PerformingTemplate-Based Reconstruction - Deformable Surface 3D Reconstruction from Monocular Images - page 41

Figure 4.11: Reconstruction of deformable surfaces made of different materials undergoing complex
deformations. In all four cases, we show the reconstructed 3D mesh overlaid on the input image and
below a side view of the same mesh. © 2011 IEEE.

assigned a penalty proportional to its Mahalanobis distance to the mean shape. To avoid optimizing
the individual patches independently, and therefore having to enforce consistency a posteriori , the
technique exploits the fact that the local mode weights ˜ c can be directly obtained from the vertex

coordinates as

= S T ( ˜ ˜ c x −˜ x 0 ) , (4.13)

S is the matrix of local modes, and

where x is the vector containing the mesh vertices associated

to a single patch. Note that this amounts to marginalizing out the mode weights, as is done in
probabilistic PCA Tipping and Bishop [ 1999 ]. The resulting regularization term for a single patch
can then be expressed as

˜

L 1 / 2

c =

L 1 / 2 S T (

x 0 ) 2 ˜ x ˜ −˜ , (4.14)