MP3: Image Filtering

In this lab, you'll filter an image using weighted-average filters and edge-detection filters.

In order to make sure everything works, you might want to go to the command line, and run

pip install -r requirements.txt

This will install the modules that are used on the autograder, including numpy, h5py, and the gradescope utilities.

import submitted
import importlib
import numpy as np
import matplotlib.pyplot as plt
import h5py
from PIL import Image
import matplotlib.figure
%matplotlib inline

---

Part 0: Load the data, and show the image

imagefile = 'image.jpg'
original = np.asarray(Image.open(imagefile)).astype('float64')
print('The image size is:')
print(original.shape)

The image size is:
(600, 800, 3)

imagefile = 'image.jpg'
original = np.asarray(Image.open(imagefile)).astype('float64')
fig = plt.figure(figsize=(10, 8))
ax = fig.subplots()
ax.imshow(original.astype(int))
ax.set_title('Here is the original image!')

Text(0.5, 1.0, 'Here is the original image!')

fig = plt.figure(figsize=(10,8))
axs = fig.subplots(3,1,sharex=True)
axs[0].plot(original[300,:,0])
axs[0].set_title('Row 300 of the image, red color plane')
axs[1].plot(original[300,:,1])
axs[1].set_title('Row 300 of the image, green color plane')
axs[2].plot(original[300,:,2])
axs[2].set_title('Row 300 of the image, blue color plane')

Text(0.5, 1.0, 'Row 300 of the image, blue color plane')

Part 1: Create a rectangular filter, and show it in a plot

importlib.reload(submitted)
help(submitted.todo_rectangular_filter)

Help on function todo_rectangular_filter in module submitted:

todo_rectangular_filter()
    Create 11 samples of a 7-sample rectangular filter,
    h[n]=1/7 for -3 <= n <= 3, h[n]=0 otherwise.
    Return 11 samples of both n_rect and h_rect.

importlib.reload(submitted)
(n_rect, h_rect) = submitted.todo_rectangular_filter()
fig = plt.figure(figsize=(10,8))
axs = fig.subplots(3,1,sharex=True)
axs[0].stem(n_rect, h_rect, markerfmt='D')
axs[0].set_title('Rectangular smoothing filter')
(n_step, x_step) = submitted.unit_step()
axs[1].stem(n_step, x_step, markerfmt='D')
axs[1].set_title('Unit step')
y_step = np.convolve(x_step, h_rect, mode='same')
axs[2].stem(n_step, y_step, markerfmt='D')
axs[2].set_title('Rectangular-smoothed Unit step')

Text(0.5, 1.0, 'Rectangular-smoothed Unit step')

Part 2: Use h_rect to smooth the rows of the image

importlib.reload(submitted)
help(submitted.todo_convolve_rows)

Help on function todo_convolve_rows in module submitted:

todo_convolve_rows(X, h)
    Convolve Y=X*h along the rows.
    Use mode='same', so that the output, Y, has the same size as X.

importlib.reload(submitted)
smoothed_rows = submitted.todo_convolve_rows(original, h_rect)
fig = plt.figure(figsize=(10, 8))
ax = fig.subplots()
ax.clear()
ax.imshow(smoothed_rows.astype(int))
ax.set_title('Image with rows smoothed by a length-7 rectangular smoother')

Text(0.5, 1.0, 'Image with rows smoothed by a length-7 rectangular smoother')

Part 3: Use h_rect to smooth the columns of the image

importlib.reload(submitted)
help(submitted.todo_convolve_columns)

Help on function todo_convolve_columns in module submitted:

todo_convolve_columns(X, h)
    Convolve Y=X*h along the columns.
    Use mode='same', so that the output, Y, has the same size as X.

importlib.reload(submitted)
smoothed_image = submitted.todo_convolve_columns(smoothed_rows, h_rect)
fig = plt.figure(figsize=(10,8))
ax = fig.subplots()
ax.clear()
ax.imshow(smoothed_image.astype(int))
ax.set_title('Image with both rows and columns smoothed by a length-7 rectangle')

Text(0.5, 1.0, 'Image with both rows and columns smoothed by a length-7 rectangle')

Step 4: create a backward-difference filter, and show it in a plot

importlib.reload(submitted)
help(submitted.todo_backward_difference)

