part0
import numpy as np
import cv2
def video_volume(image_list):
'''Create a video volume from the image list.
Input:
image_list - a list of frames. Each element of the list contains a numpy
array of a colored image. You may assume that each frame has the same shape,
(rows, cols, 3).
Output:
output - a single 4d numpy array. This array should have dimensions
(time, rows, cols, 3) and dtype np.uint8.
'''
output = None
# Insert your code here ------------------------------------------------------
(rows, cols, color) = image_list[0].shape
output = np.zeros((len(image_list), rows, cols, color), dtype = np.uint8)
for i in range(len(image_list)):
output[i,:,:,:] = image_list[i]
# ----------------------------------------------------------------------------
return output
def ssd(video_volume):
'''Compute the sum of squared distances for each pair of frames in video volume.
Input:
video_volume - as returned by your video_volume function.
Output:
output - a square 2d numpy array of dtype float. output[i,j] should contain
the dsum of square differences between frames i and j.
'''
output = None
# Insert your code here ------------------------------------------------------
(frames, rows, cols, color) = video_volume.shape
output = np.zeros((frames, frames), dtype = np.float)
for i in range(frames):
image_i = video_volume[i,:,:,:]
for j in range(frames):
image_j = video_volume[j,:,:,:]
total = 0.0
diff_image = image_i - image_j
for k1 in range(rows):
for k2 in range(cols):
for k3 in range(3):
total = total + diff_image[k1,k2,k3] * diff_image[k1,k2,k3]
output[i, j] = total
# ----------------------------------------------------------------------------
return output
def test():
'''This script will perform a unit test on your function, and provide useful
output.
'''
image_list1 = [np.array([[[ 0, 0, 0],
[ 0, 0, 0]]], dtype = np.uint8),
np.array([[[ 1, 1, 1],
[ 1, 1, 1]]], dtype = np.uint8),
np.array([[[ 2, 2, 2],
[ 2, 2, 2]]], dtype = np.uint8),
np.array([[[ 3, 3, 3],
[ 3, 3, 3]]], dtype = np.uint8)]
video_volume1 = np.array([[[[ 0, 0, 0],
[ 0, 0, 0]]],
[[[ 1, 1, 1],
[ 1, 1, 1]]],
[[[ 2, 2, 2],
[ 2, 2, 2]]],
[[[ 3, 3, 3],
[ 3, 3, 3]]]], dtype = np.uint8)
image_list2 =[np.array([[[2, 2, 2],
[2, 2, 2],
[2, 2, 2],
[2, 2, 2]],
[[2, 2, 2],
[2, 2, 2],
[2, 2, 2],
[2, 2, 2]]], dtype = np.uint8),
np.array([[[1, 1, 1],
[1, 1, 1],
[1, 1, 1],
[1, 1, 1]],
[[1, 1, 1],
[1, 1, 1],
[1, 1, 1],
[1, 1, 1]]], dtype = np.uint8),
np.array([[[0, 0, 0],
[0, 0, 0],
[0, 0, 0],
[0, 0, 0]],
[[0, 0, 0],
[0, 0, 0],
[0, 0, 0],
[0, 0, 0]]], dtype = np.uint8)]
video_volume2 = np.array([[[[2, 2, 2],
[2, 2, 2],
[2, 2, 2],
[2, 2, 2]],
[[2, 2, 2],
[2, 2, 2],
[2, 2, 2],
[2, 2, 2]]],
[[[1, 1, 1],
[1, 1, 1],
[1, 1, 1],
[1, 1, 1]],
[[1, 1, 1],
[1, 1, 1],
[1, 1, 1],
[1, 1, 1]]],
[[[0, 0, 0],
[0, 0, 0],
[0, 0, 0],
[0, 0, 0]],
[[0, 0, 0],
[0, 0, 0],
[0, 0, 0],
[0, 0, 0]]]], dtype = np.uint8)
diff1 = np.array([[ 0., 6., 24., 54.],
[ 6., 0., 6., 24.],
[ 24., 6., 0., 6.],
[ 54., 24., 6., 0.]], dtype = np.float)
diff2 = np.array([[ 0., 24., 96.],
[ 24., 0., 24.],
[ 96., 24., 0.]], dtype = np.float)
if __name__ == "__main__":
print 'Evaluating video_volume.'
