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.. "auto_examples/2_kernels/multi_scale_gaussianRKHS.py"
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.. only:: html
.. note::
:class: sphx-glr-download-link-note
:ref:`Go to the end <sphx_glr_download_auto_examples_2_kernels_multi_scale_gaussianRKHS.py>`
to download the full example code.
.. rst-class:: sphx-glr-example-title
.. _sphx_glr_auto_examples_2_kernels_multi_scale_gaussianRKHS.py:
.. _mutli_scale_gaussianRK:
A multiscale Gaussian Reproducing Kernel
================================================
In this file we look more closely at the multiscale Gaussian Reproducing Kernel (RK)
The multiscale Gaussian RK is build as a sum of Gaussian RK with different
sigma. In Metamorphosis, the RK is used to parameterize V the space of
acceptable vector fields. In practice, it has a smoothing effect on vector fields.
However, Gaussian RK erase the high frequency of the vector fields which
prevents the vector fields to match small details in the images. To overcome this
issue, we can use a multiscale Gaussian RK, that will keep smoothing properties
while keeping some high frequency information. It can be seen as a compromise
kernel.
The formal definition is:
Let $\Gamma = { \sigma_1, \sigma_2, \ldots, \sigma_n}$ be a list of standard deviations.
.. math::
\mathrm{kernel}_\Gamma = \sum_{\sigma \in \Gamma} \frac {1}{nk_\sigma} \exp\left(\frac{-x^2}{2 \sigma^2}\right)
where $n$ is the number of elements in $\Gamma$.
if normalised is True, $k$ is equal to:
.. math::
k_\sigma = \sum_{x \in Omega} \exp\left(\frac{-x^2}{2 \sigma^2}\right)
else, $k$ is equal to 1.
.. GENERATED FROM PYTHON SOURCE LINES 34-35
First let's Import the necessary libraries
.. GENERATED FROM PYTHON SOURCE LINES 35-46
.. code-block:: Python
import numpy as np
from demeter.constants import DLT_KW_IMAGE
import matplotlib.pyplot as plt
import torch
import demeter.utils.reproducing_kernels as rk
import demeter.utils.torchbox as tb
plt.rcParams['text.usetex'] = False
.. GENERATED FROM PYTHON SOURCE LINES 47-52
To use the multi-scale Gaussian RK, we need to choose
a list of sigma. The larger the sigma, the more the RK will
smooth the vector fields. The smaller the sigma, the more the RK will
keep the high frequency information of the vector fields.
Feel free to change the sigma_list to see the effect of the RK on next figures
.. GENERATED FROM PYTHON SOURCE LINES 52-63
.. code-block:: Python
sigma_list = [
# (1,1),
(2,2),
(5,5),
(10,10),
# (16,16)
# (20,20),
]
normalize = True
.. GENERATED FROM PYTHON SOURCE LINES 64-67
First let's see the kernels of the Gaussian Reproducing Kernel for different sigma
superimposed on the Multi-scale Gaussian Reproducing Kernel. We plot a 1D slice
of the 2D kernels at its center.
