I am creating a circular mask in python as follows:
import numpy as np
def make_mask(image, radius, center=(0, 0)):
r, c, d = image.shape
y, x = np.ogrid[-center[0]:r-center[0], -center[1]:r-center[1]]
mask = x*x + y*y <= radius*radius
array = np.zeros((r, c))
array[mask] = 1
return array
This returns a mask of shape (r, c). What I would like to do is have a weighted mask where the weight is 1 at the center of the image (given by the center parameter) and decreasing linearly towards the edge of the image. So, his should be an added weight calculated between 0 and 1 (0 not included) in the line. I was thinking this should be something like:
distance = (center[0] - x)**2 + (center[1] - y)**2
# weigh it inversely to distance from center
mask = (x*x + y*y) * 1.0/distance
However, this will result in divide by 0 and the mask would not be between 0 and 1 either.
First, if you want to weight to be linear, you need to take the square root of what you have for distance (ie, what you're calling "distance" isn't the distance from the center but the square of that, so you should rename it to something like R_squared). So:
R_squared = (center[0] - x)**2 + (center[1] - y)**2 # what you have for distance
r = sqrt(R_squared)
Then, since it starts off as 0 where you want it to be 1, add 1 to it; but now that you've added 1 scale the value so it's 1 where you want the result to be 0. Say you want it to be 0 at a distance L from then center, then your equation is:
weight = 1 - r/L
Here this will be 1 where r==0 and 0 where r==L.
Related
I need to draw a circle in a 2D numpy array given [i,j] as indexes of the array, and r as the radius of the circle. Each time a condition is met at index [i,j], a circle should be drawn with that as the center point, increasing all values inside the circle by +1. I want to avoid the for-loops at the end where I draw the circle (where I use p,q to index) because I have to draw possibly millions of circles. Is there a way without for loops? I also don't want to import another library for just a single task.
Here is my current implementation:
for i in range(array_shape[0]):
for j in range(array_shape[1]):
if (condition): # Draw circle if condition is fulfilled
# Create a square of pixels with side lengths equal to radius of circle
x_square_min = i-r
x_square_max = i+r+1
y_square_min = j-r
y_square_max = j+r+1
# Clamp this square to the edges of the array so circles near edges don't wrap around
if x_square_min < 0:
x_square_min = 0
if y_square_min < 0:
y_square_min = 0
if x_square_max > array_shape[0]:
x_square_max = array_shape[0]
if y_square_max > array_shape[1]:
y_square_max = array_shape[1]
# Now loop over the box and draw circle inside of it
for p in range(x_square_min , x_square_max):
for q in range(y_square_min , y_square_max):
if (p - i) ** 2 + (q - j) ** 2 <= r ** 2:
new_array[p,q] += 1 # Incrementing because need to have possibility of
# overlapping circles
If you're using the same radius for every single circle, you can simplify things significantly by only calculating the circle coordinates once and then adding the center coordinates to the circle points when needed. Here's the code:
# The main array of values is called array.
shape = array.shape
row_indices = np.arange(0, shape[0], 1)
col_indices = np.arange(0, shape[1], 1)
# Returns xy coordinates for a circle with a given radius, centered at (0,0).
def points_in_circle(radius):
a = np.arange(radius + 1)
for x, y in zip(*np.where(a[:,np.newaxis]**2 + a**2 <= radius**2)):
yield from set(((x, y), (x, -y), (-x, y), (-x, -y),))
# Set the radius value before running code.
radius = RADIUS
circle_r = np.array(list(points_in_circle(radius)))
# Note that I'm using x as the row number and y as the column number.
# Center of circle is at (x_center, y_center). shape_0 and shape_1 refer to the main array
# so we can get rid of coordinates outside the bounds of array.
def add_center_to_circle(circle_points, x_center, y_center, shape_0, shape_1):
circle = np.copy(circle_points)
circle[:, 0] += x_center
circle[:, 1] += y_center
# Get rid of rows where coordinates are below 0 (can't be indexed)
bad_rows = np.array(np.where(circle < 0)).T[:, 0]
circle = np.delete(circle, bad_rows, axis=0)
# Get rid of rows that are outside the upper bounds of the array.
circle = circle[circle[:, 0] < shape_0, :]
circle = circle[circle[:, 1] < shape_1, :]
return circle
for x in row_indices:
for y in col_indices:
# You need to set CONDITION before running the code.
if CONDITION:
# Because circle_r is the same for all circles, it doesn't need to be recalculated all the time. All you need to do is add x and y to circle_r each time CONDITION is met.
circle_coords = add_center_to_circle(circle_r, x, y, shape[0], shape[1])
array[tuple(circle_coords.T)] += 1
When I set radius = 10, array = np.random.rand(1200).reshape(40, 30) and replaced if CONDITION with if (x == 20 and y == 20) or (x == 25 and y == 20), I got this, which seems to be what you want:
Let me know if you have any questions.
