214 lines
7.9 KiB
Python
214 lines
7.9 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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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 = 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_nms(museum, SIGMA, THETA)
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plt.imshow(museum_edges, cmap='gray')
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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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