for img_list, true_out in zip((image_list1, image_list2), (video_volume1, video_volume2)):
if __name__ == "__main__":
print "input:\n{}\n".format(img_list)
usr_out = video_volume(img_list)
if not type(usr_out) == type(true_out):
if __name__ == "__main__":
print "Error- video_volume has type {}. Expected type is {}.".format(
type(usr_out), type(true_out))
return False
if not usr_out.shape == true_out.shape:
if __name__ == "__main__":
print "Error- video_volume has shape {}. Expected shape is {}.".format(
usr_out.shape, true_out.shape)
return False
if not usr_out.dtype == true_out.dtype:
if __name__ == "__main__":
print "Error- video_volume has dtype {}. Expected dtype is {}.".format(
usr_out.dtype, true_out.dtype)
return False
if not np.all(usr_out == true_out):
if __name__ == "__main__":
print "Error- video_volume has value:\n{}\nExpected value:\n{}".format(
usr_out, true_out)
return False
if __name__ == "__main__":
print "video_volume passed."
print "evaluating ssd"
for vid_volume, true_out in zip((video_volume1, video_volume2), (diff1, diff2)):
if __name__ == "__main__":
print "input:\n{}\n".format(vid_volume)
usr_out = ssd(vid_volume)
if not type(usr_out) == type(true_out):
if __name__ == "__main__":
print "Error- ssd has type {}. Expected type is {}.".format(
type(usr_out), type(true_out))
return False
if not usr_out.shape == true_out.shape:
if __name__ == "__main__":
print "Error- ssd has shape {}. Expected shape is {}.".format(
usr_out.shape, true_out.shape)
return False
if not usr_out.dtype == true_out.dtype:
if __name__ == "__main__":
print "Error- ssd has dtype {}. Expected dtype is {}.".format(
usr_out.dtype, true_out.dtype)
return False
if not np.all(np.abs(usr_out - true_out) < 1.):
if __name__ == "__main__":
print "Error- ssd has value:\n{}\nExpected value:\n{}".format(
usr_out, true_out)
return False
if __name__ == "__main__":
print "All unit tests successful."
return True
if __name__ == "__main__":
print "Performing unit tests.(ssd will be accepted if the output is within 1 of the expected output.)"
np.set_printoptions(precision=1)
test()
part1
import numpy as np
import cv2
import scipy.signal
def binomial_filter_5():
''' Create a binomial filter of length 4.
'''
return np.array([ 0.0625, 0.25 , 0.375 , 0.25 , 0.0625], dtype=float)
def diff2(diff1):
'''Compute the transition costs between frames, taking dynamics into account.
Input:
diff1 - a difference matrix as produced by your ssd function.
Output:
output - a new difference matrix that takes preceding and following frames into
account. The cost of transitioning from i to j should weight the
surrounding frames according to the binomial filter provided to you
above. So, the difference between i and j has weight 0.375, the frames
immediately following weight 0.25, etc...
The output difference matrix should have the same dtype as the input,
but be 4 rows and columns smaller, corresponding to only the frames
that have valid dynamics.
Hint: there is a very efficient way to do this with 2d convolution. Think about
the coordinates you are using as you consider the preceding and following
frame pairings.
'''
output = None
# Insert your code here ------------------------------------------------------
w_1d = binomial_filter_5()
(w_size) = w_1d.shape
(rows, cols) = diff1.shape
output = np.zeros((rows - 4, cols - 4), dtype = np.float)
for i in range(2, rows - 2, 1):
for j in range(2, cols - 2, 1):
total = 0.0;
for k in range(-2, 3, 1):
total = total + diff1[i + k, j + k]*w_1d[k + 2]
output[i - 2, j - 2] = total
#kernel = np.outer(w_1d, w_1d)
#output = scipy.signal.convolve2d(diff1, kernel, 'valid')
# ----------------------------------------------------------------------------
return output
def test():
'''This script will perform a unit test on your function, and provide useful
output.