Adding each circle can be vectorized. This solution iterates over the coordinates where the condition is met. On a 2-core colab instance ~60k circles with radius 30 can be added per second.
import numpy as np
np.random.seed(42)
arr = np.random.rand(400,300)
r = 30
xx, yy = np.mgrid[-r:r+1, -r:r+1]
circle = xx**2 + yy**2 <= r**2
condition = np.where(arr > .999) # np.where(arr > .5) to benchmark 60k circles
for x,y in zip(*condition):
# valid indices of the array
i = slice(max(x-r,0), min(x+r+1, arr.shape[0]))
j = slice(max(y-r,0), min(y+r+1, arr.shape[1]))
# visible slice of the circle
ci = slice(abs(min(x-r, 0)), circle.shape[0] - abs(min(arr.shape[0]-(x+r+1), 0)))
cj = slice(abs(min(y-r, 0)), circle.shape[1] - abs(min(arr.shape[1]-(y+r+1), 0)))
arr[i, j] += circle[ci, cj]
Visualizing np.array arr
import matplotlib.pyplot as plt
plt.figure(figsize=(8,8))
plt.imshow(arr)
plt.show()
I am trying to generate two (1D) points x1, x2 chosen randomly, and independently, from the uniform distribution U(-1,1) such that the euclidean distance between them is less than a certain value, dist. Here is one solution, but I'm looking for something more efficient:
def point_pair(low_=-1, hight_=1, dist = 0.001):
while(1):
x = np.random.uniform(low=low_, high=hight_)
y = np.random.uniform(low=low_, high=hight_)
length = np.linalg.norm(x-y)
if length <= dist:
return x,y
return 0,0
To generate two scalars whose magnitudes are close to each other:
import numpy as np
def point_pair(low, high, dist):
delta = np.random.uniform(-dist, dist)
a = np.random.uniform(low + dist, high - dist)
b = np.random.choice((-1, 1)) * a + delta
return a, b
Your scalar a is generated almost in the same way, but where the bounds are not [-1, 1) but [-1 + dist, 1 - dist). For b, you generate a new scalar which is uniformly distributed from -dist to +dist. This represents the bounds on the maximum distance away from a in either direction on the real line that you allow. Then b is simply k*a + delta, where again, delta is any value between -dist and +dist, and k is either -1 or 1.
This will ensure that both a and b are in [-1, 1) and that their magnitudes are similar, or ||a| - |b|| <= dist.
Note
The np.random.uniform(low, high) function always returns values in [low, high) so if you want to also include your upper bound, you'll need to use a different method.
use polar coordinates, your random numbers are angle and distance.
I am trying to implement a bilateral filter from the paper Fast Bilateral Filteringfor the Display of High-Dynamic-Range Images. The equation (from the paper) that implements the bilateral filter is given as :
According to what I understood,
f is a Gaussian filter
g is a Gaussian filter
p is a pixel in a given image window
s is the current pixel
Ip is the intensity at the current pixel
With this, I wrote the code to implement these equations, given as :
import cv2
import numpy as np
img = cv2.imread("fish.png")
# image of width 239 and height 200
bl_img = cv2.cvtColor(img, cv2.COLOR_BGR2GRAY)
i = cv2.magnitude(
cv2.Sobel(bl_img, cv2.CV_64F, 1, 0, ksize=3),
cv2.Sobel(bl_img, cv2.CV_64F, 0, 1, ksize=3)
)
f = cv2.getGaussianKernel(5, 0.1, cv2.CV_64F)
g = cv2.getGaussianKernel(5, 0.1, cv2.CV_64F)
rows, cols, _ = img.shape
filtered = np.zeros(img.shape, dtype=img.dtype)
for r in range(rows):
for c in range(cols):
ks = []
for index in [-2,-1,1,2]:
if index + c > 0 and index + c < cols-1:
p = img[r][index + c]
s = img[r][c]
i_p = i[index+c]
i_s = i[c]
ks.append(
(f * (p-s)) * (g * (i_p * i_s)) # EQUATION 7
)
ks = np.sum(np.array(ks))
js = []
for index in [-2, -1, 1, 2]:
if index + c > 0 and index + c < cols -1:
p = img[r][index + c]
s = img[r][c]
i_p = i[index+c]
i_s = i[c]
js.append((f * (p-s)) * (g * (i_p * i_s)) * i_p) # EQUATION 6
js = np.sum(np.asarray(js))
js = js / ks
filtered[r][c] = js
cv2.imwrite("f.png", filtered)
But as I run this code I get an error saying:
Traceback (most recent call last):
File "bft.py", line 33, in <module>
(f * (p-s)) * (g * (i_p * i_s))
ValueError: operands could not be broadcast together with shapes (5,3) (5,239)
Did I incorrectly implement the equations? What am I missing?
