Develop Reference

How to improve disparity map of stereo-pair images (python)

I captured the following 2 pictures from my mobile:Image1 , Image2
the camera was calibrated and I used this code to reconstruct 3D cloud point:
'''
Created by Omar Padierna "Para11ax" on Jan 1 2019
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
'''
import cv2
import numpy as np
import glob
from tqdm import tqdm
import PIL.ExifTags
import PIL.Image
from matplotlib import pyplot as plt
#=====================================
# Function declarations
#=====================================
#Function to create point cloud file
def create_output(vertices, colors, filename):
colors = colors.reshape(-1,3)
vertices = np.hstack([vertices.reshape(-1,3),colors])
ply_header = '''ply
format ascii 1.0
element vertex %(vert_num)d
property float x
property float y
property float z
property uchar red
property uchar green
property uchar blue
end_header
'''
with open(filename, 'w') as f:
f.write(ply_header %dict(vert_num=len(vertices)))
np.savetxt(f,vertices,'%f %f %f %d %d %d')
#Function that Downsamples image x number (reduce_factor) of times.
def downsample_image(image, reduce_factor):
for i in range(0,reduce_factor):
#Check if image is color or grayscale
if len(image.shape) > 2:
row,col = image.shape[:2]
else:
row,col = image.shape
image = cv2.pyrDown(image, dstsize= (col//2, row // 2))
return image
#=========================================================
# Stereo 3D reconstruction
#=========================================================
#Load camera parameters
ret = np.load('D:/Books/Pav Man/3DReconstruction-master/Reconstruction/camera_params/ret.npy')
K = np.load('D:/Books/Pav Man/3DReconstruction-master/Reconstruction/camera_params/K.npy')
dist = np.load('D:/Books/Pav Man/3DReconstruction-master/Reconstruction/camera_params/dist.npy')
#Specify image paths
img_path1 = 'D:/Books/Pav Man/3DReconstruction-master/Reconstruction/reconstruct_this/TestVi4.jpg'
img_path2 = 'D:/Books/Pav Man/3DReconstruction-master/Reconstruction/reconstruct_this/TestVi5.jpg'
#Load pictures
img_1 = cv2.imread(img_path1)
img_2 = cv2.imread(img_path2)
#Get height and width. Note: It assumes that both pictures are the same size. They HAVE to be same size and height.
h,w = img_2.shape[:2]
#Get optimal camera matrix for better undistortion
new_camera_matrix, roi = cv2.getOptimalNewCameraMatrix(K,dist,(w,h),1,(w,h))
#Undistort images
img_1_undistorted = cv2.undistort(img_1, K, dist, None, new_camera_matrix)
img_2_undistorted = cv2.undistort(img_2, K, dist, None, new_camera_matrix)
#Downsample each image 3 times (because they're too big)
img_1_downsampled = downsample_image(img_1_undistorted,3)
img_2_downsampled = downsample_image(img_2_undistorted,3)
#cv2.imwrite('undistorted_left.jpg', img_1_downsampled)
#cv2.imwrite('undistorted_right.jpg', img_2_downsampled)
#Set disparity parameters
#Note: disparity range is tuned according to specific parameters obtained through trial and error.
win_size = 1
min_disp = -1
max_disp = abs(min_disp) * 9 #min_disp * 9
num_disp = max_disp - min_disp # Needs to be divisible by 16
#Create Block matching object.
stereo = cv2.StereoSGBM_create(minDisparity= min_disp,
numDisparities = num_disp,
blockSize = 5,
uniquenessRatio = 5,
speckleWindowSize = 1,
speckleRange = 5,
disp12MaxDiff = 2,
P1 = 8*3*win_size**2,#8*3*win_size**2,
P2 =32*3*win_size**2) #32*3*win_size**2)
#Compute disparity map
print ("\nComputing the disparity map...")
disparity_map = stereo.compute(img_1_downsampled, img_2_downsampled)
#Show disparity map before generating 3D cloud to verify that point cloud will be usable.
plt.imshow(disparity_map,'gray')
plt.show()
#Generate point cloud.
print ("\nGenerating the 3D map...")
#Get new downsampled width and height
h,w = img_2_downsampled.shape[:2]
#Load focal length.
focal_length = np.load('D:/Books/Pav Man/3DReconstruction-master/Reconstruction/camera_params/FocalLength.npy')
#Perspective transformation matrix
#This transformation matrix is from the openCV documentation, didn't seem to work for me.
Q = np.float32([[1,0,0,-w/2.0],
[0,-1,0,h/2.0],
[0,0,0,-focal_length],
[0,0,1,0]])
#This transformation matrix is derived from Prof. Didier Stricker's power point presentation on computer vision.
#Link : https://ags.cs.uni-kl.de/fileadmin/inf_ags/3dcv-ws14-15/3DCV_lec01_camera.pdf
Q2 = np.float32([[1,0,0,0],
[0,-1,0,0],
[0,0,focal_length*0.05,0], #Focal length multiplication obtained experimentally.
[0,0,0,1]])
#Reproject points into 3D
points_3D = cv2.reprojectImageTo3D(disparity_map, Q2)
#Get color points
colors = cv2.cvtColor(img_1_downsampled, cv2.COLOR_BGR2RGB)
#Get rid of points with value 0 (i.e no depth)
mask_map = disparity_map > disparity_map.min()
#Mask colors and points.
output_points = points_3D[mask_map]
output_colors = colors[mask_map]
#Define name for output file
output_file = 'D:/Books/Pav Man/3DReconstruction-master/Reconstruction/reconstructed.ply'
#Generate point cloud
print ("\n Creating the output file... \n")
create_output(output_points, output_colors, output_file)
the following images are for disparity map and 3D model:
Disparity map
, Model
as you can see in the Model image, there are empty areas (red areas), how I can fill this area with points, and how to improve the disparity map.

