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Created Aug 29, 2019
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import cv2
import numpy as np
import math
cx = 88 #principal point x coord
cy = 109 #principal point y coord
w = 178 #image width
h = 218 #image height
near = 10 #near plane
far = 20 #far plane
fovy = 45.0/360.0*2.0*np.pi #45° in radians
f = h / 2*math.tan(fovy/2) #cf (NOTE: focal length is in pixels)
#we compute the OpenCV camera matrix
camera_mtx = np.array([
[f, 0, cx],
[0., f, cy],
], dtype=np.float64)
#we compute the corresponding opengl projection matrix
#NOTE: K00 = K11 = f, K10 = 0.0, K02 = cx, K12 = cy, K22 = 1.0
opengl_mtx = np.array([
[2*f/w, 0.0, (w - 2*cx)/w, 0.0],
[0.0, -2*f/h, (h - 2*cy)/h, 0.0],
[0.0, 0.0, (-far - near) / (far - near), -2.0*far*near/(far-near)],
[0.0, 0.0, -1.0, 0.0]
#point is in opencv camera space (along Oz axis)
point = np.array([1.0, 2.0, 15.0]) #Note: coords must be floats
#### OpenCV projection
screen_point, _ = cv2.projectPoints(np.array([point]), np.zeros(3), np.zeros(3), camera_mtx, np.zeros(5))
#Note: we obtain the same result with this: (that's what cv2.projectPoints basically does: multiply points with camera matrix and then divide result by z coord)
#### OpenGL projection
#we flip the point z coord, because in opengl camera is oriented along -Oz axis
point[2] = -point[2]
point2 = np.hstack([point,1.0]) #we add vertex w coord (usually done in vertex shader before multiplying by projection matrix)
#we get the point in clip space
clip_point =
#NOTE: what follows "simulates" what happens in OpenGL after the vertex shader.
#This is necessary so that we can make sure our projection matrix will yield the correct result when used in OpenGL
#we get the point in NDC
ndc_point = clip_point / clip_point[3]
#we get the screen coordinates
viewport_point = (ndc_point + 1.0)/2.0 * np.array([w, h, 1.0, 1.0])
#opencv Oy convention is opposite of OpenGL so we reverse y coord
viewport_point[1] = h - viewport_point[1]
#Now you can see that viewport_point and screen_point have the same x/y coordinates!
#This means you can now, from OpenCv camera matrix, use OpenGl to render stuff on top of the image,
#thanks to the opengl projection matrix, computed from opencv camera matrix
#NOTE: when near plane is small (a few units) and when focal length is small (ex: 10-12),
#both results tend to diverge. I'm not sure why the formula starts falling apart at extreme values.
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