf I} - {\bf Z}_y + {\bf U}_y \|_2^2
\end{equation*}

All the terms are effectively the same: a minimization over the residual of a kernel convolved the the target image with respect to a measured value. The minimum is found by expanding out the term and finding the target image that sets the gradient of the expanded term to zero.

The solution for the first term in the Fourier domain is earlier in the post as:

\begin{equation*}
{\bf I} = \mathcal{F}^{-1}\left[ \frac{\mathcal{F}({\bf K})^*\mathcal{F}(\bf B)}{\mathcal{F}({\bf K})^2} \right]
\end{equation*}

The remaining terms follow the same pattern and sum in the numerator and denominator of the solve:

\begin{equation*}
{\bf I} = \mathcal{F}^{-1}\left[ \frac{
\mathcal{F}({\bf K})^*\mathcal{F}(\bf B)
+\rho \mathcal{F}({\bf D}_x)^*\mathcal{F}({\bf Z}_x - {\bf U}_x)
+\rho \mathcal{F}({\bf D}_y)^*\mathcal{F}({\bf Z}_y - {\bf U}_y)
}{\mathcal{F}({\bf K})^2 + \rho \mathcal{F}({\bf D}_x)^2 + \rho \mathcal{F}({\bf D}_y)^2} \right]
\end{equation*}

This involves 9 Fourier transforms, but only the terms containing \({\bf Z}_x,{\bf U}_x\) actually change during the solve (similarly for the y subscripts). By caching the first term of the numerator, the entire denominator, as well as \(\mathcal{F}({\bf D}_x)\mbox{ & }\mathcal{F}({\bf D}_y)\) this can be reduced to three transforms. It can further be reduced by evaluating the derivative filters \({\bf D}_x\mbox{ & }{\bf D}_y\) in the spatial domain, since they are very inexpensive finite difference filters. The following gets the \({\bf I}\) update down to one forward and one inverse transform once contants are cached:

\begin{align*}
{\bf V} = \rho \left( {\bf D}^T_x \otimes ({\bf Z}_x - {\bf U}_x) + {\bf D}^T_y \otimes ({\bf Z}_y - {\bf U}_y)\right) \\
{\bf I} = \mathcal{F}^{-1}\left[ \frac{
\mathcal{F}({\bf K})^*\mathcal{F}(\bf B) + \mathcal{F}({\bf V})
}{\mathcal{F}({\bf K})^2 + \rho \mathcal{F}({\bf D}_x)^2 + \rho \mathcal{F}({\bf D}_y)^2} \right]
\end{align*}

**Important note:** the \({\bf D}^T_x \otimes\) terms in \({\bf V}\) use the **transposed** filters (mixing notation terribly). This is the complex conjugate of the filter in the Fourier domain version, or matrix transpose if the convolution operation were represented in matrix form. Expressed as spatial domain convolution, the transpose simply requires mirroring the kernels horizontally and vertically about the central pixel.

### The \({\bf Z}\) Updates

The next step of ADMM is to update the splitting variable \({\bf Z}_x\mbox{ & }{\bf Z}_y\). These terms only appear in one coupling terms and one \(L_1\) term each. The updates for the two \({\bf Z}\) and two \({\bf U}\) variables are the same except the specific filter that is used, so I'll only cover one. Here's the update needed for \({\bf Z}_x\):

\begin{equation*}
{\bf Z}_x = \mbox{argmin}_{{\bf Z}_x} \frac{\rho}{2} \| {\bf D}_x \otimes {\bf I} - {\bf Z}_x + {\bf U}_x \|_2^2 + \gamma \| {\bf Z}_x \|_1
\end{equation*}

It's important to note that there are no operators applied to \({\bf Z}_x\), meaning that each pixel can be processed independently. The terms can be rearranged and the weight factors collected onto the \(L_1\) term to give the equivalent minimization:

\begin{equation*}
{\bf Z}_x = \mbox{argmin}_{{\bf Z}_x} \frac{\gamma}{\rho} \| {\bf Z}_x \|_1 + \frac{1}{2} \| {\bf Z}_x - ({\bf D}_x \otimes {\bf I} + {\bf U}_x ) \|_2^2
\end{equation*}

However, a complication is (still) the \(L_1\) term. This is non-smooth and tricky to deal with but the form above allows a non-smooth optimization method known as **proximal operators** to be used. The proximal operator of a scaled function \(\lambda f(q)\) is defined as:

\begin{equation*}
\mbox{prox}_{\lambda f}({\bf q}) = \mbox{argmin}_{\bf p} f({\bf p}) + \frac{1}{2\lambda} \| {\bf p} - {\bf q} \|_2^2
\end{equation*}