'''
d1 = np.zeros((9,9), dtype = float)
d1[4,4] = 1
d2 = np.eye(5, dtype = float)
out1 = np.array([[ 0.0625, 0. , 0. , 0. , 0. ],
[ 0. , 0.25 , 0. , 0. , 0. ],
[ 0. , 0. , 0.375 , 0. , 0. ],
[ 0. , 0. , 0. , 0.25 , 0. ],
[ 0. , 0. , 0. , 0. , 0.0625]], dtype = float)
out2 = np.array([[1.]], dtype = float)
if __name__ == "__main__":
print 'Evaluating diff2.'
for d, true_out in zip((d1, d2), (out1, out2)):
if __name__ == "__main__":
print "input:\n{}\n".format(d)
usr_out = diff2(d)
if not type(usr_out) == type(true_out):
if __name__ == "__main__":
print "Error- diff2 output has type {}. Expected type is {}.".format(
type(usr_out), type(true_out))
return False
if not usr_out.shape == true_out.shape:
if __name__ == "__main__":
print "Error- diff2 output has shape {}. Expected shape is {}.".format(
usr_out.shape, true_out.shape)
return False
if not usr_out.dtype == true_out.dtype:
if __name__ == "__main__":
print "Error- diff2 output has dtype {}. Expected dtype is {}.".format(
usr_out.dtype, true_out.dtype)
return False
if not np.all(np.abs(usr_out - true_out) < 0.05):
if __name__ == "__main__":
print "Error- diff2 output has value:\n{}\nExpected value:\n{}".format(
usr_out, true_out)
return False
if __name__ == "__main__":
print "diff2 passed."
if __name__ == "__main__":
print "All unit tests successful."
return True
if __name__ == "__main__":
print "Performing unit tests. (Your output will be accepted if it is within .05 of the true output)"
np.set_printoptions(precision=1)
test()
part2
import numpy as np
import cv2
import sys
import scipy.signal
def find_biggest_loop(diff2, alpha):
'''Given the difference matrix, find the longest and smoothest loop that we can.
Inputs:
diff2 - a square 2d numpy array of dtype float. Each cell contains the cost
of transitioning from frame i to frame j in the input video.
alpha - a parameter for how heavily you should weigh the size of the loop
relative to the transition cost of the loop. Larger alphas favor
longer loops.
Outputs:
s - the beginning frame of the longest loop.
f - the final frame of the longest loop.
s, f will the indices in the diff2 matrix that give the maximum score
according to the following metric:
alpha*(f-s) - diff2[f,s]
'''
s = None
f = None
# Insert your code here ------------------------------------------------------
(rows, cols) = diff2.shape
maxValue = -10000
for ss in range(rows):
for ff in range(cols):
val = alpha*(ff-ss) - diff2[ff,ss]
if val > maxValue:
maxValue = val
s = ss
f = ff
# ----------------------------------------------------------------------------
return s, f
def synthesize_loop(video_volume, s, f):
'''Pull out the given loop from video.
Input:
video_volume - a (t, row, col, 3) array, as created by your video_volume function.
i - the index of the starting frame.
j - the index of the ending frame.
Output:
output - a list of arrays of size (row, col, 3) and dtype np.uint8, similar to
the original input the video_volume function in part0.
'''
output = []
# Insert your code here ------------------------------------------------------
for i in range(s, f + 1, 1):
output.append(video_volume[i, :, :, :])
# ----------------------------------------------------------------------------
return output
def test():
'''This script will perform a unit test on your function, and provide useful
output.
'''
d1 = np.ones((5,5), dtype = float)
a1 = 1
out1 = (0,4)
d2 = np.array([[ 0., 1., 1., 5.],
[ 1., 0., 3., 4.],
[ 1., 3., 0., 5.],
[ 5., 4., 5., 0.]])
a2 = 1
out2 = (0,2)
d3 = np.array([[ 0., 1., 4.],
[ 1., 0., 1.],
[ 4., 1., 0.]])
a3 = 2
out3 = (0,1)
if __name__ == "__main__":
print 'Evaluating find_biggest_loop'
for d, a, true_out in zip((d1, d2, d3), (a1, a2, a3), (out1, out2, out3)):
if __name__ == "__main__":
print "input:\n{}\n".format(d)
print "alpha = {}".format(a)
usr_out = find_biggest_loop(d, a)
if not usr_out == true_out:
if __name__ == "__main__":
print "Error- find_biggest_loop is {}. Expected output is {}.".format(
usr_out, true_out)
return False
if __name__ == "__main__":
print "find_biggest_loop passed."