The non-confident region(ie., algorithm not sure what this the correct disparity) is marked as black pixels in the disparity map. This is much expected behaviour.
You have to do some post processing to fill the map. Use Guided filter to complete map. If matlab provides any guided filter you can try once. In OpenCV "WLS" is the most common guided filter to get a filled map.

Get rotational shift using phase correlation and log polar transform

I have been working on a script which calculates the rotational shift between two images using cv2's phaseCorrelate method.
I have two images, the second is a 90 degree rotated version of the first image. After loading in the images, I convert them to log-polar before passing them into the phaseCorrelate function.
From what I have read, I believe that this should yield a rotational shift between two images.
The code below describes the implementation.
#bitwise right binary shift function
def rshift(val, n): return (val % 0x100000000)
base_img = cv2.imread('img1.jpg')
cur_img = cv2.imread('dataa//t_sv_1.jpg')
curr_img = rotateImage(cur_img, 90)
rows,cols,chan = base_img.shape
x, y, c = curr_img.shape
#convert images to valid type
ref32 = np.float32(cv2.cvtColor(base_img, cv2.COLOR_BGR2GRAY))
curr32 = np.float32(cv2.cvtColor(curr_img, cv2.COLOR_BGR2GRAY))
value = np.sqrt(((rows/2.0)**2.0)+((cols/2.0)**2.0))
value2 = np.sqrt(((x/2.0)**2.0)+((y/2.0)**2.0))
polar_image = cv2.linearPolar(ref32,(rows/2, cols/2), value, cv2.WARP_FILL_OUTLIERS)
log_img = cv2.linearPolar(curr32,(x/2, y/2), value2, cv2.WARP_FILL_OUTLIERS)
shift = cv2.phaseCorrelate(polar_image, log_img)
sx = shift[0][0]
sy = shift[0][1]
sf = shift[1]
polar_image = polar_image.astype(np.uint8)
log_img = log_img.astype(np.uint8)
cv2.imshow("Polar Image", polar_image)
cv2.imshow('polar', log_img)
#get rotation from shift along y axis
rotation = sy * 180 / (rshift(y, 1));
print(rotation)
cv2.waitKey(0)
cv2.destroyAllWindows()
I am unsure how to interpret the results of this function. The expected outcome is a value similar to 90 degrees, however, I get the value below.
Output: -0.00717516014538333
How can I make the output correct?

