title: The coordinate system of pinhole camera model
body: <p>Recently, I have been studying the pinhole camera model, but I was confused with the model provided by OpenCV and the "<em>Multiple View geometry in computer vision</em>"  textbook. </p>
  
  <p>I know that the following photo is a simplified model which switch the position of the image plane and the camera frame.  For better illustration and understanding, and taking consideration of the principal point (u0,v0), the relation between two frames is <code>x=f(X/Z)+u0</code> and <code>y=f(Y/Z)+v0</code>.</p>
  
  <p><img src="https://i.stack.imgur.com/WpXO3.png" alt="enter image description here"></p>
  
  <p>However,I was really confused because normally the image coordinate is in the form of the 4th quadrant coordinate as the following one!</p>
  
  <p>Could I directly substitute the (x,y) in the following definition to the above "equivalent" pinhole model which is not really persuasive? </p>
  
  <p><img src="https://i.stack.imgur.com/3kNtS.gif" alt="enter image description here"> </p>
  
  <p>Besides, If an object is in the region (+X,+Y) quadrant in the camera coordinate (of course, Z>f), in the equivalent model, it should appear on the right-half plane of the image coordinate. However, such object in the image taken by a normal camera, it is supposed to be located on the left-half. Therefore, for me this model is not reasonable.</p>
  
  <p>Finally, I tried to derive based on the original model as the following one.</p>
  
  <p><img src="https://i.stack.imgur.com/jlNPf.png" alt="enter image description here"></p>
  
  <p>The result is <code>x1=-f(X/Z)</code> and <code>y1=-f(Y/Z)</code>.</p>
  
  <p>Then, I tried to find the relation between (x2,y2)-coordinate and the camera coordinate.
  The result is <code>x2=-f(X/Z)+u0</code> and <code>y2=-f(Y/Z)+v0</code>.</p>
  
  <p>Between (x3,y3)-coordinate and the camera coordinate, the result is <code>x3=-f(X/Z)+u0</code> and <code>y3=f(Y/Z)+v0</code>.</p>
  
  <p>No matter which coordinate system I tried, none of them is in the form of <code>x=f(X/Z)+u0</code> and <code>y=f(Y/Z)+v0</code>, which are provided by some CV textbooks.</p>
  
  <p>Besides, the projection results on (x2,y2)-coordinate or (x3,y3)-coordinate are also not reasonable because of the same reason: an object in the (+X,+Y,+Z) region in the camera coordinate should "appear" on the left-half plane of the image taken by a camera.</p>
  
  <p>Could anyone indicate what I misunderstood?</p>