Transferring Disparity Values

We have implemented image-based warping as described in [16]. (This implementation is not used in the examples shown in this report, since those examples are planar.) McMillan does not describe how to compute disparity values for the new image. Here is a derivation, following McMillan's notation in Chapter 3.

Let P₁ and P₂ be projection matrices for the original and target camera position, and t be the translation between the camera centers. (P₂ = R P₁, where R is the rotation between cameras.) Let r₁ and r₂ be the distances from a scene point $\dot{X}$ to each camera center. Let x₁ and x₂ be the image coordinates of $\dot{X}$.We therefore have:

$$\dot{X} = \dot{C}_1 + r_1 \frac{P_1 \bar{x}_1}{\vert\vert P_... ... r_2 \frac{P_2 \bar{x}_2}{\vert\vert P_2 \bar{x}_2 \vert\vert}$$

(This is similar to McMillan's Equation 3-2). Substituting the Equation before Equation 3-10 ($\delta(\bar{x})=\frac{\vert\vert P \bar{x} \vert\vert}{r}$), and rearranging, gives:

$$\delta(\bar{x}_2) \left ( \frac{P_1 \bar{x}_1}{\delta(\bar{x}_1)} - t \right ) = P_2 \bar{x}_2$$

Taking the magnitude of both sides gives:

$$\delta(\bar{x}_2) \vert\vert \frac{P_1 \bar{x}_1}{\delta(\bar{x}_1)} - t \vert\vert = \vert\vert P_2 \bar{x}_2 \vert\vert$$

$$\delta(\bar{x}_2) = \frac{\vert\vert P_2 \bar{x}_2\vert\vert... ...rt\vert \frac{P_1 \bar{x}_1}{\delta(\bar{x}_1)} - t\vert\vert}$$

Note that $\vert\vert P_2 \bar{x}_2 \vert\vert = \vert\vert R P_1 \bar{x}_2 \vert\vert = \vert\vert P_1 \bar{x}_2 \vert\vert$, since R is orthonormal. (Rotating a vector doesn't change the magnitude.)

The LDI [22] formulation of image based rendering is much simpler (it is essentially based on depth), and can transfer depth much faster.