A method, typically referred to as the Fourier Mellin transform, and published as:
B. Srinivasa Reddy and B.N. Chatterji, "An FFT-Based Technique for Translation, Rotation, and Scale-Invariant Image Registration", IEEE Trans. on Image Proc. 5(8):1266-1271, 1996
uses the FFT and the log-polar transform to obtain the translation, rotation and scaling of one image to match the other. I find this tutorial to be very clear and informative, I will give a summary here:
Compute the magnitude of the FFT of the two images (apply a windowing function first to avoid issues with periodicity of the FFT).
Compute the log-polar transform of the magnitude of the frequency-domain images (typically a high-pass filter is applied first, but I have not seen its usefulness).
Compute the cross-correlation (actually phase correlation) between the two. This leads to a knowledge of scale and rotation.
Apply the scaling and rotation to one of the original input images.
Compute the cross-correlation (actually phase correlation) of the original input images, after correction for scaling and rotation. This leads to knowledge of the translation.
This works because:
The magnitude of the FFT is translation-invariant, we can solely focus on scaling and rotation without worrying about translation. Note that the rotation of the image is identical to the rotation of the FFT, and that scaling of the image is inverse to the scaling of the FFT.
The log-polar transform converts rotation into a vertical translation, and scaling into a horizontal translation. Phase correlation allows us to determine these translations. Converting them to a rotation and scaling is non-trivial (especially the scaling is hard to get right, but a bit of math shows the way).
If the tutorial linked above is not clear enough, one can look at the C++ code that comes with it, or at this other Python code.
OP is interested only in the rotation aspect of the method above. If we can assume that the translation is 0 (this means we know around which point the rotation was made, if we don't know the origin we need to estimate it as a translation), then we don't need to compute the magnitude of the FFT (remember it is used to make the problem translation invariant), we can apply the log-polar transform directly to the images. But note that we need to use the center of rotation as the origin for the log-polar transform. If we additionally assume that the scaling is 1, we can further simplify things by taking the linear-polar transform. That is, we logarithmic scaling of the radius axis is only necessary to estimate scaling.
OP is doing this more or less correctly, I believe. Where OP's code goes wrong is in the extent of the radius axis in the polar transform. By going all the way to the extreme corners of the image, OpenCV needs to fill in parts of the transformed image with zeros. These parts are dictated by the shape of the image, not by the contents of the image. That is, both polar images contain exactly the same sharp, high-contrast curve between image content and filled-in zeros. The phase correlation is aligning these curves, leading to an estimate of 0 degree rotation. The image content is more or less ignored because its contrast is much lower.
Instead, make the extent of the radius axis that of the largest circle that fits completely inside the image. This way, no parts of the output need to be filled with zeros, and the phase correlation can focus on the actual image content. Furthermore, considering the two images are rotated versions of each other, it is likely that the data in the corners of the images do not match, there is no need to take that into account at all!
Here is code I implemented quickly based on OP's code. I read in Lena, rotated the image by 38 degrees, computed the linear-polar transform of the original and rotated images, then the phase correlation between these two, and then determined a rotation angle based on the vertical translation. The result was 37.99560, plenty close to 38.
import cv2
import numpy as np
base_img = cv2.imread('lena512color.tif')
base_img = np.float32(cv2.cvtColor(base_img, cv2.COLOR_BGR2GRAY)) / 255.0
(h, w) = base_img.shape
(cX, cY) = (w // 2, h // 2)
angle = 38
M = cv2.getRotationMatrix2D((cX, cY), angle, 1.0)
curr_img = cv2.warpAffine(base_img, M, (w, h))
cv2.imshow("base_img", base_img)
cv2.imshow("curr_img", curr_img)
base_polar = cv2.linearPolar(base_img,(cX, cY), min(cX, cY), 0)
curr_polar = cv2.linearPolar(curr_img,(cX, cY), min(cX, cY), 0)
cv2.imshow("base_polar", base_polar)
cv2.imshow("curr_polar", curr_polar)
(sx, sy), sf = cv2.phaseCorrelate(base_polar, curr_polar)
rotation = -sy / h * 360;
print(rotation)
cv2.waitKey(0)
cv2.destroyAllWindows()
These are the four image windows shown by the code:

I created a figure that shows the phase correlation values for multiple rotations. This has been edited to reflect Cris Luengo's comment. The image is cropped to get rid of the edges of the square insert.
import cv2
import numpy as np
paths = ["lena.png", "rotate45.png", "rotate90.png", "rotate135.png", "rotate180.png"]
import os
os.chdir('/home/stephen/Desktop/rotations/')
images, rotations, polar = [],[], []
for image_path in paths:
alignedImage = cv2.imread('lena.png')
rotatedImage = cv2.imread(image_path)
rows,cols,chan = alignedImage.shape
x, y, c = rotatedImage.shape
x,y,w,h = 220,220,360,360
alignedImage = alignedImage[y:y+h, x:x+h].copy()
rotatedImage = rotatedImage[y:y+h, x:x+h].copy()
#convert images to valid type
ref32 = np.float32(cv2.cvtColor(alignedImage, cv2.COLOR_BGR2GRAY))
curr32 = np.float32(cv2.cvtColor(rotatedImage, cv2.COLOR_BGR2GRAY))
value = np.sqrt(((rows/2.0)**2.0)+((cols/2.0)**2.0))
value2 = np.sqrt(((x/2.0)**2.0)+((y/2.0)**2.0))
polar_image = cv2.linearPolar(ref32,(rows/2, cols/2), value, cv2.WARP_FILL_OUTLIERS)
log_img = cv2.linearPolar(curr32,(x/2, y/2), value2, cv2.WARP_FILL_OUTLIERS)
shift = cv2.phaseCorrelate(polar_image, log_img)
(sx, sy), sf = shift
polar_image = polar_image.astype(np.uint8)
log_img = log_img.astype(np.uint8)
sx, sy, sf = round(sx, 4), round(sy, 4), round(sf, 4)
text = image_path + "\n" + "sx: " + str(sx) + " \nsy: " + str(sy) + " \nsf: " + str(sf)
images.append(rotatedImage)
rotations.append(text)
polar.append(polar_image)