Help on function todo_backward_difference in module submitted:

todo_backward_difference()
    Create 11 samples of a backward difference filter, h[n],
    such that the convolution y=x*h gives, as output, y[n]=x[n]-x[n-1].
    Note that you'll need to set h[n]=0 for most n; only a couple of
    samples will be nonzero.
    Return the n_array and the h_array.

importlib.reload(submitted)
(n_diff, h_diff) = submitted.todo_backward_difference()
fig = plt.figure(figsize=(10,8))
axs = fig.subplots(3,1,sharex=True)
axs[0].stem(n_diff, h_diff, markerfmt='D')
axs[0].set_title('Backward-Difference filter')
(n_step, x_step) = submitted.unit_step()
axs[1].stem(n_step, x_step, markerfmt='D')
axs[1].set_title('Unit step')
y_step = np.convolve(x_step, h_diff, mode='same')
axs[2].stem(n_step, y_step, markerfmt='D')
axs[2].set_title('Differenced Unit step')

Text(0.5, 1.0, 'Differenced Unit step')

Step 5: create a Gaussian smoothing filter, and show it in a plot

importlib.reload(submitted)
help(submitted.todo_gaussian_smoother)

Help on function todo_gaussian_smoother in module submitted:

todo_gaussian_smoother(stdev=1)
    Create 11 samples of a Gaussian filter with a standard deviation of stdev.
    You might want to call the function gaussian(), which is defined at the
    top of this file.
    Return the n_array and the h_array.

importlib.reload(submitted)
stdev = 2
(n_gauss, h_gauss) = submitted.todo_gaussian_smoother(stdev)
fig = plt.figure(figsize=(10,8))
axs = fig.subplots(3,1,sharex=True)
axs[0].stem(n_gauss, h_gauss, markerfmt='D')
axs[0].set_title('Gaussian smoothing filter')
(n_step, x_step) = submitted.unit_step()
axs[1].stem(n_step, x_step, markerfmt='D')
axs[1].set_title('Unit step')
y_step = np.convolve(x_step, h_gauss, mode='same')
axs[2].stem(n_step, y_step, markerfmt='D')
axs[2].set_title('Gaussian-smoothed Unit step')

Text(0.5, 1.0, 'Gaussian-smoothed Unit step')

importlib.reload(submitted)
smoothed_rows = submitted.todo_convolve_rows(original, h_gauss)
smoothed_image = submitted.todo_convolve_columns(smoothed_rows, h_gauss)
fig = plt.figure(figsize=(10, 8))
ax = fig.subplots()
ax.clear()
ax.imshow(smoothed_rows.astype(int))
ax.set_title('Image with rows smoothed by stdev=%g Gaussian smoother'%(stdev))

Text(0.5, 1.0, 'Image with rows smoothed by stdev=2 Gaussian smoother')

Step 6: create a difference-of-Gaussians filter, and show it in a plot

importlib.reload(submitted)
help(submitted.todo_difference_of_gaussians)

Help on function todo_difference_of_gaussians in module submitted:

todo_difference_of_gaussians()
    Create 11 samples of a difference-of-Gaussians filter, h[n] is the
    difference between a unit-variance Gaussian centered at mu=0,
    minus a unit-variance Gaussian centered at mu=1.
    Return the n_array and the h_array.

importlib.reload(submitted)
(n_dog, h_dog) = submitted.todo_difference_of_gaussians()
fig = plt.figure(figsize=(10,8))
axs = fig.subplots(3,1,sharex=True)
axs[0].stem(n_dog, h_dog, markerfmt='D')
axs[0].set_title('Difference-of-Gaussians filter')
(n_step, x_step) = submitted.unit_step()
axs[1].stem(n_step, x_step, markerfmt='D')
axs[1].set_title('Unit step')
y_step = np.convolve(x_step, h_dog, mode='same')
axs[2].stem(n_step, y_step, markerfmt='D')
axs[2].set_title('DoG-filtered Unit step')

Text(0.5, 1.0, 'DoG-filtered Unit step')

Step 7: Use DoG filter to compute Gx and Gy, then normalize

importlib.reload(submitted)
help(submitted.todo_normalize_colors)

Help on function todo_normalize_colors in module submitted:

todo_normalize_colors(X)
    Normalize the color planes of the image, so that
    each color plane has a maximum value of 1.0, and a minimum value of 0.0.