There are various issues with your code. Foremost, the equation is interpreted in a wrong way. f(p-s) means evaluating the function f at p-s. f is the Gaussian. Likewise with g. The section of the code would look like this:
weight = gaussian(p - s, sigma_f) * gaussian(i_p - i_s, sigma_g)
ks.append(weight)
js.append(weight * i_p)
Note that the two loops can be merged, this way you avoid some duplicated computation. gaussian(x, sigma) would be a function that computes the Gaussian weight at x. You need to define two sigmas, sigma_f and sigma_g, the spatial and the tonal sigma respectively.
The second issue is in the definition of p and s. These are the coordinates of the pixel, not the value of the image at the pixel. i_p and i_s are the value of the image at those locations. p-s is basically the spatial distance between the pixel at (r,c) and the given neighbor.
The third issue is the loop over the neighborhood. The neighborhood is all pixels where gaussian(p - s, sigma_f) is not negligible. So how large the neighborhood is depends on the chosen sigma_f. You should take it at least to be ceil(2*sigma_f). Say sigma_f is 2, then you want the neighborhood to go from -4 to 4 (9 pixels). But this neighborhood is two dimensional, not one-dimensional as in your code. So you need two loops:
for ii in range(-ceil(2*sigma_f), ceil(2*sigma_f)+1):
if ii + c > 0 and ii + c < cols-1:
for jj in range(-ceil(2*sigma_f), ceil(2*sigma_f)+1):
if jj + r > 0 and jj + r < rows-1:
# compute weight here
Note that now, p-s is computed with math.sqrt(ii**2 + jj**2). But also note that the Gaussian uses x**2, so you could skip the computation of the square root by passing x**2 into your gaussian function.
I have a numpy array filled with intensity readings at different radii in a uniform circle (for context, this is a 1D radiative transfer project for protostellar formation models: while much better models exist, my supervisor wasnts me to have the experience of producing one so I understand how others work).
I want to take that 1d array, and "rotate" it through a circle, forming a 2D array of intensities that could then be shown with imshow (or, with a bit of work, aplpy). The final array needs to be 2d, and the projection needs to be Cartesian, not polar.
I can do it with nested for loops, and I can do it with lookup tables, but I have a feeling there must be a neat way of doing it in numpy or something.
Any ideas?
EDIT:
I have had to go back and recreate my (frankly horrible) mess of for loops and if statements that I had before. If I really tried, I could probably get rid of one of the loops and one of the if statements by condensing things down. However, the aim is not to make it work with for loops, but see if there is a built in way to rotate the array.
impB is an array that differs slightly from what I stated it was before. Its actually just a list of radii where particles are detected. I then bin those into radius bins to get the intensity (or frequency if you prefer) in each radius. R is the scale factor for my radius as I run the model in a dimensionless way. iRes is a resolution scale factor, essentially how often I want to sample my radial bins. Everything else should be clear.
