Generating Bird Eye View of distorted image in OpenCV without knowing camera properties

I'm trying to transform this soccer field image

to a bird eye view and have attempted initially with warpPerspective only (OpenCV in python) but results with

which shows that there is distortion in the original image.

I've researched ways to undistort images but from what I've read so far is that I need to do camera calibration. The issue here is that I do not have any information on the properties of the camera used to record the soccer field footage (and this is the only angle which makes it difficult to do camera calibration).

I'm looking for any advice on how to go about undistort-ing the image to have a perfect bird eye view of the field without knowing the camera properties. Any next steps or resources I can learn from are also appreciated.

This is the code I have so far in Python:

import cv2 as cv
import numpy as np

img = cv.imread("/Users/laulpee/Desktop/python/Screen Shot 2022-05-20 at 2.44.15 AM.png")
print(img.shape)

# Field Size 60' x 90'
h = 900
w = 1200


# 4 Points on Original Image
pt1 = np.float32([[2125,382],[1441,502],[3236,773],[3252,530]])

# 4 Corresponding Points of Desired Bird Eye View Image
pt2 = np.float32([[0,0],[0,h],[w,h],[w,0]])


matrix = cv.getPerspectiveTransform(pt1, pt2)
output = cv.warpPerspective(img,matrix,(w,h))

for i in range(0,4):
    cv.circle(img,(int(pt1[i][0]),int(pt1[i][1])),5,(0,0,255),cv.FILLED)

window1 = cv.namedWindow("w1")
cv.imshow(window1,img)
cv.waitKey(0)
cv.destroyWindow(window1)


window2 = cv.namedWindow("w2")
cv.imshow(window2,output)
cv.waitKey(0)
cv.destroyWindow(window2)

In general, you need to "calibrate" the camera. That means to estimate lens distortion coefficients, optical center, focal lengths, perhaps even shear coefficients. All of that depends on the camera sensor and the lens in front of it (including focus and zoom). That is usually done with calibration patterns.

In place of a proper calibration, you can assume some defaults.

im = cv.imread("L91xP.jpg")
(height, width) = im.shape[:2]
assert (width, height) == (1280, 720), "or whatever else"

K = np.eye(3)
K[0,0] = K[1,1] = 1000 # 500-5000 is common
K[0:2, 2] = (width-1)/2, (height-1)/2
# array([[1000. ,    0. ,  639.5],
#        [   0. , 1000. ,  359.5],
#        [   0. ,    0. ,    1. ]])

Center is in the center, focal length is some moderate value. A wrong focal length here will only affect the distortion coefficients, and as a factor at that.

Distortion coefficients can be assumed to be all 0. You can tweak them and watch what happens.

dc = np.float32([-0.54,  0.28,  0.  ,  0.  ,  0.  ]) # k1, k2, p1, p2, k3

Undistortion... can be applied to entire pictures or to individual points.

Points:

• either cv.undistortImagePoints(impts, K, dc) (newish API because undistortPoints did more than some people need)
• or cv.perspectiveTransform(cv.undistortPoints(impts, K, dc), K) (perspectiveTransform undoes some of the work of undistortPoints)

Images:

• im_undistorted = cv.undistort(im, K, dc)

Now you have image and points without lens distortion.

modelpts = np.float32([
    [45.,  0.],
    [90.,  0.],
    [90., 60.],
    [45., 60.]]) * 15 # 15 pixels per foot

impts = [
 [511.54881, 184.64497],
 [758.16124, 141.19525],
 [1159.37185, 191.21864],
 [1153.4168, 276.2696]
]

impts_undist = np.float32([
    [ 508.38733,  180.3246 ],
    [ 762.08234,  133.98148],
    [1271.5339 ,  154.91203],
    [1250.6611 ,  260.52057]]).reshape((-1, 1, 2))

Perspective transform requires at least four pairs of points. In each pair, one point is defined in the one perspective (side of the field), and the other point is defined in the other perspective (top-down/"model").

H = cv.getPerspectiveTransform(impts_undist, modelpts)

You can chain some more transformations to that homography (H), like translation/scaling in either image space, to move the picture where you want it. That's just matrix multiplication.

# add some in the X and Y dimension
Tscale = np.array([
    [  1.,   0.,  75.], # arbitrary values
    [  0.,   1.,  25.],
    [  0.,   0.,   1.]])

And then you apply the homography to the undistorted input image:

topdown = cv.warpPerspective(im_undistorted, H, dsize=(90*15, 60*15))

Those are the building blocks. You can then build something interactive using createTrackbar to mess with the distortion coefficients until the output looks straight-ish.

Don't expect that to become perfect. Besides distortion coefficients, the optical center might not really be where it's supposed to. And the picked points on the side view may be off by a pixel or so, but that translates into several feet at such a shallow angle and distance across the field.

It's really best to get a calibration pattern and wave it (well... hold very still!) in front of the camera. I'd recommend "ChArUco" boards. They're the easiest to yield usable results because with those you don't need to keep the entire board in view.

Here are some pictures:

• input as you've given it...

• undistorted:

• top-down view: (to get some more of the surroundings, multiply some translation in front of the homography like H2 = T @ H to move that to the bottom right a little, and give warpPerspective a larger dsize)