Setup: OpenCV
For this assignment you will need to use a popular and powerful library known as OpenCV. To do this on hub, you will need to open a terminal (File->New Launcher->Terminal) and enter pip install opencv-python. If you are trying to run the notebook somewhere else and can’t figure out how to install the library, ask on piazza and provide details.
Assignment: Solving Image Maze
Given a maze as an image with a start and end point, we would like to write code to solve the maze.
An image is a 2D matrix of pixels of a particular size that depends on its resolution. Each pixel has a color which is given by its Red, Green and Blue (RGB) values.
Given an image, we will view it as a graph where each pixel of the image is a vertex and edges connect a pixel to its neighbor. The weight of an edge should be very small if the pixel colors are similar (i.e, the differences between r, g and b values are close to zero) and correspondingly large as the pixel colors diverge.
Next, given a source pixel (i0,j0)(i0,j0) and destination pixel, (i1,j1)(i1,j1), we wish find the shortest weight path from source to destination.
You should use the Dijkstra’s algorithm modified in two ways:
- It can exit as soon as the destination is reached.
- A 1000 x 1000 pixel image gives rise to a graph with million vertices. Storing such a graph as an adjacency list is going to be very memory intensive. Instead, your goal will be to generate the vertices and edges on-the-fly.
We will use opencv library, a popular computer vision library to load, and manipulate images of mazes.
Manipulating Images
You can directly manipulate images in python in many ways. The opencv library is considered a standard for numerous image manipulation tasks.
Here we load an image maze.png and you can see it nicely plotted with coordinates. We then show you two pixels shown in red and blue. The goal here is to detect a path from one of the colored circle to the other, in the maze without crossing the black pixels.
from matplotlib import pyplot as plt
import cv2
# You can read png, jpg and other file types
img = cv2.imread('maze.png') # read an image from a file using opencv (cv2) library
# you can annotate images
cv2.circle(img,(5,220), 3, (255,0,0), -1) # add a circle centered at (5, 220) radius 3, color red (RGB: 255,0,0)
cv2.circle(img, (5,5), 3, (0,0,255), -1) # add a circle centered at (5,5) radius 3, color red (RGB: 0,0,255)
plt.imshow(img) # show the image on the screen
plt.title('Amazing')
plt.show()
img = cv2.imread('maze-solution.png') # read an image from a file using opencv (cv2) library
# you can annotate images
cv2.circle(img,(5,220), 3, (255,0,0), -1) # add a circle centered at (5, 220) radius 3, color red (RGB: 255,0,0)
cv2.circle(img, (5,5), 3, (0,0,255), -1) # add a circle centered at (5,5) radius 3, color red (RGB: 0,0,255)
plt.imshow(img) # show the image on the screen
plt.title('Amazing Solution ')
plt.show()
Given an image it is simple to read the color at a pixel. Let us read the color at pixel (645, 67)
print('Image size (height, width, num layers) is', img.shape)
px = img[145, 67] # img[y,x] is the color of the pixel of x,y
print(px)
cv2.circle(img, (80, 18), 3, (198,31,4),-1) # Draw a colored circle centered at (80, 18)
px1 = img[18, 80] # It is important to note that rows of the image are y values and columns are x values.
print(px1)
px2 = img[80, 18] # This is important to note that indexing the img data structure takes y, x values.
# Most opencv functions will require (x,y) coordinates for pixel as is natural.
print(px2)Image size (height, width, num layers) is (225, 225, 3) [255 255 255] [198 31 4] [0 0 0]
The pixel color is expressed in RGB format. R is the red value from 0 -> 255, G is the green value 0 -> 255 and B is the blue value from 0 -> 255
We will now define a edge weight function for an edge in the image
import math
def fixPixelValues(px):
# convert the RGB values into floating point to avoid an overflow that will give me wrong answers
return [ float(px[0]), float(px[1]), float(px[2]) ]
## Given (x,y) coordinates of two neighboring pixels, calculate the edge weight.
# We take the squared euclidean distance between the pixel values and add 0.1
def getEdgeWeight(img, u, v):
# get edge weight for edge between u, v
# First make sure that the edge is legit
i0,j0 = u[0], u[1]
i1,j1 = v[0], v[1]
height, width, _ = img.shape
assert i0 >= 0 and j0 >= 0 and i0 < width and j0 < height # pixel position valid?
assert i1 >= 0 and j1 >= 0 and i1 < width and j1 < height # pixel position valid?
assert -1 <= i0 - i1 <= 1 # edge between node and neighbor?