Here's an approach to determine the rotational shift between two images in degrees. The idea is to find the skew angle for each image in relation to a horizontal line. If we can find this skewed angle then we can calculate the angle difference between the two images. Here are some example images to illustrate this concept
Original unrotated image
Rotated counterclockwise by 10 degrees (neg_10) and counterclockwise by 35 degrees (neg_35)
Rotated clockwise by 7.9 degrees (pos_7_9) and clockwise by 21 degrees (pos_21)
For each image, we want to determine the skew angle in relation to a horizontal line with negative being rotated counterclockwise and positive being rotated clockwise
Here's the helper function to determine this skew angle
def compute_angle(image):
# Convert to grayscale, invert, and Otsu's threshold
gray = cv2.cvtColor(image, cv2.COLOR_BGR2GRAY)
gray = 255 - gray
thresh = cv2.threshold(gray, 0, 255, cv2.THRESH_BINARY + cv2.THRESH_OTSU)[1]
# Find coordinates of all pixel values greater than zero
# then compute minimum rotated bounding box of all coordinates
coords = np.column_stack(np.where(thresh > 0))
angle = cv2.minAreaRect(coords)[-1]
# The cv2.minAreaRect() function returns values in the range
# [-90, 0) so need to correct angle
if angle < -45:
angle = -(90 + angle)
else:
angle = -angle
# Rotate image to horizontal position
(h, w) = image.shape[:2]
center = (w // 2, h // 2)
M = cv2.getRotationMatrix2D(center, angle, 1.0)
rotated = cv2.warpAffine(image, M, (w, h), flags=cv2.INTER_CUBIC, \
borderMode=cv2.BORDER_REPLICATE)
return (angle, rotated)
After determining the skew angle for each image, we can simply calculate the difference
angle1, rotated1 = compute_angle(image1)
angle2, rotated2 = compute_angle(image2)
# Both angles are positive
if angle1 >= 0 and angle2 >= 0:
difference_angle = abs(angle1 - angle2)
# One positive, one negative
elif (angle1 < 0 and angle2 > 0) or (angle1 > 0 and angle2 < 0):
difference_angle = abs(angle1) + abs(angle2)
# Both negative
elif angle1 < 0 and angle2 < 0:
angle1 = abs(angle1)
angle2 = abs(angle2)
difference_angle = max(angle1, angle2) - min(angle1, angle2)
Here's the step by step walk through of whats going on. Using pos_21 and neg_10, the compute_angle() function will return the skew angle and the normalized image
For pos_21, we normalize the image and determine the skew angle. Left (before) -> right (after)
20.99871826171875
Similarly for neg_10, we also normalize the image and determine the skew angle. Left (before) -> right (after)
-10.007980346679688
Now that we have both angles, we can compute the difference angle. Here's the result
31.006698608398438
Here's results with other combinations. With neg_10 and neg_35 we get
24.984039306640625
With pos_7_9 and pos_21,
13.09155559539795
Finally with pos_7_9 and neg_35,
42.89918231964111
Here's the full code
import cv2
import numpy as np
def rotational_shift(image1, image2):
def compute_angle(image):
# Convert to grayscale, invert, and Otsu's threshold
gray = cv2.cvtColor(image, cv2.COLOR_BGR2GRAY)
gray = 255 - gray
thresh = cv2.threshold(gray, 0, 255, cv2.THRESH_BINARY + cv2.THRESH_OTSU)[1]
# Find coordinates of all pixel values greater than zero
# then compute minimum rotated bounding box of all coordinates
coords = np.column_stack(np.where(thresh > 0))
angle = cv2.minAreaRect(coords)[-1]
# The cv2.minAreaRect() function returns values in the range
# [-90, 0) so need to correct angle
if angle < -45:
angle = -(90 + angle)
else:
angle = -angle
# Rotate image to horizontal position
(h, w) = image.shape[:2]
center = (w // 2, h // 2)
M = cv2.getRotationMatrix2D(center, angle, 1.0)
rotated = cv2.warpAffine(image, M, (w, h), flags=cv2.INTER_CUBIC, \
borderMode=cv2.BORDER_REPLICATE)
return (angle, rotated)
angle1, rotated1 = compute_angle(image1)
angle2, rotated2 = compute_angle(image2)
# Both angles are positive
if angle1 >= 0 and angle2 >= 0:
difference_angle = abs(angle1 - angle2)
# One positive, one negative
elif (angle1 < 0 and angle2 > 0) or (angle1 > 0 and angle2 < 0):
difference_angle = abs(angle1) + abs(angle2)
# Both negative
elif angle1 < 0 and angle2 < 0:
angle1 = abs(angle1)
angle2 = abs(angle2)
difference_angle = max(angle1, angle2) - min(angle1, angle2)
return (difference_angle, rotated1, rotated2)
if __name__ == '__main__':
image1 = cv2.imread('pos_7_9.png')
image2 = cv2.imread('neg_35.png')
angle, rotated1, rotated2 = rotational_shift(image1, image2)
print(angle)

