Question: I want to check if an object (defined by four corners in 3D space) is inside the Field of View of a camera pose. I saw this solution and tried to implement it, but I missed something, can you please tell me how to fix it?  the provided 4 points are 2 inside, 2 outside camera frustum.  import numpy as np from typing import Tuple class CameraFrustum: def __init__( self, d_dist: float = 0.3, fov: Tuple[float, float] = (50, 40) ): self.d_dist = d_dist self.fov = fov self.frustum_vectors = None self.n_sight = None self.u_hvec = None self.v_vvec = None def compute_frustum_vectors(self, cam_pose: np.ndarray): fov_horizontal, fov_vertical = np.radians(self.fov[0] / 2), np.radians( self.fov[1] / 2 ) self.cam_position = cam_pose[:3, 3] cam_orientation = cam_pose[:3, :3] base_vectors = np.array( [ [np.tan(fov_horizontal), np.tan(fov_vertical), 1], [-np.tan(fov_horizontal), np.tan(fov_vertical), 1], [-np.tan(fov_horizontal), -np.tan(fov_vertical), 1], [np.tan(fov_horizontal), -np.tan(fov_vertical), 1], ] ) base_vectors /= np.linalg.norm(base_vectors, axis=1, keepdims=True) self.frustum_vectors = np.dot(base_vectors, cam_orientation.T) self.n_sight = np.mean(self.frustum_vectors, axis=0) self.u_hvec = np.cross( np.mean(self.frustum_vectors[:2], axis=0), self.n_sight ) self.v_vvec = np.cross( np.mean(self.frustum_vectors[1:3], axis=0), self.n_sight ) def project_point( self, p_point: np.ndarray, cam_orientation: np.ndarray ) -> bool: if self.frustum_vectors is None: self.compute_frustum_vectors(cam_orientation) # p_point_vec = p_point - self.cam_position p_point_vec /= np.linalg.norm(p_point_vec, axis=-1, keepdims=True) # d_prime = np.dot(p_point_vec, self.n_sight) if abs(d_prime) < 1e-6: print("point is not in front of the camera") return False elif d_prime < self.d_dist: print("point is too close to camera") return False # p_prime_vec = self.d_dist *( p_point_vec / d_prime ) - self.d_dist * self.n_sight u_prime = np.dot(p_prime_vec, self.u_hvec) v_prime = np.dot(p_prime_vec, self.v_vvec) # width = 2 * self.d_dist * np.tan(np.radians(self.fov[0]) / 2) height = 2 * self.d_dist * np.tan(np.radians(self.fov[1]) / 2) u_min, u_max = -width / 2, width / 2 v_min, v_max = -height / 2, height / 2 if not (u_min < u_prime < u_max): return False if not (v_min < v_prime < v_max): return False return True cam_frustum = CameraFrustum() pts = np.array( [ [1.54320189, -0.35068437, -0.48266792], [1.52144436, 0.44898697, -0.48990338], [0.32197813, 0.41622155, -0.50429738], [0.34373566, -0.38344979, -0.49706192], ] ) cam_pose = np.array( [ [-0.02719692, 0.9447125, -0.3271947, 1.25978471], [0.99958918, 0.02274412, 0.0, 0.03276859], [-0.00904433, -0.32711006, -0.94495695, 0.4514743], [0.0, 0.0, 0.0, 1.0], ] ) for pt in pts: res = cam_frustum.project_point(pt, cam_pose) print(res)   Can you please tell me how can I fix this? thanks. I tried to implement this as follows

Answer: EDIT: pending a response from the OP. There is a problem with your cam_pose matrix. The [0:3,0:3] components (first three rows and first three columns) should be a rotation matrix. However, it isn't: the first and third columns aren't orthogonal. Well, no matter how I try to do it, I think all those points lie outside the frustum (really a pyramid). Could you check that those are the particular points that you intended (because the picture in your post suggests that you also tried other camera locations). It would be really good if somebody else looked at this independently. Maybe there's some distinction between the camera position and the focal point that I don't know about. (I'm currently making no distinction.) I tried two things - Method 1: fix your code; Method 2: reverse the transformation and calculate the angles to compare with FOV/2. I also cobbled together something to plot the points. Method 1: Try to fix your code. You calculate a local basis triad, n_sight, u_hvec and v_vvec. These need to be normalised, because you use projections onto them to calculate the relevant coordinate lengths d_prime, u_prime and v_prime. Your method of finding self.h_uvec and self.v_vvec is unnecessarily complicated and hard to envisage (even though correct). I've tried to indicate where these are relative to the local frustrum points below.  