radJ = np.ndarray(shape=(2*iRes, 2*iRes)) # Create array of 2xRadius square
for i in range(iRes):
n = len(impB[np.where(impB[:] < ((i+1.) * (R / iRes)))]) # Count number of things within this radius +1
m = len(impB[np.where(impB[:] <= ((i) * (R / iRes)))]) # Count number of things in this radius
a = (((i + 1) * (R / iRes))**2 - ((i) * (R / iRes))**2) * math.pi # A normalisation factor based on area.....dont ask
for x in range(iRes):
for y in range(iRes):
if (x**2 + y**2) < (i * iRes)**2:
if (x**2 + y**2) >= (i * iRes)**2: # Checks for radius, and puts in cartesian space
radJ[x+iRes,y+iRes] = (n-m) / a # Put in actual intensity bins
radJ[x+iRes,-y+iRes] = (n-m) / a
radJ[-x+iRes,y+iRes] = (n-m) / a
radJ[-x+iRes,-y+iRes] = (n-m) / a
Nested loops are a simple approach for that. With ri_data_r and y containing your radius values (difference to the middle pixel) and the array for rotation, respectively, I would suggest:
from scipy import interpolate
import numpy as np
y = np.random.rand(100)
ri_data_r = np.linspace(-len(y)/2,len(y)/2,len(y))
interpol_index = interpolate.interp1d(ri_data_r, y)
xv = np.arange(-1, 1, 0.01) # adjust your matrix values here
X, Y = np.meshgrid(xv, xv)
profilegrid = np.ones(X.shape, float)
for i, x in enumerate(X[0, :]):
for k, y in enumerate(Y[:, 0]):
current_radius = np.sqrt(x ** 2 + y ** 2)
profilegrid[i, k] = interpol_index(current_radius)
print(profilegrid)
This will give you exactly what you are looking for. You just have to take in your array and calculate an symmetric array ri_data_r that has the same length as your data array and contains the distance between the actual data and the middle of the array. The code is doing this automatically.
I stumbled upon this question in a different context and I hope I understood it right. Here are two other ways of doing this. The first uses skimage.transform.warp with interpolation of desired order (here we use order=0 Nearest-neighbor). This method is slower but more precise and needs less memory then the second method.
The second one does not use interpolation, therefore is faster but also less precise and needs way more memory because it stores each 2D array containing one tilt until the end, where they are averaged with np.nanmean().
The difference between both solutions stemmed from the problem of handling the center of the final image where the tilts overlap the most, i.e. the first one would just add values with each tilt ending up out of the original range. This was "solved" by clipping the matrix in each step to a global_min and global_max (consult the code). The second one solves it by taking the mean of the tilts where they overlap, which forces us to use the np.nan.
Please, read the Example of usage and Sanity check sections in order to understand the plot titles.
Solution 1:
import numpy as np
from skimage.transform import warp
def rotate_vector(vector, deg_angle):
# Credit goes to skimage.transform.radon
assert vector.ndim == 1, 'Pass only 1D vectors, e.g. use array.ravel()'
center = vector.size // 2
square = np.zeros((vector.size, vector.size))
square[center,:] = vector
rad_angle = np.deg2rad(deg_angle)
cos_a, sin_a = np.cos(rad_angle), np.sin(rad_angle)
R = np.array([[cos_a, sin_a, -center * (cos_a + sin_a - 1)],
[-sin_a, cos_a, -center * (cos_a - sin_a - 1)],
[0, 0, 1]])
# Approx. 80% of time is spent in this function
return warp(square, R, clip=False, output_shape=((vector.size, vector.size)))
def place_vectors(vectors, deg_angles):
matrix = np.zeros((vectors.shape[-1], vectors.shape[-1]))
global_min, global_max = 0, 0
for i, deg_angle in enumerate(deg_angles):
tilt = rotate_vector(vectors[i], deg_angle)
global_min = tilt.min() if global_min > tilt.min() else global_min
global_max = tilt.max() if global_max < tilt.max() else global_max
matrix += tilt
matrix = np.clip(matrix, global_min, global_max)
return matrix
Solution 2:
Credit for the idea goes to my colleague Michael Scherbela.
import numpy as np
def rotate_vector(vector, deg_angle):
assert vector.ndim == 1, 'Pass only 1D vectors, e.g. use array.ravel()'
square = np.ones([vector.size, vector.size]) * np.nan
radius = vector.size // 2
r_values = np.linspace(-radius, radius, vector.size)
rad_angle = np.deg2rad(deg_angle)
ind_x = np.round(np.cos(rad_angle) * r_values + vector.size/2).astype(np.int)
ind_y = np.round(np.sin(rad_angle) * r_values + vector.size/2).astype(np.int)
ind_x = np.clip(ind_x, 0, vector.size-1)
ind_y = np.clip(ind_y, 0, vector.size-1)
square[ind_y, ind_x] = vector
return square
def place_vectors(vectors, deg_angles):
matrices = []
for deg_angle, vector in zip(deg_angles, vectors):
matrices.append(rotate_vector(vector, deg_angle))
matrix = np.nanmean(np.array(matrices), axis=0)
return np.nan_to_num(matrix, copy=False, nan=0.0)
Example of usage:
r = 100 # Radius of the circle, i.e. half the length of the vector
n = int(np.pi * r / 8) # Number of vectors, e.g. number of tilts in tomography
v = np.ones(2*r) # One vector, e.g. one tilt in tomography
V = np.array([v]*n) # All vectors, e.g. a sinogram in tomography
# Rotate 1D vector to a specific angle (output is 2D)
angle = 45
rotated = rotate_vector(v, angle)
# Rotate each row of a 2D array according to its angle (output is 2D)
angles = np.linspace(-90, 90, num=n, endpoint=False)
inplace = place_vectors(V, angles)
Sanity check:
These are just simple checks which by no means cover all possible edge cases. Depending on your use case you might want to extend the checks and adjust the method.