assert -1 <= j0 - j1 <= 1
px1 = fixPixelValues(img[j0,i0])
px2 = fixPixelValues(img[j1,i1])
return 0.1 + (px1[0] - px2[0])**2 + (px1[1] - px2[1])**2 + (px1[2]- px2[2])**2
# This is a useful function that given a list of (x,y) values,
# draw a series of red lines between each coordinate and next to
# show the path in the image
def drawPath(img, path, pThick=2):
v = path[0]
x0, y0 = v[0], v[1]
for v in path:
x, y = v[0], v[1]
cv2.line(img,(x,y), (x0,y0), (255,0,0),pThick)
x0, y0 = x,y
# Example
img = cv2.imread('maze.png') # read an image from a file using opencv (cv2) library
drawPath(img, [ (15, 15), (150, 15), (150, 85), (75, 85), (75, 195)])
plt.imshow(img) # show the image on the screen
plt.title('Illustration of drawPath')
plt.show()
Step 1: Compute Single Source Shortest Path For an Image
Given an image, compute the shortest path between source and destination pixels by modifying Dijkstra’s algorithm. Your challenge is to implement it without needing to create the entire the adjacency list for the graph in the first place. However, for simplicity you can try a first cut implementation of a generic Dijkstra algorithm over graphs represented as adjacency matrix or list.
class Vertex: # This is the outline for a vertex data structure
def __init__ (self, i, j):
self.x = i # The x coordinate
self.y = j # The y coordinate
self.d = float('inf') # the shortest path estimate
self.processed = False # Has this vertex's final shortest path distance been computed
self.idx_in_priority_queue = -1 # The index of this vertex in the queue
self.pi = None # the parent vertex.
# Dijkstra's algorithm requires a priority queue so that the
# minimum weight vertex can be found efficiently.
# However, we provide you with a list data structure as a stand in for priority queue
class SimpleQueue:
# CONSTRUCTOR
def __init__(self):
self.q = []
# Insert a vertex into the queue
def insert(self, v):
v.idx_in_priority_queue = len(self.q)
self.q.append(v)
# Find the vertex with the smallest distance estimate, and
# delete it from the queue
# return the vertex
def get_and_delete_min(self):
n = len(self.q)
assert n > 0
min_pos = 0
for i in range(n):
if self.q[i].d < self.q[min_pos].d:
min_pos = i
v = self.q[min_pos]
del self.q[min_pos]
return v
# Is the queue empty?
def is_empty(self):
return len(self.q) == 0
# Notify the queue that the weight of vertex v has been updated.
def update_vertex_weight(self, v):
pass # Nothing to do, for a simple list, we ignore this notification.
# However, if you want Dijkstra efficiently,
# you may want to implement a priority queue.
# We provide you the signature for a priority queue.
# Feel free to implement extra functions if you wish
class PriorityQueue:
# Constructor:
def __init__(self):
pass # YOUR CODE HERE
def insert(self, v):
pass # YOUR CODE HERE
def get_and_delete_min(self):
pass # YOUR CODE HERE: must return a vertex
def is_empty(self):
pass # YOUR CODE HERE: must return true/false
def update_vertex_weight(self, v):
pass # YOUR CODE HERE
def computeShortestPath(img, source, dest):
# IMPLEMENT DIJKSTRA
path = [source, dest]
return pathimg = cv2.imread('maze.png') # read an image from a file using opencv (cv2) library
# you can annotate images
cv2.circle(img,(5,220), 3, (255,0,0), -1) # add a circle centered at (5, 220) radius 3, color red (RGB: 255,0,0)
cv2.circle(img, (5,5), 3, (0,0,255), -1) # add a circle centered at (5,5) radius 3, color red (RGB: 0,0,255)
plt.imshow(img) # show the image on the screen
plt.title('Amazing')
plt.show()img = cv2.imread('maze.png') # read an image from a file using opencv (cv2) library
p = computeShortestPath(img, (5,220), (5,5))drawPath(img, p, 2)
plt.imshow(img) # show the image on the screen
plt.title('Amazing')
plt.show()
cv2.imwrite('maze-solution.png', img)
True
img = cv2.imread('maze2.JPG') # read an image from a file using opencv (cv2) library
cv2.circle(img,(250,470), 10, (255,0,0), -1) # add a circle centered at (600, 70) radius 10, color red (RGB: 255,0,0)
cv2.circle(img, (20,100), 10, (255,0,0), -1) # add a circle centered at (790,200) radius 10, color red (RGB: 255,0,0)
plt.imshow(img) # show the image on the screen
plt.title('Amazing 2')
plt.show()
img = cv2.imread('maze2.JPG') # read an image from a file using opencv (cv2) library
p = computeShortestPath(img, (250,470), (20,100))Debug: reached destination. d= 119.79999999999747
drawPath(img,p)
plt.imshow(img) # show the image on the screen
plt.title('Amazing2')
plt.show()
img = cv2.imread('maze3.JPG')
cv2.circle(img,(70,1750), 15, (255,0,0), -1) # add a circle centered at (600, 70) radius 10, color red (RGB: 255,0,0)
cv2.circle(img, (900,500), 15, (0,255,255), -1) # add a circle centered at (790,200) radius 10, color red (RGB: 255,0,0)
plt.imshow(img) # show the image on the screen
plt.title('Amazing 3')
plt.show()
img = cv2.imread('maze3.JPG') # read an image from a file using opencv (cv2) library
p = computeShortestPath(img, (70,1750), (900,500))Debug: reached destination. d= 2595.1999999987474
drawPath(img,p,10)
plt.imshow(img) # show the image on the screen
plt.title('Amazing2 Solution')
plt.show()
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