You don't need self.dist in find u_prime and v_prime (despite what your link says) because you have already got a cross-wise plane at distance d_prime (or you would have if n_sight was normalised to unit length). Your projection plane is at distance n.p from the camera and the vector p_prime_vec is the components of the original displacement perpendicular to n_sight. You shouldn't normalise p_point_vec (or you will get distance d_prime wrong).  I've left some debugging print statements in the code below: if you know the order of points (I eventually worked them out) they may help you. import numpy as np import math from typing import Tuple import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D class CameraFrustum: def __init__( self, d_dist: float = 0.3, fov: Tuple[float, float] = (50, 40) ): # original self.d_dist = d_dist self.fov = fov self.frustum_vectors = None self.n_sight = None self.u_hvec = None self.v_vvec = None def compute_frustum_vectors(self, cam_pose: np.ndarray): fov_horizontal, fov_vertical = np.radians(self.fov[0] / 2), np.radians( self.fov[1] / 2 ) self.cam_position = cam_pose[:3, 3] cam_orientation = cam_pose[:3, :3] base_vectors = np.array( [ [np.tan(fov_horizontal), np.tan(fov_vertical), 1], [-np.tan(fov_horizontal), np.tan(fov_vertical), 1], [-np.tan(fov_horizontal), -np.tan(fov_vertical), 1], [np.tan(fov_horizontal), -np.tan(fov_vertical), 1], ] ) base_vectors /= np.linalg.norm(base_vectors, axis=1, keepdims=True) self.frustum_vectors = np.dot(base_vectors, cam_orientation.T) self.n_sight = np.mean(self.frustum_vectors, axis=0) self.u_hvec = self.frustum_vectors[0] - self.frustum_vectors[1] ##### much easier self.v_vvec = self.frustum_vectors[0] - self.frustum_vectors[3] self.n_sight /= np.linalg.norm( self.n_sight ) ##### normalise basis vectors to unit length self.u_hvec /= np.linalg.norm( self.u_hvec ) self.v_vvec /= np.linalg.norm( self.v_vvec ) print( 'n_sight = ', self.n_sight ) # check unit-vector directions print( 'u_hvec = ', self.u_hvec ) print( 'v_vvec = ', self.v_vvec ) def project_point( self, p_point: np.ndarray, cam_orientation: np.ndarray ) -> bool: if self.frustum_vectors is None: self.compute_frustum_vectors(cam_orientation) p_point_vec = p_point - self.cam_position d_prime = np.dot(p_point_vec, self.n_sight) # p.n = plane distance from the camera if abs(d_prime) < 1e-6: print("Point is not in front of the camera") return False elif d_prime < self.d_dist: print("Point is too close to the camera") return False p_prime_vec = p_point_vec - self.n_sight * d_prime # p - (p.n)n = displacement from centreline u_prime = np.dot(p_prime_vec, self.u_hvec) v_prime = np.dot(p_prime_vec, self.v_vvec) u_max = d_prime * np.tan(np.radians(self.fov[0]) / 2); u_min = -u_max v_max = d_prime * np.tan(np.radians(self.fov[1]) / 2); v_min = -v_max print( "u_prime, v_prime, u_max, v_max=", u_prime, v_prime, u_max, v_max ) # check rotated coordinates if not (u_min < u_prime < u_max): return False if not (v_min < v_prime < v_max): return False return True cam_frustum = CameraFrustum() pts = np.array( [ [1.54320189, -0.35068437, -0.48266792], [1.52144436, 0.44898697, -0.48990338], [0.32197813, 