different ways to calculate the essential matrix

essential_mat.py

p1 = np.random.random((100, 2)) # feature points in camera 1
p2 = np.random.random((100, 2)) # feature points in camera 1

# camera matrices
k1 = np.random.random((3, 3))
k2 = np.random.random((3, 3))


# Method 1: calculate essential matrix (E) from the fundamental matrix (F)

F, _ = cv2.findFundamentalMat(p1, p2, cv2.FM_LMEDS)
E = k2.T @ F @ k1  # -> the singular value of E is typically not (sigma, sigma, 0)


# Method 2: calculate E directly from normalised feature points

p1h = np.concatenate((p1.T, np.ones(n).reshape((1, n)))) # homogeneous representation
p2h = np.concatenate((p2.T, np.ones(n).reshape((1, n))))
p1hn = np.linalg.inv(c1.k) @ p1h  # normalisation
p2hn = np.linalg.inv(c2.k) @ p2h
E, _ = cv2.findFundamentalMat(p1hn.T, p2hn.T)

# Method 3: use opencv's 5-point algorithm

E, _ = cv2.findEssentialMat(
    p1, p2,
    cameraMatrix1=c1.k,
    distCoeffs1=np.zeros(5),
    cameraMatrix2=c2.k,
    distCoeffs2=np.zeros(5)
)  # -> the singular value of E is (sigma, sigma, 0)