Wrap image around a circle

What I'm trying to do in this example is wrap an image around a circle, like below.
To wrap the image I simply calculated the x,y coordinates using trig.
The problem is the calculated X and Y positions are rounded to make them integers. This causes the blank pixels in seen the wrapped image above. The x,y positions have to be an integer because they are positions in lists.
I've done this again in the code following but without any images to make things easier to see. All I've done is create two arrays with binary values, one array is black the other white, then wrapped one onto the other.
The output of the code is.
import math as m
from PIL import Image # only used for showing output as image
width = 254.0
height = 24.0
Ro = 40.0
img = [[1 for x in range(int(width))] for y in range(int(height))]
cir = [[0 for x in range(int(Ro * 2))] for y in range(int(Ro * 2))]
def shom_im(img): # for showing data as image
list_image = [item for sublist in img for item in sublist]
new_image = Image.new("1", (len(img[0]), len(img)))
new_image.putdata(list_image)
new_image.show()
increment = m.radians(360 / width)
rad = Ro - 0.5
for i, row in enumerate(img):
hyp = rad - i
for j, column in enumerate(row):
alpha = j * increment
x = m.cos(alpha) * hyp + rad
y = m.sin(alpha) * hyp + rad
# put value from original image to its position in new image
cir[int(round(y))][int(round(x))] = img[i][j]
shom_im(cir)
I later found out about the Midpoint Circle Algorithm but I had worse result with that
from PIL import Image # only used for showing output as image
width, height = 254, 24
ro = 40
img = [[(0, 0, 0, 1) for x in range(int(width))]
for y in range(int(height))]
cir = [[(0, 0, 0, 255) for x in range(int(ro * 2))] for y in range(int(ro * 2))]
def shom_im(img): # for showing data as image
list_image = [item for sublist in img for item in sublist]
new_image = Image.new("RGBA", (len(img[0]), len(img)))
new_image.putdata(list_image)
new_image.show()
def putpixel(x0, y0):
global cir
cir[y0][x0] = (255, 255, 255, 255)
def drawcircle(x0, y0, radius):
x = radius
y = 0
err = 0
while (x >= y):
putpixel(x0 + x, y0 + y)
putpixel(x0 + y, y0 + x)
putpixel(x0 - y, y0 + x)
putpixel(x0 - x, y0 + y)
putpixel(x0 - x, y0 - y)
putpixel(x0 - y, y0 - x)
putpixel(x0 + y, y0 - x)
putpixel(x0 + x, y0 - y)
y += 1
err += 1 + 2 * y
if (2 * (err - x) + 1 > 0):
x -= 1
err += 1 - 2 * x
for i, row in enumerate(img):
rad = ro - i
drawcircle(int(ro - 1), int(ro - 1), rad)
shom_im(cir)
Can anybody suggest a way to eliminate the blank pixels?

