python - 为什么用透视相机翻转轴？

Question

我正在尝试使用右手坐标系在 Python 中实现一个简单的透视相机，其中 +x 轴在右，+y 轴在上，+z 轴在屏幕外。

我有一些代码可以将点从 3D 世界坐标投影到 2D 图像坐标。为了测试它，我尝试沿 +X、+Y 和 +Z 轴投影三个单位矢量并渲染它们，但是当我这样做时，所有的点似乎都在“相机后面”，我希望看到这样的东西:

当我取消注释 l = -l 行时，所有出现的轴都翻转了，当我围绕原点旋转相机时，它们看不到在正确的平面上旋转。

这是显示问题的我的代码。我有什么误解吗？

import numpy as np
import cv2

def compute_focal(angle, dimension):
    return dimension / 2.0 / np.tan( np.radians(angle) / .2)

# Positive camera at c looking at p with up=u http://ksimek.github.io/2012/08/22/extrinsic/
def lookat(c, p, u):
    l = p - c
    l = l / np.linalg.norm(l)
    s = np.cross(l, u)
    s = s / np.linalg.norm(s)
    u = np.cross(s, l)

    # uncomment this and the axis will appear by are all flipped
    # l = -l

    R = np.vstack( (s, u, -l))
    Rc = R.T
    return Rc

# project 3D point into camera define by projection matrix
def projectPoint(P, point):
    xw, yw, zw = point
    W = np.array([ [xw, yw, zw, 1] ]).T
    xi, yi, zi = P.dot(W).flatten()

    if zi < 0.0:
        print("point {},{},{} is behind the camera!".format(xi, yi, zi))

    xi = int(xi + 0.5)
    yi = int(yi + 0.5)
    return xi, yi

theta = 0

while True:
    # used to rotate the camera around the y-axis looking at origin
    theta += 1

    w  = h  = 500
    fx = fy = compute_focal(w, 45.)
    cx = w / 2.
    cy = h / 2.

    K = np.array([ [fx, 0., cx], [0., fy, cy], [0., 0., 1.] ], dtype='float32')

    # position of the camera in world coordintes 1-unit from the origin rotating around the y-axis looking at the origin
    C = np.array([ np.sin(np.radians(theta)), 0, np.cos(np.radians(theta)) ])
    # pointing towards the origin
    P = np.array([ 0.0, 0.0, 0.0 ])
    # up direction is along the positive y-axis
    U = np.array([ 0, 1, 0 ])
    Rc = lookat(C, P, U)

    img = np.zeros((h, w, 3), dtype='uint8')

    # create the projection matrix from camera position
    R = Rc.T
    t = R.dot( -np.reshape(C, (3, 1)) )
    P = K.dot(np.hstack([R, t]))

    # draw and project positive principle axes
    x0, y0 = projectPoint(P, (0, 0, 0))
    x1, y1 = projectPoint(P, (1, 0, 0))
    x2, y2 = projectPoint(P, (0, 1, 0))
    x3, y3 = projectPoint(P, (0, 0, 1))

    # x-axis red
    cv2.line(img, (x0, y0), (x1, y1), [0, 0, 255], 1)
    # y-axis green
    cv2.line(img, (x0, y0), (x2, y2), [0, 255, 0], 1)
    # z-axis blue
    cv2.line(img, (x0, y0), (x3, y3), [255, 0, 0], 1)

    # flip image because opencv images have origin in top left
    img = np.flipud(img)

    cv2.imshow("camera", img)
    cv2.waitKey(1)

Answer 1

如果我改为将内在矩阵更改为:

 K = np.array([
            [fx, 0., -cx],
            [0., fy, -cy],
            [0., 0., -1.]
        ], dtype='float32')

这些条目被否定。我不确定为什么，但我在一些示例代码中看到并尝试了它并且它有效，尽管它非常不直观。