314 lines
12 KiB
Python
314 lines
12 KiB
Python
import numpy as np
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import numpy.typing as npt
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from matplotlib import pyplot as plt
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import cv2
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import uz_framework.image as uz_image
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import uz_framework.text as uz_text
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##############################################
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# EXCERCISE 1: Exercise 1: Image derivatives #
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##############################################
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def ex1():
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#one_a()
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#one_b()
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#one_c()
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one_d()
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def one_a() -> None:
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"""
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Follow the equations above and derive the equations used to compute first and
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second derivatives with respect to y: Iy(x, y), Iyy(x, y), as well as the mixed derivative
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Ixy(x, y)
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"""
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def one_b() -> None:
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"""
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Implement a function that computes the derivative of a 1-D Gaussian kernel
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Implement the function gaussdx(sigma) that works the same as function gauss
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from the previous assignment. Don’t forget to normalize the kernel. Be careful as
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the derivative is an odd function, so a simple sum will not do. Instead normalize the
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kernel by dividing the values such that the sum of absolute values is 1. Effectively,
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you have to divide each value by sum(abs(gx(x))).
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"""
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sigmas = [0.5, 1, 2]
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for sigma in sigmas:
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kernel = uz_image.gaussdx(sigma)
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print(kernel)
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def one_c() -> None:
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"""
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The properties of the filter can be analyzed by using an impulse response function.
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This is performed as a convolution of the filter with a Dirac delta function. The
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discrete version of the Dirac function is constructed as a finite image that has all
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elements set to 0 except the central element, which is set to a high value (e.g. 1).
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Generate a 1-D Gaussian kernel G and a Gaussian derivative kernel D.
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What happens if you apply the following operations to the impulse image?
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(a) First convolution with G and then convolution with GT
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(b) First convolution with G and then convolution with DT
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(c) First convolution with D and then convolution with GT
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(d) First convolution with GT and then convolution with D.
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(e) First convolution with DT and then convolution with G.
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Is the order of operations important? Display the images of the impulse responses
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for different combinations of operations.
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"""
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impulse = uz_image.generate_dirac_impulse(50)
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gauss = np.array([uz_image.get_gaussian_kernel(3)])
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gaussdx = np.array([uz_image.gaussdx(3)])
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# Becouse CV2 applies the correlation instead of convolution, we need to flip the kernels
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gauss = np.flip(gauss, axis=1)
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gaussdx = np.flip(gaussdx, axis=1)
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fig, axs = plt.subplots(2, 3)
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# Plot impulse only
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axs[0, 0].imshow(impulse, cmap='gray')
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axs[0, 0].set_title('Impulse')
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# Plot impulse after convolution with G and GT
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g_gt_impulse = impulse.copy()
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g_gt_impulse = cv2.filter2D(g_gt_impulse, cv2.CV_64F, gauss)
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g_gt_impulse = cv2.filter2D(g_gt_impulse, cv2.CV_64F, gauss.T)
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axs[1, 0].imshow(g_gt_impulse, cmap='gray')
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axs[1, 0].set_title('impulse * G * GT')
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# Plot impulse after convolution with G and DT
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g_dt_impulse = impulse.copy()
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g_dt_impulse = cv2.filter2D(g_dt_impulse, cv2.CV_64F, gauss)
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g_dt_impulse = cv2.filter2D(g_dt_impulse, cv2.CV_64F, gaussdx.T)
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axs[0, 1].imshow(g_dt_impulse, cmap='gray')
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axs[0, 1].set_title('impulse * G * DT')
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# Plot impulse after convolution with D and GT
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d_gt_impulse = impulse.copy()
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d_gt_impulse = cv2.filter2D(d_gt_impulse, cv2.CV_64F, gaussdx)
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d_gt_impulse = cv2.filter2D(d_gt_impulse, cv2.CV_64F, gauss.T)
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axs[0, 2].imshow(d_gt_impulse, cmap='gray')
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axs[0, 2].set_title('impulse * D * GT')
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# Plot impulse after convolution with GT and D
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gt_d_impulse = impulse.copy()
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gt_d_impulse = cv2.filter2D(gt_d_impulse, cv2.CV_64F, gauss.T)
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gt_d_impulse = cv2.filter2D(gt_d_impulse, cv2.CV_64F, gaussdx)
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axs[1, 1].imshow(gt_d_impulse, cmap='gray')
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axs[1, 1].set_title('impulse * GT * D')
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# Plot impulse after convolution with DT and G
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dt_g_impulse = impulse.copy()
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dt_g_impulse = cv2.filter2D(dt_g_impulse, cv2.CV_64F, gaussdx.T)
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dt_g_impulse = cv2.filter2D(dt_g_impulse, cv2.CV_64F, gauss)
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axs[1, 2].imshow(dt_g_impulse, cmap='gray')
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axs[1, 2].set_title('impulse * DT * G')
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plt.show()
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def one_d() -> None:
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"""
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Implement a function that uses functions gauss and gaussdx to compute both
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partial derivatives of a given image with respect to x and with respect to y.
