contextual_enhancement.py

# -*- coding: utf-8 -*-
"""
==============================================
Crossing-preserving contextual enhancement
==============================================

This demo presents an example of crossing-preserving contextual enhancement of
FOD/ODF fields [Meesters2016]_, implementing the contextual PDE framework
of [Portegies2015a]_ for processing HARDI data. The aim is to enhance the
alignment of elongated structures in the data such that crossing/junctions are
maintained while reducing noise and small incoherent structures. This is
achieved via a hypo-elliptic 2nd order PDE in the domain of coupled positions
and orientations :math:`\mathbb{R}^3 \rtimes S^2`. This domain carries a
non-flat geometrical differential structure that allows including a notion of
alignment between neighboring points.

Let :math:`({\bf y},{\bf n}) \in \mathbb{R}^3\rtimes S^2` where
:math:`{\bf y} \in \mathbb{R}^{3}` denotes the spatial part, and
:math:`{\bf n} \in S^2` the angular part.
Let :math:`W:\mathbb{R}^3\rtimes S^2\times \mathbb{R}^{+} \to \mathbb{R}` be
the function representing the evolution of FOD/ODF field. Then, the contextual
PDE with evolution time :math:`t\geq 0` is given by:

.. math::

  \begin{cases}
  \frac{\partial}{\partial t} W({\bf y},{\bf n},t) &= ((D^{33}({\bf n} \cdot
          \nabla)^2 + D^{44} \Delta_{S^2})W)({\bf y},{\bf n},t)
  \\ W({\bf y},{\bf n},0) &= U({\bf y},{\bf n})
  \end{cases},

where:

* :math:`D^{33}>0` is  the coefficient for the spatial smoothing (which goes only in the direction of :math:`n`);

* :math:`D^{44}>0` is the coefficient for the angular smoothing (here :math:`\Delta_{S^2}` denotes the Laplace-Beltrami operator on the sphere :math:`S^2`);

* :math:`U:\mathbb{R}^3\rtimes S^2 \to \mathbb{R}` is the initial condition given by the noisy FOD/ODF’s field.

This equation is solved via a shift-twist convolution (denoted by :math:`\ast_{\mathbb{R}^3\rtimes S^2}`) with its corresponding kernel :math:`P_t:\mathbb{R}^3\rtimes S^2 \to \mathbb{R}^+`:

.. math::

  W({\bf y},{\bf n},t) = (P_t \ast_{\mathbb{R}^3 \rtimes S^2} U)({\bf y},{\bf n})
  = \int_{\mathbb{R}^3} \int_{S^2} P_t (R^T_{{\bf n}^\prime}({\bf y}-{\bf y}^\prime),
   R^T_{{\bf n}^\prime} {\bf n} ) U({\bf y}^\prime, {\bf n}^\prime)

Here, :math:`R_{\bf n}` is any 3D rotation that maps the vector :math:`(0,0,1)`
onto :math:`{\bf n}`.

Note that the shift-twist convolution differs from a Euclidean convolution and
takes into account the non-flat structure of the space :math:`\mathbb{R}^3\rtimes S^2`.

The kernel :math:`P_t` has a stochastic interpretation [DuitsAndFranken2011]_.
It can be seen as the limiting distribution obtained by accumulating random
walks of particles in the position/orientation domain, where in each step the
particles can (randomly) move forward/backward along their current orientation,
and (randomly) change their orientation. This is an extension to the 3D case of
the process for contour enhancement of 2D images.

.. figure:: _static/stochastic_process.jpg
   :scale: 60 %
   :align: center

   The random motion of particles (a) and its corresponding probability map
   (b) in 2D. The 3D kernel is shown on the right. Adapted from
   [Portegies2015a]_.

In practice, as the exact analytical formulas for the kernel :math:`P_t`
are unknown, we use the approximation given in [Portegies2015b]_.

"""

"""
The enhancement is evaluated on the Stanford HARDI dataset
(150 orientations, b=2000 $s/mm^2$) where Rician noise is added. Constrained
spherical deconvolution is used to model the fiber orientations.