0.41622155, -0.50429738], [0.34373566, -0.38344979, -0.49706192] ] ) cam_pose = np.array( [ [-0.02719692, 0.9447125, -0.3271947, 1.25978471], [ 0.99958918, 0.02274412, 0.0, 0.03276859], [-0.00904433,-0.32711006,-0.94495695, 0.4514743 ], [0.0, 0.0, 0.0, 1.0], ] ) for pt in pts: res = cam_frustum.project_point(pt, cam_pose) print(res)  Output n_sight = [-0.3271947 0. -0.94495695] u_hvec = [-0.02719692 0.99958918 -0.00904433] v_vvec = [ 0.9447125 0.02274412 -0.32711006] u_prime, v_prime, u_max, v_max= -0.3963363671027199 0.5645937705948938 0.3683791237780479 0.2875334205549486 False u_prime, v_prime, u_max, v_max= 0.40342016151710725 0.5645937726848527 0.37488698183988656 0.29261304252107834 False u_prime, v_prime, u_max, v_max= 0.3963363671027199 -0.5645937705948936 0.5642361967573459 0.44040705127553315 False u_prime, v_prime, u_max, v_max= -0.4034201615171073 -0.5645937726848521 0.5577283386955073 0.4353274293094035 False  Method 2 - Invert the translation and rotation You can simply reverse the transformation and calculate the relevant angles, confirming if abs(angle) < FOV/2 in the relevant direction. import numpy as np import math pts = np.array( [ [1.54320189, -0.35068437, -0.48266792], [1.52144436, 0.44898697, -0.48990338], [0.32197813, 0.41622155, -0.50429738], [0.34373566, -0.38344979, -0.49706192] ] ) cam_pose = np.array( [ [-0.02719692, 0.9447125, -0.3271947, 1.25978471], [ 0.99958918, 0.02274412, 0.0, 0.03276859], [-0.00904433,-0.32711006,-0.94495695, 0.4514743 ], [0.0, 0.0, 0.0, 1.0], ] ) fov = ( 50, 40 ) origin = cam_pose[0:3,3] rotate = cam_pose[0:3,0:3] reverse = np.linalg.inv( rotate ) print( 'Angles' ) for p in pts: q = np.dot( p - origin, reverse.T ) angle1 = math.atan2( q[0], q[2] ) * 180 / np.pi angle2 = math.atan2( q[1], q[2] ) * 180 / np.pi print( f'{angle1:7.2f} {angle2:7.2f} ', abs( angle1 ) < fov[0] / 2 and abs( angle2 ) < fov[1] / 2 )  Output: Angles -26.45 35.32 False 26.86 35.32 False 18.24 -25.14 False -18.54 -25.14 False  Plotting If anyone wants to try it you can have a go with what I cobbled together to plot it. (You need to rotate it by hand in 3d). I haven't done much more than apply the same rotation and camera (or, at least, focal point) as you have. import numpy as np import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D pts = np.array( [ [1.54320189, -0.35068437, -0.48266792], [1.52144436, 0.44898697, -0.48990338], [0.32197813, 0.41622155, -0.50429738], [0.34373566, -0.38344979, -0.49706192] ] ) cam_pose = np.array( [ [-0.02719692, 0.9447125, -0.3271947, 1.25978471], [ 0.99958918, 0.02274412, 0.0, 0.03276859], [-0.00904433,-0.32711006,-0.94495695, 0.4514743 ], [0.0, 0.0, 0.0, 1.0], ] ) fov = ( 50, 40 ) origin = cam_pose[0:3,3] rotate = cam_pose[0:3,0:3] fov_horizontal, fov_vertical = np.radians(fov[0]/2), np.radians(fov[1]/2) base_vectors = np.array( [ [np.tan(fov_horizontal), np.tan(fov_vertical), 1], [-np.tan(fov_horizontal), np.tan(fov_vertical), 1], [-np.tan(fov_horizontal), -np.tan(fov_vertical), 1], [np.tan(fov_horizontal), -np.tan(fov_vertical), 1], ] ) base_vectors /= np.linalg.norm(base_vectors, axis=1, keepdims=True) frustum_vectors = np.dot(base_vectors, rotate.T) pp = frustum_vectors + origin fig = plt.figure(figsize=(10, 8)) ax = fig.add_subplot(111, projection='3d') for xyz in pts: ax.scatter( xyz[0], xyz[1], xyz[2], color='r' ) ax.scatter( origin[0], origin[1], origin[2], color='b' ) for p in pp: ax.plot( [origin[0],p[0]], [origin[1],p[1]], [origin[2],p[2]], color='g' ) ax.set_xlabel('X') ax.set_ylabel('Y') ax.set_zlabel('Z') plt.show()  Seen from one direction parallel to one face of the frustum - two points are outside this face.  Looking parallel to the opposite face - the other two points are outside.