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Similarly, implement a function that returns partial second order derivatives of a
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given image.
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Additionally, implement the function gradient_magnitude that accepts a grayscale
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image I and returns both derivative magnitudes and derivative angles. Magnitude
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is calculated as m(x, y) = sqrt(Ix(x,y)^2 + Iy(x, y)^2) and angles are calculated as
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φ(x, y) = arctan(Iy(x, y)/Ix(x, y))
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Hint: Use function np.arctan2 to avoid division by zero for calculating the arctangent function.
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Use all the implemented functions on the same image and display the results in the
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same window.
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"""
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museum = uz_image.imread_gray('./images/museum.jpg', uz_image.ImageType.float64)
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museum_x, museum_y = uz_image.derive_image_first_order(museum, 1)
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(museum_xx, museum_xy) , (_, museum_yy) = uz_image.derive_image_second_order(museum, 1)
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derivative_magnitude, derivative_angle = uz_image.gradient_magnitude(museum, 1)
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fig, axs = plt.subplots(2, 4)
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fig.suptitle('Museum')
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axs[0,0].imshow(museum, cmap='gray')
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axs[0,0].set_title('Original')
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axs[0, 1].imshow(museum_x, cmap='gray')
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axs[0, 1].set_title('I_x')
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axs[0, 2].imshow(museum_y, cmap='gray')
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axs[0, 2].set_title('I_y')
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axs[1, 0].imshow(museum_xx, cmap='gray')
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axs[1, 0].set_title('I_xx')
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axs[1, 1].imshow(museum_xy, cmap='gray')
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axs[1, 1].set_title('I_xy')
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axs[1, 2].imshow(museum_yy, cmap='gray')
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axs[1, 2].set_title('I_yy')
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axs[0, 3].imshow(derivative_magnitude, cmap='gray')
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axs[0, 3].set_title('I_mag')
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axs[1, 3].imshow(derivative_angle, cmap='gray')
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axs[1, 3].set_title('I_dir')
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plt.show()
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############################################
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# EXCERCISE 2: Exercise 1: Edges in images #
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############################################
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def ex2():
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#two_a()
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two_b()
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two_c()
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def two_a():
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"""
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Firstly, create a function findedges that accepts an image I, and the parameters
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sigma and theta.
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The function should create a binary matrix Ie that only keeps pixels higher than
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threshold theta:
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Ie(x, y) =
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1 ; Imag(x, y) ≥ ϑ
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0 ; otherwise (6)
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Test the function with the image museum.png and display the results for different
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values of the parameter theta. Can you set the parameter so that all the edges in
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the image are clearly visible?
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"""
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SIGMA = 0.2
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THETA = 0.16
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museum = uz_image.imread_gray('./images/museum.jpg', uz_image.ImageType.float64)
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museum_edges = uz_image.find_edges_primitive(museum, SIGMA, THETA)
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plt.imshow(museum_edges, cmap='gray')
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plt.show()
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def two_b():
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"""
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Using magnitude produces only a first approximation of detected edges. Unfortunately,
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these are often wide and we would like to only return edges one pixel wide.
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Therefore, you will implement non-maxima suppression based on the image derivative magnitudes and angles.
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Iterate through all the pixels and for each search its
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8-neighborhood. Check the neighboring pixels parallel to the gradient direction and
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set the current pixel to 0 if it is not the largest in the neighborhood (based on
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derivative magnitude). You only need to compute the comparison to actual pixels,
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interpolating to more accuracy is not required.
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"""
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SIGMA = 1
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THETA = 0.01
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museum = uz_image.imread_gray('./images/museum.jpg', uz_image.ImageType.float64)
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museum_edges = uz_image.find_edges_nms(museum, SIGMA, THETA)
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plt.imshow(museum_edges, cmap='gray')
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plt.show()
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def two_c():
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"""
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The final step of Canny’s algorithm is edge tracking by hysteresis.
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Add the final step after performing non-maxima suppression along edges. Hysteresis
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uses two thresholds tlow < thigh, keeps all pixels above thigh and discards all pixels
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below tlow. The pixels between the thresholds are kept only if they are connected to
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a pixel above thigh.
