Shape From Shading Models - Biomedical Image Analysis - page 275

this method discretizes the reflectance map in a different way. Like Pentland's
method, the surface orientation ( p , q ) is approximated by its linear approx-
imation using the forward difference formula (5.36), while unlike Pentland's
method, the reflectance map is then directly linearized in terms of the depth Z
using Taylor series expansion. Finally, Newton's iteration method is applied to
the discretized equation to get a numerical approximation to the depth Z . In
what follows, we will derive this scheme step by step.
To begin with, we rewrite the irradiance equation (5.2) in the following for-
mat:

0 = f = I − R . (5.42)

Replacing p and q by their linear approximation using the forward difference
formulas (5.36), we obtain

0 = f ( I ( x , y ) , Z ( x , y ) , Z ( x − 1 , y ) , Z ( x , y − 1))
= I ( x , y ) − R ( Z ( x , y ) − Z ( x − 1 , y ) , Z ( x , y ) − Z ( x , y − 1)) . (5.43)

If we take the Taylor series expansion about a given depth map Z n − 1 ,weget
the following equation:

0 = f ( I ( x , y ) , Z ( x , y ) , Z ( x − 1 , y ) , Z ( x , y − 1))

≈ f ( I ( x , y ) , Z n − 1 ( x , y ) , Z n − 1 ( x − 1 , y ) , Z n − 1 ( x , y − 1))

( Z ( x , y ) − Z n − 1 ( x , y )) +

× ∂ f ( I ( x , y ) , Z n − 1 ( x , y ) , Z n − 1 ( x − 1 , y ) , Z n − 1 ( x , y − 1))
∂ Z ( x , y )

( Z ( x − 1 , y ) − Z n − 1 ( x − 1 , y ))

+

× ∂ f ( I ( x , y ) , Z n − 1 ( x , y ) , Z n − 1 ( x − 1 , y ) , Z n − 1 ( x , y − 1))
∂ Z ( x − 1 , y )

( Z ( x , y − 1) − Z n − 1 ( x , y − 1)) +

× ∂ f ( I ( x , y ) , Z n − 1 ( x , y ) , Z n − 1 ( x − 1 , y ) , Z n − 1 ( x , y − 1))
∂ Z ( x , y − 1)

(5.44)

.

Given an initial value Z 0 ( x , y ), and using the iterative formula:

Z n ( x , y − 1) = Z n − 1 ( x , y − 1) ,
Z n ( x − 1 , y ) = Z n − 1 ( x − 1 , y ) ,