You are having problems filling up your circle because you are approaching this from the wrong way – quite literally.
When mapping from a source to a target, you need to fill your target, and map each translated pixel from this into the source image. Then, there is no chance at all you miss a pixel, and, equally, you will never draw (nor lookup) a pixel more than once.
The following is a bit rough-and-ready, it only serves as a concept example. I first wrote some code to draw a filled circle, top to bottom. Then I added some more code to remove the center part (and added a variable Ri, for "inner radius"). This leads to a solid ring, where all pixels are only drawn once: top to bottom, left to right.
How you exactly draw the ring is not actually important! I used trig at first because I thought of re-using the angle bit, but it can be done with Pythagorus' as well, and even with Bresenham's circle routine. All you need to keep in mind is that you iterate over the target rows and columns, not the source. This provides actual x,y coordinates that you can feed into the remapping procedure.
With the above done and working, I wrote the trig functions to translate from the coordinates I would put a pixel at into the original image. For this, I created a test image containing some text:
and a good thing that was, too, as in the first attempt I got the text twice (once left, once right) and mirrored – that needed a few minor tweaks. Also note the background grid. I added that to check if the 'top' and 'bottom' lines – the outermost and innermost circles – got drawn correctly.
Running my code with this image and Ro,Ri at 100 and 50, I get this result:
You can see that the trig functions make it start at the rightmost point, move clockwise, and have the top of the image pointing outwards. All can be trivially adjusted, but this way it mimics the orientation that you want your image drawn.
This is the result with your iris-image, using 33 for the inner radius:
and here is a nice animation, showing the stability of the mapping:
Finally, then, my code is:
import math as m
from PIL import Image
Ro = 100.0
Ri = 50.0
# img = [[1 for x in range(int(width))] for y in range(int(height))]
cir = [[0 for x in range(int(Ro * 2))] for y in range(int(Ro * 2))]
# image = Image.open('0vWEI.png')
image = Image.open('this-is-a-test.png')
# data = image.convert('RGB')
pixels = image.load()
width, height = image.size
def shom_im(img): # for showing data as image
list_image = [item for sublist in img for item in sublist]
new_image = Image.new("RGB", (len(img[0]), len(img)))
new_image.putdata(list_image)
new_image.save("result1.png","PNG")
new_image.show()
for i in range(int(Ro)):
# outer_radius = Ro*m.cos(m.asin(i/Ro))
outer_radius = m.sqrt(Ro*Ro - i*i)
for j in range(-int(outer_radius),int(outer_radius)):
if i < Ri:
# inner_radius = Ri*m.cos(m.asin(i/Ri))
inner_radius = m.sqrt(Ri*Ri - i*i)
else:
inner_radius = -1
if j < -inner_radius or j > inner_radius:
# this is the destination
# solid:
# cir[int(Ro-i)][int(Ro+j)] = (255,255,255)
# cir[int(Ro+i)][int(Ro+j)] = (255,255,255)
# textured:
x = Ro+j
y = Ro-i
# calculate source
angle = m.atan2(y-Ro,x-Ro)/2
distance = m.sqrt((y-Ro)*(y-Ro) + (x-Ro)*(x-Ro))
distance = m.floor((distance-Ri+1)*(height-1)/(Ro-Ri))
# if distance >= height:
# distance = height-1
cir[int(y)][int(x)] = pixels[int(width*angle/m.pi) % width, height-distance-1]
y = Ro+i
# calculate source
angle = m.atan2(y-Ro,x-Ro)/2
distance = m.sqrt((y-Ro)*(y-Ro) + (x-Ro)*(x-Ro))
distance = m.floor((distance-Ri+1)*(height-1)/(Ro-Ri))
# if distance >= height:
# distance = height-1
cir[int(y)][int(x)] = pixels[int(width*angle/m.pi) % width, height-distance-1]
shom_im(cir)
The commented-out lines draw a solid white ring. Note the various tweaks here and there to get the best result. For instance, the distance is measured from the center of the ring, and so returns a low value for close to the center and the largest values for the outside of the circle. Mapping that directly back onto the target image would display the text with its top "inwards", pointing to the inner hole. So I inverted this mapping with height - distance - 1, where the -1 is to make it map from 0 to height again.
A similar fix is in the calculation of distance itself; without the tweaks Ri+1 and height-1 either the innermost or the outermost row would not get drawn, indicating that the calculation is just one pixel off (which was exactly the purpose of that grid).

I think what you need is a noise filter. There are many implementations from which I think Gaussian filter would give a good result. You can find a list of filters here. If it gets blurred too much:
keep your first calculated image
calculate filtered image
copy fixed pixels from filtered image to first calculated image
Here is a crude average filter written by hand:
cir_R = int(Ro*2) # outer circle 2*r
inner_r = int(Ro - 0.5 - len(img)) # inner circle r
for i in range(1, cir_R-1):
for j in range(1, cir_R-1):
if cir[i][j] == 0: # missing pixel
dx = int(i-Ro)
dy = int(j-Ro)
pix_r2 = dx*dx + dy*dy # distance to center
if pix_r2 <= Ro*Ro and pix_r2 >= inner_r*inner_r:
cir[i][j] = (cir[i-1][j] + cir[i+1][j] + cir[i][j-1] +
cir[i][j+1])/4
shom_im(cir)
and the result:
This basically scans between two ranges checks for missing pixels and replaces them with average of 4 pixels adjacent to it. In this black white case it is all white.
Hope it helps!

Area of overlapping circles

I have the following Python code to generate random circles in order to simulate Brownian motion. I need to find the total area of the small red circles so that I can compare it to the total area of a larger blue circle. Since the circles are generated randomly, many of them overlap making it difficult to find the area. I have read many other responses related to this question about pixel painting, etc. What is the best way to find the area of these circles? I do not want to modify the generation of the circles, I just need to find the total area of the red circles on the plot.
The code to generate the circles I need is as follows (Python v. 2.7.6):
import matplotlib.pyplot as plt
import numpy as np
new_line = []
new_angle = []
x_c = [0]
y_c = [0]
x_real = []
y_real = []
xy_dist = []
circ = []
range_value = 101
for x in range(0,range_value):
mu, sigma = 0, 1
new_line = np.random.normal(mu, sigma, 1)
new_angle = np.random.uniform(0, 360)*np.pi/180
x_c.append(new_line*np.cos(new_angle))
y_c.append(new_line*np.sin(new_angle))
x_real = np.cumsum(x_c)
y_real = np.cumsum(y_c)
a = np.mean(x_real)
b = np.mean(y_real)
i = 0
while i<=range_value:
xy_dist.append(np.sqrt((x_real[i]-a)**2+(y_real[i]-b)**2))
i += 1
circ_rad = max(xy_dist)
small_rad = 0.2
fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)
circ1 = plt.Circle((a,b), radius=circ_rad+small_rad, color='b')
ax.add_patch(circ1)
j = 0
while j<=range_value:
circ = plt.Circle((x_real[j], y_real[j]), radius=small_rad, color='r', fill=True)
ax.add_patch(circ)
j += 1
plt.axis('auto')
plt.show()