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Hint: Since we are looking for connected components containing at least one pixel
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above thigh, you could use something like cv2.connectedComponentsWithStats to
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extract them. Try to avoid explicit for loops as much as possible
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"""
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SIGMA = 1
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THETA = 0.02
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T_LOW = 0.04
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T_HIGH = 0.16
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museum = uz_image.imread_gray('./images/museum.jpg', uz_image.ImageType.float64)
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connected = uz_image.find_edges_canny(museum, SIGMA, THETA, T_LOW, T_HIGH)
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plt.imshow(connected, cmap='gray')
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plt.show()
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############################################
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# EXCERCISE 2: Exercise 1: Edges in images #
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############################################
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def ex3():
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#three_a()
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three_b()
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#three_c()
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def three_a():
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"""
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Create an accumulator array defined by the resolution on ρ and ϑ values. Calculate the
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sinusoid that represents all the lines that pass through some nonzero point.
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Increment the corresponding cells in the accumulator array. Experiment with different
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positions of the nonzero point to see how the sinusoid changes. You can set
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the number of accumulator bins on each axis to 300 to begin with.
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"""
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x_y_values = np.array([[10, 10], [30, 60], [50, 20], [80, 90]])
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fig, axs = plt.subplots(2, 2)
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fig.suptitle('Trasformation of points into hugh space')
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for i in range(0, 4):
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accumulator = uz_image.hough_transform_a_point(x_y_values[i][0], x_y_values[i][1], 300)
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axs[i // 2, i % 2].imshow(accumulator)
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axs[i // 2, i % 2].set_title(f'x = {x_y_values[i][0]}, y = {x_y_values[i][1]}')
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plt.show()
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def three_b():
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"""
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Implement the function hough_find_lines that accepts a binary image, the number
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of bins for ϑ and ρ (allow the possibility of them being different) and a threshold.
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Create an accumulator matrix A for the parameter space (ρ, ϑ). Parameter ϑ is
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defined in the interval from −π/2 to π/2, ρ is defined on the interval from −D to
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D, where D is the length of the image diagonal. For each nonzero pixel in the image,
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generate a curve in the (ρ, ϑ) space by using the equation (7) for all possible values
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of ϑ and increase the corresponding cells in A. Display the accumulator matrix. Test
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the method on your own synthetic images ((e.g. 100 × 100 black image, with two
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white pixels at (10, 10) and (10, 20)).
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Finally, test your function on two synthetic images oneline.png and rectangle.png. First, you should obtain an edge map for each image using either your
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function findedges or some inbuilt function. Run your implementation of the Hough
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algorithm on the resulting edge maps.
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"""
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SIGMA = 1
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THETA = 0.02
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T_LOW = 0.04
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T_HIGH = 0.16
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synthetic_image = np.zeros((100, 100))
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synthetic_image[10, 10] = 1
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synthetic_image[10, 20] = 1
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oneline_image = uz_image.imread_gray('./images/oneline.png', uz_image.ImageType.float64)
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rectangle_image = uz_image.imread_gray('./images/rectangle.png', uz_image.ImageType.float64)
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oneline_image_edges = uz_image.find_edges_canny(oneline_image, SIGMA, THETA, T_LOW, T_HIGH)
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rectangle_image_edges = uz_image.find_edges_canny(rectangle_image, SIGMA, THETA, T_LOW, T_HIGH)
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fig, axs = plt.subplots(3, 3)
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axs[0, 0].imshow(synthetic_image, cmap='gray')
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axs[0, 0].set_title('Synthetic image')
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axs[2, 0].imshow(uz_image.hough_find_lines(synthetic_image, 200, 200, 0.2))
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axs[0, 1].imshow(oneline_image, cmap='gray')
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axs[0, 1].set_title('Oneline image')
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axs[1, 1].imshow(oneline_image_edges, cmap='gray')
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axs[0, 2].imshow(rectangle_image, cmap='gray')
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axs[0, 2].set_title('Rectangle image')
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axs[1, 2].imshow(rectangle_image_edges, cmap='gray')
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axs[1, 0].set_visible(False)
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axs[2, 1].imshow(uz_image.hough_find_lines(oneline_image_edges, 200, 200, 0.2))
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axs[2, 2].imshow(uz_image.hough_find_lines(rectangle_image_edges, 200, 200, 0.2))
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plt.show()
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# ######## #
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# SOLUTION #
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# ######## #
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def main():
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#ex1()
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#ex2()
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ex3()
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if __name__ == '__main__':
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main()
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