The similarities with the \({\bf Z}_x\) minimization are strong. By setting \(f({\bf Z}_x)=\|{\bf Z}_x\|_1\mbox{ & }\lambda = \frac{\gamma}{\rho}\mbox{ & }{\bf q} = {\bf D}_x \otimes {\bf I} + {\bf U}_x\) the proximal operator is recovered. This is great because the closed form solution for \(\mbox{prox}_{\lambda\|{\bf p}\|_1}(\bf{q})\) is known and trivial to implement:

\begin{equation*}
\mbox{prox}_{\lambda\|{\bf p}\|_1}({\bf q}) = \mbox{sign}({\bf q}) \mbox{max}( 0, |{\bf q}| - \lambda)
\end{equation*}

This is known as the *soft-thresholding operator*, or \(L_1\) *shrinkage operator* and is used **everywhere**. It can be implemented in python as:

def soft_threshold( q, lam ):
return numpy.sign(q)*numpy.maximum( numpy.abs(q) - lam, 0.0 )

In my view, it's the most potent line of code that exists in signal processing.

### The \({\bf U}\) Updates

The \({\bf U}\) updates simply add the current constraint violation to the Lagrange multipliers \({\bf U}\). It's kind of like the integral term in a PID controller.

The update was actually given already in the overview above but is repeated here for completeness:

\begin{align*}
{\bf U}_x = {\bf U}_x + {\bf D}_x \otimes {\bf I} - {\bf Z}_x \\
{\bf U}_y = {\bf U}_y + {\bf D}_x \otimes {\bf I} - {\bf Z}_y
\end{align*}

This is identical to the formula ealier, I've just used the \({\bf D}_x \otimes {\bf I}\) notation rather than \(\frac{\partial}{\partial x} {\bf I}\).

## Summary

That's all the math that's needed. It looks like a lot but ends up being surprisingly little code. In the next part I work through the implementation.

In image reconstruction tasks, it's often convenient to perform filtering operations in either the spatial or frequency domain. Both have their advantages and disadvantages. For the spatial domain, small filters such as finite differences can be evaluated extremely quickly and include local coefficients but objective functions often result in large systems of equations. In contrast, many objective functions diagonalize to yield pixel-wise solutions in the frequency domain but spatial variation cannot be incorporated directly.

As a result, it's common to express reconstruction tasks with individual terms in each domain. The domains are then coupled through penalties or constraints that can be expressed naturally either domain and lead to efficient solvers.

By way of example: in the total-variation regularized inpainting example below, the data term expresses a least-squares penalty on a (fixed) random selection of pixels in the reconstruction \({\bf I}\) and measurements \({\bf Y}\) via the selection mask \({\bf S}\) (\(\odot\) is pixelwise multiplication). This term cannot be expressed efficiently in the Fourier domain. However the regularizers (e.g. \(\lambda \| \frac{\partial}{\partial x} {\bf I} \|_1\)) **can and should be** since otherwise they couple every pixel in the image together. This is bad because it results in very costly solves.

\begin{equation*}
{\bf I} = \mbox{argmin}_{\bf I} \frac{1}{2} \| S \odot \left( {\bf I} - {\bf Y} \right) \|_2^2
+ \lambda \| \frac{\partial}{\partial x} {\bf I} \|_1 + + \lambda \| \frac{\partial}{\partial y} {\bf I} \|_1
\end{equation*}

Solving this efficiently requires splitting the problem into separate solve that couple through constraints and Lagrange multipliers (or more simply through penalties). How to do that is an entire topic in itself but the key thing is that **filtering is performed in both domains and must be consistent** to produce sensible results. If this is not the case, all sorts of weird things happen: images shifting each iteration, convergence stalling, etc.

This sounds (and is) easy but requires a bit of attention to detail.

## Brass Tacks

The starting point here is a definition of a sparse kernel. By sparse, I mean a kernel with very few non-zero weights, generally less than 10-20 or so. This includes common filters like Laplacians, edge filters and finite difference stencils.