"""

import numpy as np
from dipy.data import fetch_stanford_hardi, read_stanford_hardi
from dipy.sims.voxel import add_noise

# Read data
fetch_stanford_hardi()
img, gtab = read_stanford_hardi()
data = img.get_data()

# Add Rician noise
from dipy.segment.mask import median_otsu
b0_slice = data[:, :, :, 1]
b0_mask, mask = median_otsu(b0_slice)
np.random.seed(1)
data_noisy = add_noise(data, 10.0, np.mean(b0_slice[mask]), noise_type='rician')

# Select a small part of it.
padding = 3  # Include a larger region to avoid boundary effects
data_small = data[25-padding:40+padding, 65-padding:80+padding, 35:42]
data_noisy_small = data_noisy[25-padding:40+padding, 65-padding:80+padding, 35:42]

"""
Enables/disables interactive visualization
"""

interactive = False

"""
Fit an initial model to the data, in this case Constrained Spherical
Deconvolution is used.
"""

# Perform CSD on the original data
from dipy.reconst.csdeconv import auto_response
from dipy.reconst.csdeconv import ConstrainedSphericalDeconvModel
response, ratio = auto_response(gtab, data, roi_radius=10, fa_thr=0.7)
csd_model_orig = ConstrainedSphericalDeconvModel(gtab, response)
csd_fit_orig = csd_model_orig.fit(data_small)
csd_shm_orig = csd_fit_orig.shm_coeff

# Perform CSD on the original data + noise
response, ratio = auto_response(gtab, data_noisy, roi_radius=10, fa_thr=0.7)
csd_model_noisy = ConstrainedSphericalDeconvModel(gtab, response)
csd_fit_noisy = csd_model_noisy.fit(data_noisy_small)
csd_shm_noisy = csd_fit_noisy.shm_coeff

"""
Inspired by [Rodrigues2010]_, a lookup-table is created, containing
rotated versions of the kernel :math:`P_t` sampled over a discrete set of
orientations. In order to ensure rotationally invariant processing, the discrete
orientations are required to be equally distributed over a sphere. By default,
a sphere with 100 directions is used.

"""

from dipy.denoise.enhancement_kernel import EnhancementKernel
from dipy.denoise.shift_twist_convolution import convolve

# Create lookup table
D33 = 1.0
D44 = 0.02
t = 1
k = EnhancementKernel(D33, D44, t)

"""
Visualize the kernel
"""

from dipy.viz import window, actor
from dipy.data import get_sphere
from dipy.reconst.shm import sf_to_sh, sh_to_sf
ren = window.Renderer()

# convolve kernel with delta spike
spike = np.zeros((7, 7, 7, k.get_orientations().shape[0]), dtype=np.float64)
spike[3, 3, 3, 0] = 1
spike_shm_conv = convolve(sf_to_sh(spike, k.get_sphere(), sh_order=8), k,
                          sh_order=8, test_mode=True)

sphere = get_sphere('symmetric724')
spike_sf_conv = sh_to_sf(spike_shm_conv, sphere, sh_order=8)
model_kernel = actor.odf_slicer(spike_sf_conv * 6,
                                sphere=sphere,
                                norm=False,
                                scale=0.4)
model_kernel.display(x=3)
ren.add(model_kernel)
ren.set_camera(position=(30, 0, 0), focal_point=(0, 0, 0), view_up=(0, 0, 1))
window.record(ren, out_path='kernel.png', size=(900, 900))
if interactive:
    window.show(ren)

"""
.. figure:: kernel.png
   :align: center

   Visualization of the contour enhancement kernel.
"""

"""
Shift-twist convolution is applied on the noisy data
"""

# Perform convolution
csd_shm_enh = convolve(csd_shm_noisy, k, sh_order=8)


"""
The Sharpening Deconvolution Transform is applied to sharpen the ODF field.
"""