.. GENERATED FROM PYTHON SOURCE LINES 67-110
.. code-block:: Python
def see_kernels_filter_2D(sigma_list,ax=None, force_xticks=None):
mono_gauss = [
rk.GaussianRKHS(s,'constant',normalized=normalize)
for s in sigma_list
]
multi_RKHS = rk.Multi_scale_GaussianRKHS(sigma_list, normalized=normalize)
font_size= 20
if ax is None:
fig,ax = plt.subplots(figsize=(10,5))
kernel_multi = multi_RKHS.kernel
_,h,w = kernel_multi.shape
tick_max = w if force_xticks is None else force_xticks
x_scale = (torch.arange(w) - w/2)
ax.plot(x_scale,kernel_multi[0,h//2],label=str(sigma_list),c='black')
style = ['--','-.',':']
for i,s in enumerate(sigma_list):
kernel_m = mono_gauss[i].kernel
_,hh,ww = kernel_m.shape
# x_scale = (torch.arange(ww)+((tick_max - ww)/2))
x_scale = (torch.arange(ww) - ww/2)
ax.plot(x_scale,
mono_gauss[i].kernel[0,hh//2],
label=str(s),linestyle=style[i - 3*(i//3)])
ax.set_ylabel(r'$K_\sigma(x)$',fontsize=font_size)
ax.set_xlabel(r'$x$',fontsize=font_size)
# tick_max = w if force_xticks is None else force_xticks
x_ticks = np.linspace(-tick_max/2,tick_max/2,5)
ax.set_xticks(x_ticks)
ax.tick_params(axis='both',
#which='minor',
labelsize=font_size)
ax.legend(loc='upper left',
fontsize=font_size,
bbox_to_anchor=(.75, 1.),
# ncol=len(sigma_list)+1
)
see_kernels_filter_2D(sigma_list)
.. image-sg:: /auto_examples/2_kernels/images/sphx_glr_multi_scale_gaussianRKHS_001.png
:alt: multi scale gaussianRKHS
:srcset: /auto_examples/2_kernels/images/sphx_glr_multi_scale_gaussianRKHS_001.png
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 111-114
Now let's visualize the convolution of an image by the Gaussian RKs and the
multi-scale Gaussian RK. For visualization purposes, we will use images
but keep in mind that the RK is intended to be used on vector fields.
.. GENERATED FROM PYTHON SOURCE LINES 114-146
.. code-block:: Python
def see_im_convoled_by_kernel(kernelOp,I,ax):
_,_,H,W = I.shape
if kernelOp is None:
I_g =I
else:
I_g = kernelOp(I)
ax[0].imshow(I_g[0,0], **DLT_KW_IMAGE)
ax[1].plot(I_g[0,0,H//2])
def see_mono_kernels(I,sigma_list):
fig,ax = plt.subplots(2,len(sigma_list)+2,figsize=(20,5))
fig.suptitle(f'Convolution of an image by a reproducing kernel, {sigma_list}')
see_im_convoled_by_kernel(None,I,ax[:,0])
for i,s in enumerate(sigma_list):
grk = rk.GaussianRKHS(s,'constant')
see_im_convoled_by_kernel(grk,I,ax[:,i+1])
ax[0,i+1].set_title(r'$\sigma = $'+str(grk.sigma))
mgrk = rk.Multi_scale_GaussianRKHS(sigma_list, normalized=normalize)
see_im_convoled_by_kernel(mgrk,I,ax[:,-1])
ax[0,-1].set_title(r"Multi-scale ")
# You can choose different images to see the effect of the RK on them
img_name = '01' # simple disk
img_name = 'sri24' # slice of a brain
img = tb.reg_open(img_name,size = (300,300))
see_mono_kernels(img,sigma_list)
plt.show()
.. image-sg:: /auto_examples/2_kernels/images/sphx_glr_multi_scale_gaussianRKHS_002.png
:alt: Convolution of an image by a reproducing kernel, [(2, 2), (5, 5), (10, 10)], $\sigma = $(2, 2), $\sigma = $(5, 5), $\sigma = $(10, 10), Multi-scale
:srcset: /auto_examples/2_kernels/images/sphx_glr_multi_scale_gaussianRKHS_002.png
:class: sphx-glr-single-img
.. rst-class:: sphx-glr-timing
**Total running time of the script:** (0 minutes 0.422 seconds)
.. _sphx_glr_download_auto_examples_2_kernels_multi_scale_gaussianRKHS.py:
.. only:: html
.. container:: sphx-glr-footer sphx-glr-footer-example
.. container:: sphx-glr-download sphx-glr-download-jupyter
:download:`Download Jupyter notebook: multi_scale_gaussianRKHS.ipynb <multi_scale_gaussianRKHS.ipynb>`
.. container:: sphx-glr-download sphx-glr-download-python
:download:`Download Python source code: multi_scale_gaussianRKHS.py <multi_scale_gaussianRKHS.py>`
.. container:: sphx-glr-download sphx-glr-download-zip
:download:`Download zipped: multi_scale_gaussianRKHS.zip <multi_scale_gaussianRKHS.zip>`
.. only:: html
.. rst-class:: sphx-glr-signature
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