Computer Vision Basics
How does a 3d object gets transformed into 2d image?
Before we go explain this in detail let's clarify the terms
The object (flower) is said to be in the world and its position is given by world coordinates
Now imagine The light rays from the object pass through a pinhole and get projected on a wall. The wall is the IMAGE plane and the hole it passes through is called the PINHOLE. The camera is placed at the pinhole. The camera is said to be in the CAMERA plane
The distance between the camera placed at the pinhole and the wall (image plane) is called FOCAL LENGTH
So what is the point of all this definition?
Our goal is to take the 3D coordinates (x,y,z) and first TRANSFORM them to the camera plane (3D) and PROJECT them to the image plane (2D).
In other words, we move from WORLD => CAMERA => IMAGE. This is called Forward Imaging Model. The projection from the Camera to the image plane is called Perspective Projection. We can write the forward imaging model as
\( [x_w, y_w, z_w] => [x_c, y_c, z_c] => [x_i, y_i] \)
Perspective Imaging with Pinhole
The above diagram is the same as the Pinhole diagram with labels removed for explaining the geometry. The yellow and blue triangles are similar. By the property of similar triangles (see below note), we have
\( \frac{r_i}{f} = \frac{r_o}{z} \)
Where \(r\_o = (x\_o, y\_o, z\_o )\) and \( r\_i = (x\_i, y\_i, f)\)
where \(r\_i\) and \( r\_o \) are vectors describing the point on the image and world respectively. Hence we can split and write the equation as
$$x_i/f = x_o/z$$
$$y_i/f = y_o/z$$
This is the standard pinhole camera equation. The equation has a few interesting properties.
Pinhole Properties
The straight line remains straight in the image plane
Magnification is inversely proportional to depth z_o . In other words, The further you are from the camera, the smaller you appear. Now this is obvious since we all use cameras today. But take a look at the parallel lines intersecting for a railway track. The distance between the parallel tracks becomes so small in the image plane that it looks like an intersection though, in reality, we know it's parallel throughout. The point of intersection is called the Vanishing Point
Since a pinhole is a tiny hole letting the light inside the camera capture the images, for a large object it will take a few seconds before the image is captured. Imagine clicking a camera and waiting 5 seconds to see the image!! To improve this, we substitute pinhole with LENS.
Pinhole vs Lens
We won't cover LENS and its properties, but just keep in mind, LENS is just a "pinhole" made larger. That's a simplistic view of it. Obviously with more area to capture light, now the wait time to capture the image is reduced, but the light is scattered now, and hence causes distortions in your image. Apart from the focal length we introduce distortion coefficients as one of the Intrinsic Parameters. All modern cameras use sophisticated lenses.
See below
