Isotopy: continuous deformation of homeomorphisms

The pseudocode for the isotopy is below. The swirl is more intense near the center, and decays exponentially as you approach the border.

Code

img = load_image
x,y = pixel_coordinates(img)

r,t = convert_to_polar(x,y)
radius = r/rmax

swirl = 2pi / exp(radius)  # more intense near center
near_border = where(r > 0.6*rmax)  # threshold for "near the border"

# swirl less near the border
swirl[near_border] *= exp(-40*(radius - 0.6))  # Decay factor for near the border

# linear homotopy b/w swirl and minor swirl
minor_swirl = 0.0000001
swirl_strength = (1-radius)*swirl  + radius*minor_swirl

theta += swirl_strength # add swirl to theta

# convert back to Cartesian coordinates
map_x = r * np.cos(theta) + center_x
map_y = r * np.sin(theta) + center_y

show(img, map_x, map_y)

Code

import cv2
import numpy as np
from PIL import Image

# load image and convert to float32 numpy array
img = np.array(Image.open('uv_checkerboard.png').convert('RGB'))
rows, cols = img.shape[:2]

# create the coordinate grids
# These MUST be float32 for cv2.remap
x_grid, y_grid = np.meshgrid(np.arange(cols), np.arange(rows))
map_x = x_grid.astype(np.float32)
map_y = y_grid.astype(np.float32)


# convert map_x, map_y to polar coordinates
# center the coordinates around the middle of the image
center_x = cols / 2
center_y = rows / 2
# shift the grid to be centered around (0, 0)
shifted_x = map_x - center_x
shifted_y = map_y - center_y
# convert to polar coordinates
r = np.sqrt(shifted_x**2 + shifted_y**2)
theta = np.arctan2(shifted_y, shifted_x)

# swirl the image by adding a function of r to theta
# create a swirl effect that increases with distance from the center
rmax = r.max()
img_id = '1'

main_swirl = 2*np.pi / np.exp(r/rmax)  # more intense near center
near_border = np.where(r > 0.6*rmax)  # Define a threshold for "near the border"

# gradually reduce the swirl strength as you approach the border
main_swirl[near_border] *= np.exp(-40*(r[near_border]/rmax - 0.6))  # Decay factor for near the border
# swirl that decays exponentially as you approach the border, creating a more natural swirl effect
minor_swirl = 0.0000001
swirl_strength = (1-r/rmax)*main_swirl  + r/rmax*minor_swirl

theta += swirl_strength

# convert back to Cartesian coordinates
map_x = r * np.cos(theta) + center_x
map_y = r * np.sin(theta) + center_y

# remap call
# using BORDER_WRAP makes the pixels that fall off the left side
# appear on the right side 
result = cv2.remap(
    img,
    map_x,
    map_y,
    interpolation=cv2.INTER_LINEAR,
    borderMode=cv2.BORDER_WRAP
)

# show it
Image.fromarray(result).show()

# save the image
Image.fromarray(result).save(f'swirled_checkerboard_{img_id}.png')

The swirled image is mapped onto the torus below.