print "testing synthesize_loop."
video_volume1 = np.array([[[[ 0, 0, 0],
[ 0, 0, 0]]],
[[[ 1, 1, 1],
[ 1, 1, 1]]],
[[[ 2, 2, 2],
[ 2, 2, 2]]],
[[[ 3, 3, 3],
[ 3, 3, 3]]]], dtype = np.uint8)
video_volume2 = np.array([[[[2, 2, 2],
[2, 2, 2],
[2, 2, 2],
[2, 2, 2]],
[[2, 2, 2],
[2, 2, 2],
[2, 2, 2],
[2, 2, 2]]],
[[[1, 1, 1],
[1, 1, 1],
[1, 1, 1],
[1, 1, 1]],
[[1, 1, 1],
[1, 1, 1],
[1, 1, 1],
[1, 1, 1]]],
[[[0, 0, 0],
[0, 0, 0],
[0, 0, 0],
[0, 0, 0]],
[[0, 0, 0],
[0, 0, 0],
[0, 0, 0],
[0, 0, 0]]]], dtype = np.uint8)
frames1 = (2,3)
frames2 = (1,1)
out1 = [np.array([[[ 2, 2, 2],
[ 2, 2, 2]]], dtype = np.uint8),
np.array([[[ 3, 3, 3],
[ 3, 3, 3]]], dtype = np.uint8)]
out2 = [np.array([[[1, 1, 1],
[1, 1, 1],
[1, 1, 1],
[1, 1, 1]],
[[1, 1, 1],
[1, 1, 1],
[1, 1, 1],
[1, 1, 1]]], dtype = np.uint8)]
for video_volume, frames, true_out in zip((video_volume1, video_volume2),
(frames1, frames2), (out1, out2)):
if __name__ == "__main__":
print "input:\n{}\n".format(video_volume)
print "input frames:\n{}\n".format(frames)
usr_out = synthesize_loop(video_volume, frames[0], frames[1])
if not type(usr_out) == type(true_out):
if __name__ == "__main__":
print "Error- synthesize_loop has type {}. Expected type is {}.".format(
type(usr_out), type(true_out))
return False
if not len(usr_out) == len(true_out):
if __name__ == "__main__":
print "Error - synthesize_loop has len {}. Expected len is {}.".format(
len(usr_out), len(true_out))
return False
for usr_img, true_img in zip(usr_out, true_out):
if not type(usr_img) == type(true_img):
if __name__ == "__main__":
print "Error- synthesize_loop has type {}. Expected type is {}.".format(
type(usr_img), type(true_img))
return False
if not usr_img.shape == true_img.shape:
if __name__ == "__main__":
print "Error- synthesize_loop has shape {}. Expected shape is {}.".format(
usr_img.shape, true_img.shape)
return False
if not usr_img.dtype == true_img.dtype:
if __name__ == "__main__":
print "Error- synthesize_loop has dtype {}. Expected dtype is {}.".format(
usr_img.dtype, true_img.dtype)
return False
if not np.all(usr_img == true_img):
if __name__ == "__main__":
print "Error- synthesize_loop has value:\n{}\nExpected value:\n{}".format(
usr_img, true_img)
return False
if __name__ == "__main__":
print "synthesize_loop passed."
if __name__ == "__main__":
print "All unit tests successful."
return True
if __name__ == "__main__":
print "Performing unit tests."
test()
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What the min cut tells us. By solving a maxflow problem, we can determine whether a given team is mathematically eliminated. We would also like to explain the reason for the team’s elimination to a friend in nontechnical terms (using only grade-school arithmetic). Here’s such an explanation for Detroit’s elimination in the American League East example above. With the best possible luck, Detroit finishes the season with 49 + 27 = 76 wins. Consider the subset of teams R = { New York, Baltimore, Boston, Toronto }. Collectively, they already have 75 + 71 + 69 + 63 = 278 wins; there are also 3 + 8 + 7 + 2 + 7 = 27 remaining games among them, so these four teams must win at least an additional 27 games. Thus, on average, the teams in R win at least 305 / 4 = 76.25 games. Regardless of the outcome, one team in R will win at least 77 games, thereby eliminating Detroit.