In the following code, a kernel is defined by three equal length lists containing the row-offsets, column-offsets and filter values for the kernel. In the code, the user specifies the **correlation kernel**; this produces results that match typical finite-difference kernels when the correlate_spatial or correlate_fourier methods are called. Here is the code for the kernel class:

import numpy
class SparseKernel2D:
"""Represents a kernel for a 2D convolution operation
Example:
# make a first-order finite difference along the column axis
K = SparseKernel2D( [0,0], [-1,0], [-1.0, 1.0] )
"""
def __init__( self, offy, offx, vals ):
"""Represents a kernel for a 2D convolution operation
Args:
offy (integer array/list) vertical offsets of non-zero kernel values
offx (integer array/list) horizontal offsets of non-zero kernel values
vals (float array/list) non-zero kernel values
Raises:
ValueError whenever offx, offx, vals do not have the same length
"""
if len(offy) != len(offx) or len(vals) != len(offx):
raise ValueError('offx, offy and vals must be 1D array-like with the same size')
self.offx = offx
self.offy = offy
self.vals = vals
def image( self, dim ):
"""Returns an image represenation of the kernel at the specified dimensions with the kernel centered.
Args:
dim (integer tuple) output image dimensions [height,width]
Raises:
ValueError when dim is not 2-element
Returns:
height x width image of float32 elements
"""
if len(dim) != 2:
raise ValueError('Expected a 2-element integer tuple/list for image dimensions')
cen = ( int(dim[0]/2), int(dim[1]/2) )
out = numpy.zeros( dim, dtype=numpy.float32 )
for dx,dy,val in zip( self.offx, self.offy, self.vals):
out[cen[0]+dy,cen[1]+dx] = val
return out
def spectrum( self, dim ):
"""Returns a spectrum representation of the kernel at the specified dimensions. The returned
kernel is *not* centered, i.e. has it's DC component at the corner.
Args:
dim (integer tuple) output spectrum dimensions [height,width]
Raises:
ValueError when dim is not 2-element
Returns:
height x width complex image
"""
return numpy.fft.fft2( numpy.roll( self.image(dim), [int(dim[0]/2+0.5),int(dim[1]/2+0.5)], axis=[0,1] ) )
def convolve_spatial( self, I ):
"""Convolves the input image by the kernel in the spatial domain
Args:
I (2d float32 array) input image to convolve with kernel
Raises:
ValueError when input is not a 2D ndarray or I.dtype is not numpy.floating subclass
Returns:
Convolution results as 2D ndarray of I.dtype elements
"""
if not isinstance(I,numpy.ndarray) or I.ndim != 2 or not numpy.issubdtype(I.dtype,numpy.floating):
raise ValueError('Expected a 2D ndarray or float32 values')
result = numpy.zeros( I.shape, dtype=I.dtype )
for dx,dy,val in zip(self.offx, self.offy, self.vals):
result += val*numpy.roll( I, [dy,dx], axis=[0,1] )
return result
def correlate_spatial( self, I ):
"""Correlates the input image with the kernel in the spatial domain
"""
if not isinstance(I,numpy.ndarray) or I.ndim != 2 or not numpy.issubdtype(I.dtype,numpy.floating):
raise ValueError('Expected a 2D ndarray or float32 values')
result = numpy.zeros( I.shape, dtype=I.dtype )
for dx,dy,val in zip(self.offx, self.offy, self.vals):
result += val*numpy.roll( I, [-dy,-dx], axis=[0,1] )
return result
def convolve_fourier( self, I ):
"""Convolves the input image by the kernel in the Fourier domain
Args:
I (2d float32 array) input image to convolve with kernel
Raises:
ValueError when input is not a 2D ndarray or I.dtype is not numpy.floating subclass
Returns:
Convolution results as 2D ndarray of I.dtype elements
"""
if not isinstance(I,numpy.ndarray) or I.ndim != 2 or not numpy.issubdtype(I.dtype,numpy.floating):
raise ValueError('Expected a 2D ndarray or float32 values')
fK = self.spectrum( I.shape )
fI = numpy.fft.fft2( I )
return numpy.real( numpy.fft.ifft2( fK*fI ) )
def correlate_fourier( self, I ):
"""Correlates the input image by the kernel in the Fourier domain
Args:
I (2d float32 array) input image to convolve with kernel
Raises:
ValueError when input is not a 2D ndarray or I.dtype is not numpy.floating subclass
Returns:
Convolution results as 2D ndarray of I.dtype elements
"""
if not isinstance(I,numpy.ndarray) or I.ndim != 2 or not numpy.issubdtype(I.dtype,numpy.floating):
raise ValueError('Expected a 2D ndarray or float32 values')
fK = self.spectrum( I.shape )
fI = numpy.fft.fft2( I )
return numpy.real( numpy.fft.ifft2( numpy.conj(fK)*fI ) )

## Testing & Validation

To ensure that indexing is done correctly, it's important to check that the kernel can represent an identity operation correctly for odd and even image sizes in both Fourier and spatial domains. After that, it's important that the convolution and correlation functions return the correct results.