The package Shapely might be of some use:
https://gis.stackexchange.com/questions/11987/polygon-overlay-with-shapely
http://toblerity.org/shapely/manual.html#geometric-objects

I can think of an easy way to do it thought the result will have inaccuracies:
With Python draw all your circles on a white image, filling the circles as you draw them. At the end each "pixel" of your image will have one of 2 colors: white color is the background and the other color (let's say red) means that pixel is occupied by a circle.
You then need to sum the number of red pixels and multiply them by the scale with which you draw them. You will have then the area.
This is inaccurate as there is no way of drawing a circle using square pixels, so in the mapping you lose accuracy. Keep in mind that the bigger you draw the circles, the smaller the inaccuracy becomes.

Automatically remove hot/dead pixels from an image in python

I am using numpy and scipy to process a number of images taken with a CCD camera. These images have a number of hot (and dead) pixels with very large (or small) values. These interfere with other image processing, so they need to be removed. Unfortunately, though a few of the pixels are stuck at either 0 or 255 and are always at the same value in all of the images, there are some pixels that are temporarily stuck at other values for a period of a few minutes (the data spans many hours).
I am wondering if there is a method for identifying (and removing) the hot pixels already implemented in python. If not, I am wondering what would be an efficient method for doing so. The hot/dead pixels are relatively easy to identify by comparing them with neighboring pixels. I could see writing a loop that looks at each pixel, compares its value to that of its 8 nearest neighbors. Or, it seems nicer to use some kind of convolution to produce a smoother image and then subtract this from the image containing the hot pixels, making them easier to identify.
I have tried this "blurring method" in the code below, and it works okay, but I doubt that it is the fastest. Also, it gets confused at the edge of the image (probably since the gaussian_filter function is taking a convolution and the convolution gets weird near the edge). So, is there a better way to go about this?
Example code:
import numpy as np
import matplotlib.pyplot as plt
import scipy.ndimage
plt.figure(figsize=(8,4))
ax1 = plt.subplot(121)
ax2 = plt.subplot(122)
#make a sample image
x = np.linspace(-5,5,200)
X,Y = np.meshgrid(x,x)
Z = 255*np.cos(np.sqrt(x**2 + Y**2))**2
for i in range(0,11):
#Add some hot pixels
Z[np.random.randint(low=0,high=199),np.random.randint(low=0,high=199)]= np.random.randint(low=200,high=255)
#and dead pixels
Z[np.random.randint(low=0,high=199),np.random.randint(low=0,high=199)]= np.random.randint(low=0,high=10)
#Then plot it
ax1.set_title('Raw data with hot pixels')
ax1.imshow(Z,interpolation='nearest',origin='lower')
#Now we try to find the hot pixels
blurred_Z = scipy.ndimage.gaussian_filter(Z, sigma=2)
difference = Z - blurred_Z
ax2.set_title('Difference with hot pixels identified')
ax2.imshow(difference,interpolation='nearest',origin='lower')
threshold = 15
hot_pixels = np.nonzero((difference>threshold) | (difference<-threshold))
#Don't include the hot pixels that we found near the edge:
count = 0
for y,x in zip(hot_pixels[0],hot_pixels[1]):
if (x != 0) and (x != 199) and (y != 0) and (y != 199):
ax2.plot(x,y,'ro')
count += 1
print 'Detected %i hot/dead pixels out of 20.'%count
ax2.set_xlim(0,200); ax2.set_ylim(0,200)
plt.show()
And the output:

