Merge pull request #80 from mrava87/main

Small improvement to docstring of Pnp

Author: mrava87

pyproximal/optimization/pnp.py

@@ -42,7 +42,7 @@ def PlugAndPlay(proxf, denoiser, dims, x0, tau, niter=10,
     .. math::
 
         \mathbf{x},\mathbf{z}  = \argmin_{\mathbf{x}}
-        f(\mathbf{x}) + g(\mathbf{x})
+        f(\mathbf{x}) + \lambda g(\mathbf{x})
 
     where :math:`f(\mathbf{x})` is a function that has a known proximal
     operator where :math:`g(\mathbf{x})` is a function acting as implicit
@@ -88,21 +88,33 @@ def PlugAndPlay(proxf, denoiser, dims, x0, tau, niter=10,
     .. math::
 
         \mathbf{x}^{k+1} = \prox_{\tau f}(\mathbf{z}^{k} - \mathbf{u}^{k})\\
-        \mathbf{z}^{k+1} = \operatorname{Denoise}(\mathbf{x}^{k+1} + \mathbf{u}^{k}, \tau \sigma)\\
+        \mathbf{z}^{k+1} = \operatorname{Denoise}(\mathbf{x}^{k+1} + \mathbf{u}^{k}, \tau \lambda)\\
         \mathbf{u}^{k+1} = \mathbf{u}^{k} + \mathbf{x}^{k+1} - \mathbf{z}^{k+1}
 
-    where :math:`\operatorname{Denoise}` is a denoising algorithm of choice and
-    :math:`\tau \sigma` is the denoising parameter (should be chosen to
-    represent an estimate of the noise variance). This rather peculiar step
-    originates from the intuition that the optimization process associated
-    with the z-update can be interpreted as a denoising inverse problem -
-    for this reason any denoising of choice can be used instead of a function
-    with known proximal operator.
+    where :math:`\operatorname{Denoise}` is a denoising algorithm of choice. This rather peculiar step originates
+    from the intuition that the optimization process associated with the z-update can be interpreted as a denoising
+    inverse problem, or more specifically a MAP denoiser where the noise is gaussian with zero mean and variance
+    equal to :math:`\tau \lambda`. For this reason any denoising of choice can be used instead of a function with
+    known proximal operator.
+
+    Finally, whilst the :math:`\tau \lambda` denoising parameter should be chosen to
+    represent an estimate of the noise variance (of the denoiser, not the data of the problem we wish to solve!),
+    special care must be taken when setting up the denoiser and calling this optimizer. More specifically,
+    :math:`\lambda` should not be passed to the optimizer, rather set directly in the denoiser.
+    On the other hand :math:`\tau` must be passed to the optimizer as it is also affecting the x-update;
+    when defining the denoiser, ensure that :math:`\tau` is multiplied to :math:`\lambda` as shown in the tutorial.
+
+    Alternative, as suggested in [2]_, the :math:`\tau` could be set to 1. The parameter :math:`\lambda` can then be set
+    to maximize the value of the denoiser and a second tuning parameter can be added directly to :math:`f`.
 
     .. [1] Venkatakrishnan, S. V., Bouman, C. A. and Wohlberg, B.
        "Plug-and-Play priors for model based reconstruction",
        IEEE. 2013.
 
+    .. [1] Meinhardt, T., Moeller, M, Hazirbas, C., and Cremer, D.
+       "Learning Proximal Operators: Using Denoising Networks for Regularizing Inverse Imaging Problems",
+       arXiv. 2017.
+
     """
     # Denoiser
     proxpnp = _Denoise(denoiser, dims=dims)

pyproximal/projection/Euclidean.py

@@ -37,6 +37,6 @@ def __init__(self, center, radius):
 
     def __call__(self, x):
         x = self.center +  \
-            self.radius  / (max(np.linalg.norm(x - self.center),
-                                self.radius)) * (x - self.center)
+            self.radius / (max(np.linalg.norm(x - self.center),
+                               self.radius)) * (x - self.center)
         return x