Tomasi-Kanade Factorization

Figure: Orthographic Projection

An alternative way to simplify the SfM problem is to consider a different projection model. The perspective projection case is characteristic of real cameras however, the corresponding equations are difficult to deal with. The orthographic case in Figure 6 greatly simplifies projection into the almost trivial form u=X and v=Y.

One orthographic approach which has gained popularity is the factorization method proposed by Tomasi and Kanade [55]. Once again, the result is a linear formulation however the linearity is fundamentally different from the one induced in the previous epipolar geometry approaches. The technique begins with P tracked feature points over F frames and these are all combined into a matrix W of size 2F x P. For each frame (or row of W), the P feature points are registered by subtracting off their mean (recovering and factoring out the 2D translation). The resulting 2F x P matrix $\hat{W}$ is then described as the product $\hat{W}=RS$ where R is a 2F x 3 matrix and S is 3 x P. These matrices are obtained from $\hat{W}$ via a singular value decomposition and some direct linear operations.

The algorithm is robust in many situations however it is tuned for orthographic projection, not for perspective effects. Degeneracies may occur when the camera translates forward and this forward motion parameter is not recovered by the system. Only two image-plane translations, camera yaw, roll and pitch are estimated. Therefore, it may not be applicable in some situations. The factorization method has subsequently been extended by Poelman and Kanade to the paraperspective case which is a closer approximation to perspective projection than orthographic projection [44].