This last point depends on whether the convolution or correlation filter is being defined. The difference between the two is mirroring horizontally and vertically. Personally, I find it most natural to define the correlation filter which coincides with the typical finite difference stencils. Generating some analytic data as well as some known finite difference kernels allows correct operation to be confirmed:

def TestSparseKernel2D():
"""Runs a series of test functions on the SparseKernel2D class"""
# local helper function that will raise an error if the
# results are insufficiently accurate
def MSE( x, ref ):
val = numpy.mean( numpy.power( x-ref, 2 ) )
if val > 1e-6:
raise ValueError('Values do not match!')
return val
# define a mean-squared error function
#MSE = lambda I, Ref: numpy.mean( numpy.power( I-Ref, 2 ) )
# generate an identity kernel
ident = SparseKernel2D( [0],[0],[1.0] )
# generate some test images
x,y = numpy.meshgrid( numpy.arange(0.,200.), numpy.arange(0.,100.) )
z = x+y
# check that convolving with the identity kernel
# reproduces the input for even mesh sizes
conv_z_s = ident.convolve_spatial( z )
conv_z_f = ident.convolve_fourier( z )
print('Identity test, even sizes: ',z.shape)
print('\tMSE spatial: %f'%MSE(conv_z_s,z) )
print('\tMSE fourier: %f'%MSE(conv_z_f,z) )
# check that convolving with the identity kernel
# reproduces the input for odd/even mesh sizes
tz = z[1:,:]
conv_z_s = ident.convolve_spatial( tz )
conv_z_f = ident.convolve_fourier( tz )
print('Identity test, odd/even sizes: ',tz.shape)
print('\tMSE spatial: %f'%MSE(conv_z_s,tz) )
print('\tMSE fourier: %f'%MSE(conv_z_f,tz) )
# check that convolving with the identity kernel
# reproduces the input for odd/even mesh sizes
tz = z[:,1:]
conv_z_s = ident.convolve_spatial( tz )
conv_z_f = ident.convolve_fourier( tz )
print('Identity test, even/odd sizes: ',tz.shape)
print('\tMSE spatial: %f'%MSE(conv_z_s,tz) )
print('\tMSE fourier: %f'%MSE(conv_z_f,tz) )
# check that convolving with the identity kernel
# reproduces the input for odd/odd mesh sizes
tz = z[:-1,1:]
conv_z_s = ident.convolve_spatial( tz )
conv_z_f = ident.convolve_fourier( tz )
print('Identity test, odd/odd sizes: ',tz.shape)
print('\tMSE spatial: %f'%MSE(conv_z_s,tz) )
print('\tMSE fourier: %f'%MSE(conv_z_f,tz) )
# generate first order finite difference formula
# try convolving in both the spatial and frequency
# domains
ddx = SparseKernel2D( [0,0],[-1,0],[-1.0,1.0] )
ddx_z_s = ddx.correlate_spatial( z )[1:-2,1:-2]
ddx_z_f = ddx.correlate_fourier( z )[1:-2,1:-2]
ones = numpy.ones( ddx_z_s.shape, dtype=numpy.float32 )
print('First order finite-difference d(x+y)/dx')
print('\tMSE spatial: %f' % MSE(ddx_z_s,ones) )
print('\tMSE fourier: %f' % MSE(ddx_z_f,ones) )
# generate first order finite difference formula
# try convolving in both the spatial and frequency
# domains
ddy = SparseKernel2D( [-1,0],[0,0],[-1.0,1.0] )
ddy_z_s = ddy.correlate_spatial( z )[1:-2,1:-2]
ddy_z_f = ddy.correlate_fourier( z )[1:-2,1:-2]
ones = numpy.ones( ddy_z_s.shape, dtype=numpy.float32 )
print('First order finite-difference d(x+y)/dy')
print('\tMSE spatial: %f' % MSE(ddy_z_s,ones) )
print('\tMSE fourier: %f' % MSE(ddy_z_f,ones) )
# now try the convolution functions, this simply
# flips the kernel about the central pixel
# horizontally and vertically. for the filters
# defined above, this negates the results
ddx_z_s = ddx.convolve_spatial( z )[1:-2,1:-2]
ddx_z_f = ddx.convolve_fourier( z )[1:-2,1:-2]
ones = numpy.ones( ddx_z_s.shape, dtype=numpy.float32 )
print('Convolved first order finite-difference d(x+y)/dx')
print('\tMSE spatial: %f' % MSE(ddx_z_s,-ones) )
print('\tMSE fourier: %f' % MSE(ddx_z_f,-ones) )
# generate first order finite difference formula
# try convolving in both the spatial and frequency
# domains
ddy_z_s = ddy.convolve_spatial( z )[1:-2,1:-2]
ddy_z_f = ddy.convolve_fourier( z )[1:-2,1:-2]
ones = numpy.ones( ddy_z_s.shape, dtype=numpy.float32 )
print('Convolved first order finite-difference d(x+y)/dy')
print('\tMSE spatial: %f' % MSE(ddy_z_s,-ones) )
print('\tMSE fourier: %f' % MSE(ddy_z_f,-ones) )