# Sharpen via the Sharpening Deconvolution Transform
from dipy.reconst.csdeconv import odf_sh_to_sharp
csd_shm_enh_sharp = odf_sh_to_sharp(csd_shm_enh, sphere,  sh_order=8, lambda_=0.1)

# Convert raw and enhanced data to discrete form
csd_sf_orig = sh_to_sf(csd_shm_orig, sphere, sh_order=8)
csd_sf_noisy = sh_to_sf(csd_shm_noisy, sphere, sh_order=8)
csd_sf_enh = sh_to_sf(csd_shm_enh, sphere, sh_order=8)
csd_sf_enh_sharp = sh_to_sf(csd_shm_enh_sharp, sphere, sh_order=8)

# Normalize the sharpened ODFs
csd_sf_enh_sharp = csd_sf_enh_sharp * np.amax(csd_sf_orig)/np.amax(csd_sf_enh_sharp) * 1.25

"""
The end results are visualized. It can be observed that the end result after
diffusion and sharpening is closer to the original noiseless dataset.
"""

ren = window.Renderer()

# original ODF field
fodf_spheres_org = actor.odf_slicer(csd_sf_orig,
                                    sphere=sphere,
                                    scale=0.4,
                                    norm=False)
fodf_spheres_org.display(z=3)
fodf_spheres_org.SetPosition(0, 25, 0)
ren.add(fodf_spheres_org)

# ODF field with added noise
fodf_spheres = actor.odf_slicer(csd_sf_noisy,
                                sphere=sphere,
                                scale=0.4,
                                norm=False,)
fodf_spheres.SetPosition(0, 0, 0)
ren.add(fodf_spheres)

# Enhancement of noisy ODF field
fodf_spheres_enh = actor.odf_slicer(csd_sf_enh,
                                    sphere=sphere,
                                    scale=0.4,
                                    norm=False)
fodf_spheres_enh.SetPosition(25, 0, 0)
ren.add(fodf_spheres_enh)

# Additional sharpening
fodf_spheres_enh_sharp = actor.odf_slicer(csd_sf_enh_sharp,
                                          sphere=sphere,
                                          scale=0.4,
                                          norm=False)
fodf_spheres_enh_sharp.SetPosition(25, 25, 0)
ren.add(fodf_spheres_enh_sharp)

window.record(ren, out_path='enhancements.png', size=(900, 900))
if interactive:
    window.show(ren)

"""

.. figure:: enhancements.png
   :align: center

   The results after enhancements. Top-left: original noiseless data.
   Bottom-left: original data with added Rician noise (SNR=10). Bottom-right:
   After enhancement of noisy data. Top-right: After enhancement and sharpening
   of noisy data.

References
----------

.. [Meesters2016] S. Meesters, G. Sanguinetti, E. Garyfallidis, J. Portegies,
   R. Duits. (2016) Fast implementations of contextual PDE’s for HARDI data
   processing in DIPY. ISMRM 2016 conference.

.. [Portegies2015a] J. Portegies, R. Fick, G. Sanguinetti, S. Meesters,
   G.Girard, and R. Duits. (2015) Improving Fiber Alignment in HARDI by
   Combining Contextual PDE flow with Constrained Spherical Deconvolution.
   PLoS One.

.. [Portegies2015b] J. Portegies, G. Sanguinetti, S. Meesters, and R. Duits.
   (2015) New Approximation of a Scale Space Kernel on SE(3) and Applications
   in Neuroimaging. Fifth International Conference on Scale Space and
   Variational Methods in Computer Vision.

.. [DuitsAndFranken2011] R. Duits and E. Franken (2011) Left-invariant
   diffusions on the space of positions and orientations and their application
   to crossing-preserving smoothing of HARDI images. International Journal of
   Computer Vision, 92:231-264.

.. [Rodrigues2010] P. Rodrigues, R. Duits, B. Romeny, A. Vilanova (2010).
   Accelerated Diffusion Operators for Enhancing DW-MRI. Eurographics Workshop
   on Visual Computing for Biology and Medicine. The Eurographics Association.

"""