STEREO IMAGE ACQUISITION

In order to address the loss of depth information, given 2 images of a 3D scene and correspondence in points the depth of a point can be recovered using triangulation.

so given the following model:

The relation between the 2 images is only an horizontal translation

$$x_l-x_r=b$$ $$y_l=y_r=y$$ $$z_l=z_r=z$$

The perspective projection model the relation between image points and 3D ones can be described as follows:

$$u_L=x_L\ast \frac{f}{z}$$ $$u_R=x_R\ast \frac{f}{z}$$

computing the difference:

$$u_L–u_R=b\ast \frac{f}{z}$$

It’s true to say that $u_L–u_R$ can be computed from image measurements

$$u_L–u_R=d$$

where $d$ is called disparity. than $z$ can be computed

$$d=b\ast \frac{f}{z}\Rightarrow z=b\ast \frac{f}{d}$$

THE PROBLEM OF FINDING CORRESPONDENCES

In order to apply this model the correspondences between the points of the 2 images need to be computed, this is not an easy task nether computationally or conceptually, also the models relays on the 2 images being horizontally aligned, and to obtained this some computation is needed.

fortunately, the search space for correspondences is always 1 dimensional as 2 points share always the same $y$ coordinate. To address the horizontal alignment problem a homography called rectification is used on the images to warp them and obtain something horizontally aligned.

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