This prints:

Identity test, even sizes: (100, 200)
MSE spatial: 0.000000
MSE fourier: 0.000000
Identity test, odd/even sizes: (99, 200)
MSE spatial: 0.000000
MSE fourier: 0.000000
Identity test, even/odd sizes: (100, 199)
MSE spatial: 0.000000
MSE fourier: 0.000000
Identity test, odd/odd sizes: (99, 199)
MSE spatial: 0.000000
MSE fourier: 0.000000
First order finite-difference d(x+y)/dx
MSE spatial: 0.000000
MSE fourier: 0.000000
First order finite-difference d(x+y)/dy
MSE spatial: 0.000000
MSE fourier: 0.000000
Convolved first order finite-difference d(x+y)/dx
MSE spatial: 0.000000
MSE fourier: 0.000000
Convolved first order finite-difference d(x+y)/dy
MSE spatial: 0.000000
MSE fourier: 0.000000

Bilteratal filtering is a process for removing noise from images. Like most filters, each output pixel is produced as a weighted combination of input pixels. However unlike typical filters, the weights used to combine input pixels are a function of the input pixels themselves rather than fixed in advance. This makes it a non-linear filter and is key to its effectiveness.

Given an input grayscale image \({\bf I}(y,x)\) where \(x\) & \(y\) are column and row indices, bilateral filtering defines the output image \({\bf F}\) as:

$$
{\bf F}(y,x) = \frac{ \sum_{r=y-w}^{y+w} \sum_{s=x-w}^{x+w} {\bf I}(r,s) \mathcal{N}_s(r-y,s-x) \mathcal{N}_v \left( {\bf I}(y,x)-{\bf I}(s,r) \right) }{ \sum_{r=y-w}^{y+w} \sum_{s=x-w}^{x+w} \mathcal{N}_s(r-y,s-x) \mathcal{N}_v \left( {\bf I}(y,x)-{\bf I}(s,r) \right)}
$$

This looks more involved than it actually is:

- The numerator and denominator both sum over a window that is \(2w+1 \times 2w+1\) pixels in size, centered on \(y\),\(x\). The pixel coordinates \(r\),\(s\) reference each pixel within that window.
- The pixel values of the input are combined via a weighted sum using the
*spatial proximity* in \(\mathcal{N}_s(\delta_y,\delta_x) = e^{ -\frac{\delta_x^2+\delta_y^2}{2\sigma_s^2} }\), as in standard Gaussian filtering.
- An additional factor based on the
*value proximity* \(\mathcal{N}_v \left( v \right) = e^{ -\frac{v^2}{2 \sigma_v^2} }\) is used to modify the weights. This is the nonlinear part of the filter and causes pixels with similar value to the central pixel to be weighted more strongly.
- The entire sum is normalized by the sum of the product of the
*spatial proximity* and *value proximity* weights in order to produce an output with proper range.

