"""
Convolution and frontier of the Mandelbrot set
================================================
"""

#################################
# Convolution with a 2D kernel
# ---------------------------------
# Before we begin, a reminder of the 2D convolution between two matrix can be useful.
# In our case, we will use a :math:`3\times 3` kernel. So, the convolution of a kernel with a matrix is defined as the
# sum of the conter-row-wise by row-wise product of the elements ie the last element of the kernel is multiplied by the first of the matrix,
# the penultimate of the kernel (at the left of the last) is multiplied by the seconde one of the matrix (at the right of the first) and so
# on from the antepenultimate to the first one. In a formula we have :
#
# .. math::
#
#    \begin{bmatrix} k_1 & k_2 & k_3 \\ k_4 & k_5 & k_6 \\ k_7 & k_8 & k_9 \\ \end{bmatrix} \star
#     \begin{bmatrix} c_1 & c_2 & c_3 \\ c_4 & c_5 & c_6 \\ c_7 & c_8 & c_9 \\ \end{bmatrix} =
#       k_1 c_9 + k_2c_8 + \dots + k_9c_1
#
# There are mutliple ways to get the edges of a shape using this method. We chose to use a kernel with only :math:`-1` on its borders
# and a value :math:`k_5=8=-\sum_{i=1,\ i\neq 5}^9 k_i`. We also need to pad the image in order to get all the pixels in it, and not
# forget the borders. Let's take a look at the result.


import chaoseverywhere as chaos
import numpy as np
import matplotlib.pyplot as plt
from scipy.signal import convolve2d

mandel = chaos.Mandelbrot_disp(-.5, 0, 1.5).mandelbrot()
kernel_edge_detect = np.array([[-1, -1, -1], [-1, 8, -1], [-1, -1, -1]])
pad_mandel = np.pad(mandel, ((1, 1), (1, 1)), "maximum")
bound = convolve2d(pad_mandel, kernel_edge_detect, mode='valid').astype(bool) * mandel

plt.imshow(bound, cmap='bone')
plt.axis('off')
plt.show()