Problem 2.2

to investigate higher order derivatives we can use Prewitt operators and compute convolution.
for first order derivative it has following form

with kernel

and similar idea is applied to higher order derivatives with kernels

First calculate vertical and horizontal Prewitt operators by convolution which will give us approximation of partial derivatives

and then calculate magnitude M which is "fusion" of both operators to get the final result

and to get even more clear image we can use Thresholding to get following result

we can even get additional information by calculating angle of the edge.
by calculating following expression

the color of the edges corresponds to the angle. For example, an angle of 0 is red, and an angle of is blue.

and second order Prewitt operator detects edges more clearly

and following implementation

import numpy as np
import skimage.io as io
import skimage.color as color
import matplotlib
from scipy.signal import convolve

matplotlib.use("TkAgg")

import matplotlib.pyplot as plt


def convert_to_rgb_format(img):
    if img.shape[2] == 4:
        img = color.rgba2rgb(img)
    return img


def convert_to_gray_scale(img):
    return color.rgb2gray(img)


def threshold(g, T):
    h, w = g.shape[:2]
    for j in range(h):
        for i in range(w):
            if g[j, i] >= T:
                g[j, i] = 1
            else:
                g[j, i] = 0
    return g


def prewitt_operator(g):
    h_x = np.array([[1, 0, -1], [1, 0, -1], [1, 0, -1]])
    h_y = np.array([[1, 1, 1], [0, 0, 0], [-1, -1, -1]])

    grad_x = convolve(g, h_x, mode="same")
    grad_y = convolve(g, h_y, mode="same")

    M = np.sqrt(grad_x**2 + grad_y**2)  # NOTE: magnitude M

    A = np.arctan(grad_y / grad_x)  # NOTE: angle A

    return M, A


img = io.imread("./sky.png")

img = convert_to_rgb_format(img)

g = convert_to_gray_scale(img)

M, A = prewitt_operator(g)

# M, A = prewitt_operator(M)

plt.figure(figsize=(10, 5))

plt.subplot(1, 2, 1)
plt.imshow(img)
plt.title("Original Image")

plt.subplot(1, 2, 2)
plt.imshow(threshold(M, 0.5), cmap="gray")
plt.title("Prewitt Operator with magnitude M")
plt.show()
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Problem 2.1 Problem 2.3