The mathematical definition is good for understanding but a straightforward implementation results in four nested loops: two for the rows/columns and two for the offsets in each window. In pure python this leads to considerable overhead. A better approach is to vectorize the operations by:

- Only looping over the offsets for the window
- Generating the
*spatial proximity* weights once per offset
- Shifting the entire input image by these offsets
- Generating the
*value proximity* weights in builk using the shifted image
- Accumulating an unnormalized result image and sum of weights image in bulk
- Returning the pixel-wise quotient of the result and weight sum images

Code that implements this is below:

def filter_bilateral( img_in, sigma_s, sigma_v, reg_constant=1e-8 ):
"""Simple bilateral filtering of an input image
Performs standard bilateral filtering of an input image. If padding is desired,
img_in should be padded prior to calling
Args:
img_in (ndarray) monochrome input image
sigma_s (float) spatial gaussian std. dev.
sigma_v (float) value gaussian std. dev.
reg_constant (float) optional regularization constant for pathalogical cases
Returns:
result (ndarray) output bilateral-filtered image
Raises:
ValueError whenever img_in is not a 2D float32 valued numpy.ndarray
"""
# check the input
if not isinstance( img_in, numpy.ndarray ) or img_in.dtype != 'float32' or img_in.ndim != 2:
raise ValueError('Expected a 2D numpy.ndarray with float32 elements')
# make a simple Gaussian function taking the squared radius
gaussian = lambda r2, sigma: (numpy.exp( -0.5*r2/sigma**2 )*3).astype(int)*1.0/3.0
# define the window width to be the 3 time the spatial std. dev. to
# be sure that most of the spatial kernel is actually captured
win_width = int( 3*sigma_s+1 )
# initialize the results and sum of weights to very small values for
# numerical stability. not strictly necessary but helpful to avoid
# wild values with pathological choices of parameters
wgt_sum = numpy.ones( img_in.shape )*reg_constant
result = img_in*reg_constant
# accumulate the result by circularly shifting the image across the
# window in the horizontal and vertical directions. within the inner
# loop, calculate the two weights and accumulate the weight sum and
# the unnormalized result image
for shft_x in range(-win_width,win_width+1):
for shft_y in range(-win_width,win_width+1):
# compute the spatial weight
w = gaussian( shft_x**2+shft_y**2, sigma_s )
# shift by the offsets
off = numpy.roll(img_in, [shft_y, shft_x], axis=[0,1] )
# compute the value weight
tw = w*gaussian( (off-img_in)**2, sigma_v )
# accumulate the results
result += off*tw
wgt_sum += tw
# normalize the result and return
return result/wgt_sum

An example of calling the function is below:

# use opencv for image loading
import cv2
import numpy
# function definition here....
# read the lena image, convert to float and scale to [0,1]
I = cv2.imread('images/lena.png', cv2.IMREAD_UNCHANGED ).astype(numpy.float32)/255.0
# bilateral filter the image
B = numpy.stack([
filter_bilateral( I[:,:,0], 10.0, 0.1 ),
filter_bilateral( I[:,:,1], 10.0, 0.1 ),
filter_bilateral( I[:,:,2], 10.0, 0.1 )], axis=2 )
# stack the images horizontally
O = numpy.hstack( [I,B] )
# write out the image
cv2.imwrite( 'images/lena_bilateral.png', out*255.0 )

Which produces the following, the input image is one the left and the output image is on the right.

The only downside to this is the speed. Evaluating the *value proximity* weights is costly. It can be sped up by discretizing and interpolating them from a lookup table. Since the filter also preserves edges very well, there is often the temptation to use very large spatial filter sizes. In the example above I used a value of 10 which corresponds to a **61x61** pixel kernel. This makes the filter a prime candidate to implement in C++ or CUDA for efficiency.

Reading and writing 16 bit images is very important for a wide variety of topics but is surprisingly overlooked by many imaging libraries in Python.

There are a number of different Python extensions that support 16 bit images but have surprising restrictions or complications:

**scikit-image+FreeImage plugin**: apparently reads/writes 16 bit formats but I could not find any explanation of how to actually install it.
**Pillow**: extension of the PIL library but seems to only support monochrome 16 bit formats.
**PyPNG**: reads/writes 16 bit images but is terribly slow due to pure-Python implementation
**easy_image_io**: My own foray into this, reads/writes 16 bit pngs & tiffs but requires compilation

However, I recently discovered that OpenCV will happily read/write at least 16 bit png files. Installation is very easy:

pip install opencv-python

Reading and writing is then easily done:

import cv2
I = np.ones( (10,20,3) )*65535
cv2.imwrite( 'output16.png', I.astype(np.uint16) )
I2 = cv2.imread( 'output16.png', cv2.IMREAD_UNCHANGED )
I2.shape
# prints (10, 20, 3)
I2[1,1,1]
# prints 65535

This is **by far** the easiest way I've seen to get 16 bit image IO running under python, provided you're willing to introduce the relatively large dependency of OpenCV.