importlib.reload(submitted)
tmp = submitted.todo_convolve_rows(original, h_dog)
hgrad = submitted.todo_normalize_colors(np.abs(tmp))
tmp = submitted.todo_convolve_columns(original, h_dog)
vgrad = submitted.todo_normalize_colors(np.abs(tmp))
fig = plt.figure(figsize=(14,8))
axs = fig.subplots(1,2)
axs[0].imshow(hgrad)
axs[0].set_title('Horizontal grad magnitude')
axs[1].imshow(vgrad)
axs[1].set_title('Vertical grad magnitude')

Text(0.5, 1.0, 'Vertical grad magnitude')

Step 8: Gradient magnitude

importlib.reload(submitted)
help(submitted.todo_gradient_magnitude)

Help on function todo_gradient_magnitude in module submitted:

todo_gradient_magnitude(GH, GV)
    Given the horizontal gradient GH and vertical gradient GV,
    compute and return the gradient magnitude GM=sqrt(GH**2 + GV**2)

importlib.reload(submitted)
tmp = submitted.todo_gradient_magnitude(hgrad, vgrad)        
gradient_magnitude = submitted.todo_normalize_colors(tmp)
fig = plt.figure(figsize=(10,8))
ax = fig.subplots()
ax.imshow(gradient_magnitude)
ax.set_title('Gradient magnitude')

Text(0.5, 1.0, 'Gradient magnitude')

Extra Credit

For extra credit, you can create an image pyramid.

In an image pyramid, each image is a smoothed and downsampled version of the previous image.

import extra

importlib.reload(extra)
help(extra.todo_pyramid)

Help on function todo_pyramid in module extra:

todo_pyramid(img, pyramid_length=3, stdev=1)
    img_pyramid = extra.todo_pyramid(img, pyramid_length, stdev)
    Parameters:
    img (numpy array, nrows*ncolumns*ncolors, float64): an img
    steps (scalar int): the number of steps in the pyramid to return
    stdev (scalar float): standard deviation of the Gaussian smoother to use
    
    Return:
    img_pyramid: a list, len(img_pyramid)==pyramid_length
    
    An img pyramid is a list of images, each of which is a smoothed and downsampled
    version of the image that came before it.
    
    img_pyramid[0]: a numpy array, int(nrows/2)*int(ncolumns/2)*ncolors:
       the input img is first smoothed by a Gaussian smoother in both rows and columns,
       then downsampled by a factor of two (keep only the even-numbered rows and columns).
    img_pyramid[1]: a numpy array, int(nrows/4)*int(ncolumns/4)*ncolors:
       img_pyramid[0] is first smoothed by a Gaussian smoother in both rows and columns,
       then downsampled by a factor of two (keep only the even-numbered rows and columns).
    ...
    img_pyramid[pyramid_length-1]: a numpy array, 
       int(nrows/(2^pyramid_length)*int(ncolumns/2^pyramid_length)*ncolors:
       img_pyramid[pyramid_length-2] is first smoothed by a Gaussian smoother,
       then downsampled by a factor of two (keep only the even-numbered rows and columns).
    
    You can use any functions from submitted.py, e.g., submitted.todo_gaussian_smoother,
    submitted.todo_convolve_rows, and submitted.todo_convolve_columns.  
    The autograder will use reference implementations of those functions, 
    but on your computer, grade.py will use your own implementations.

importlib.reload(extra)
image_pyramid = extra.todo_pyramid(original,3,np.sqrt(2))
print('image_pyramid contains %d images'%(len(image_pyramid)))
print('The original image has a size of',original.shape)
for n in range(3):
    print('image_pyramid[%d] has a shape of'%(n), image_pyramid[n].shape)

image_pyramid contains 3 images
The original image has a size of (600, 800, 3)
image_pyramid[0] has a shape of (300, 400, 3)
image_pyramid[1] has a shape of (150, 200, 3)
image_pyramid[2] has a shape of (75, 100, 3)

fig = plt.figure(figsize=(10,20))
axs = fig.subplots(3,1)
axs[0].imshow(image_pyramid[0].astype('int'))
axs[0].set_title('image_pyramid[0]')
axs[1].imshow(image_pyramid[1].astype('int'))
axs[1].set_title('image_pyramid[1]')
axs[2].imshow(image_pyramid[2].astype('int'))
axs[2].set_title('image_pyramid[2]')

Text(0.5, 1.0, 'image_pyramid[2]')