# I. Sanity check
# Assuming n <= πr and v = np.ones(2r)
# Then sum(inplace) should be approx. equal to (n * (2πr - n)) / π
# which is an area that should be covered by the tilts
desired_area = (n * (2 * np.pi * r - n)) / np.pi
covered_area = np.sum(inplace)
covered_frac = covered_area / desired_area
print(f'This method covered {covered_frac * 100:.2f}% '
'of the area which should be covered in total.')
# II. Sanity check
# Assuming n <= πr and v = np.ones(2r)
# Then a circle M with radius m <= r should be the largest circle which
# is fully covered by the vectors. I.e. its mean should be no less than 1.
# If n = πr then m = r.
# m = n / π
m = int(n / np.pi)
# Code for circular mask not included
mask = create_circular_mask(2*r, 2*r, center=None, radius=m)
m_area = np.mean(inplace[mask])
print(f'Full radius r={r}, radius m={m}, mean(M)={m_area:.4f}.')
Code for plotting:
import matplotlib.pyplot as plt
plt.figure(figsize=(16, 8))
plt.subplot(121)
rotated = np.nan_to_num(rotated) # not necessary in case of the first method
plt.title(
f'Output of rotate_vector(), angle={angle}°\n'
f'Sum is {np.sum(rotated):.2f} and should be {np.sum(v):.2f}')
plt.imshow(rotated, cmap=plt.cm.Greys_r)
plt.subplot(122)
plt.title(
f'Output of place_vectors(), r={r}, n={n}\n'
f'Covered {covered_frac * 100:.2f}% of the area which should be covered.\n'
f'Mean of the circle M is {m_area:.4f} and should be 1.0.')
plt.imshow(inplace)
circle=plt.Circle((r, r), m, color='r', fill=False)
plt.gcf().gca().add_artist(circle)
plt.gcf().gca().legend([circle], [f'Circle M (m={m})'])
I was using a circular image mask before and I was calculating a weight based on the distance from the centre of the circle as follows:
import numpy as np
def create_mask(image, radius, center=(0, 0)):
r, c, d = image.shape
x, y = np.ogrid[:r, :c]
distance = np.sqrt((x-center[0])**2 + (y-center[1])**2)
m = distance < radius
distance[m] = 1.0 - distance[m]/radius
array = np.zeros((r, c))
array[m] = distance[m]
return array
This was basically setting the height weight at the centre and the weight was dropping linearly towards the edges.
Now, I want to do something similar with an ellipse. Again, the ellipse can have very different radii along the two dimensions and I would like the weight to drop linearly with distance as well. However, regardless of the long or the short radii, I would like the weights to decay similarly towards the edges. I am guessing I need to include a weight based on both the radius to achieve this but was unable to figure it out.
I'm not sure about linear weights, but you can achieve a continuous array weights from 1 to 0 using (and I'm sure there's a more efficient way to do this)
ellipse = lambda x0, y0, r_x, r_y: lambda x, y: ((x - x0) / r_x)**2 + ((y - y0) / r_y)**2
def gen_ellipse(el, lower, upper, step):
coords = np.arange(lower, upper, step)
x, y = np.meshgrid(coords, coords)
mask = el(x, y)
mask[np.where(mask > 1)] = 0
return 1 - mask
For example:
> %pylab
> el = ellipse(0.0, 0.0, 0.3, 0.8)
> mask = gen_ellipse(el, -1.0, 1.0, 0.0025)
> imshow(mask, cmap=get_cmap('Greys'))
Where black is 1 and white is 0.