Basically, I think that the fastest way to deal with hot pixels is just to use a size=2 median filter. Then, poof, your hot pixels are gone and you also kill all sorts of other high-frequency sensor noise from your camera.
If you really want to remove ONLY the hot pixels, then substituting you can subtract the median filter from the original image, as I did in the question, and replace only these values with the values from the median filtered image. This doesn't work well at the edges, so if you can ignore the pixels along the edge, then this will make things a lot easier.
If you want to deal with the edges, you can use the code below. However, it is not the fastest:
import numpy as np
import matplotlib.pyplot as plt
import scipy.ndimage
plt.figure(figsize=(10,5))
ax1 = plt.subplot(121)
ax2 = plt.subplot(122)
#make some sample data
x = np.linspace(-5,5,200)
X,Y = np.meshgrid(x,x)
Z = 100*np.cos(np.sqrt(x**2 + Y**2))**2 + 50
np.random.seed(1)
for i in range(0,11):
#Add some hot pixels
Z[np.random.randint(low=0,high=199),np.random.randint(low=0,high=199)]= np.random.randint(low=200,high=255)
#and dead pixels
Z[np.random.randint(low=0,high=199),np.random.randint(low=0,high=199)]= np.random.randint(low=0,high=10)
#And some hot pixels in the corners and edges
Z[0,0] =255
Z[-1,-1] =255
Z[-1,0] =255
Z[0,-1] =255
Z[0,100] =255
Z[-1,100]=255
Z[100,0] =255
Z[100,-1]=255
#Then plot it
ax1.set_title('Raw data with hot pixels')
ax1.imshow(Z,interpolation='nearest',origin='lower')
def find_outlier_pixels(data,tolerance=3,worry_about_edges=True):
#This function finds the hot or dead pixels in a 2D dataset.
#tolerance is the number of standard deviations used to cutoff the hot pixels
#If you want to ignore the edges and greatly speed up the code, then set
#worry_about_edges to False.
#
#The function returns a list of hot pixels and also an image with with hot pixels removed
from scipy.ndimage import median_filter
blurred = median_filter(Z, size=2)
difference = data - blurred
threshold = 10*np.std(difference)
#find the hot pixels, but ignore the edges
hot_pixels = np.nonzero((np.abs(difference[1:-1,1:-1])>threshold) )
hot_pixels = np.array(hot_pixels) + 1 #because we ignored the first row and first column
fixed_image = np.copy(data) #This is the image with the hot pixels removed
for y,x in zip(hot_pixels[0],hot_pixels[1]):
fixed_image[y,x]=blurred[y,x]
if worry_about_edges == True:
height,width = np.shape(data)
###Now get the pixels on the edges (but not the corners)###
#left and right sides
for index in range(1,height-1):
#left side:
med = np.median(data[index-1:index+2,0:2])
diff = np.abs(data[index,0] - med)
if diff>threshold:
hot_pixels = np.hstack(( hot_pixels, [[index],[0]] ))
fixed_image[index,0] = med
#right side:
med = np.median(data[index-1:index+2,-2:])
diff = np.abs(data[index,-1] - med)
if diff>threshold:
hot_pixels = np.hstack(( hot_pixels, [[index],[width-1]] ))
fixed_image[index,-1] = med
#Then the top and bottom
for index in range(1,width-1):
#bottom:
med = np.median(data[0:2,index-1:index+2])
diff = np.abs(data[0,index] - med)
if diff>threshold:
hot_pixels = np.hstack(( hot_pixels, [[0],[index]] ))
fixed_image[0,index] = med
#top:
med = np.median(data[-2:,index-1:index+2])
diff = np.abs(data[-1,index] - med)
if diff>threshold:
hot_pixels = np.hstack(( hot_pixels, [[height-1],[index]] ))
fixed_image[-1,index] = med
###Then the corners###
#bottom left
med = np.median(data[0:2,0:2])
diff = np.abs(data[0,0] - med)
if diff>threshold:
hot_pixels = np.hstack(( hot_pixels, [[0],[0]] ))
fixed_image[0,0] = med
#bottom right
med = np.median(data[0:2,-2:])
diff = np.abs(data[0,-1] - med)
if diff>threshold:
hot_pixels = np.hstack(( hot_pixels, [[0],[width-1]] ))
fixed_image[0,-1] = med
#top left
med = np.median(data[-2:,0:2])
diff = np.abs(data[-1,0] - med)
if diff>threshold:
hot_pixels = np.hstack(( hot_pixels, [[height-1],[0]] ))
fixed_image[-1,0] = med
#top right
med = np.median(data[-2:,-2:])
diff = np.abs(data[-1,-1] - med)
if diff>threshold:
hot_pixels = np.hstack(( hot_pixels, [[height-1],[width-1]] ))
fixed_image[-1,-1] = med
return hot_pixels,fixed_image
hot_pixels,fixed_image = find_outlier_pixels(Z)
for y,x in zip(hot_pixels[0],hot_pixels[1]):
ax1.plot(x,y,'ro',mfc='none',mec='r',ms=10)
ax1.set_xlim(0,200)
ax1.set_ylim(0,200)
ax2.set_title('Image with hot pixels removed')
ax2.imshow(fixed_image,interpolation='nearest',origin='lower',clim=(0,255))
plt.show()
Output: