Shape From Shading Models - Biomedical Image Analysis - page 280

which satisfies the irradiance equation (5.2):

R ( p , q ) = I ( x , y ) ,

where p = ∂ Z /∂ x and q = ∂ Z /∂ y , Z ( x , y ) is the height of image at ( x , y ).
Notice that, for each pixel, the right side of Eq. (5.2) is given values and in
the left side p and q are free variables. Therefore, we write p = p ( x , y ) and
q = q ( x , y ). Now we rewrite the energy equation (5.49) as

F ( p , q , Z ) dx dy ,

Energy = (5.50)

where F ( p , q , Z ) is the sum of the following three parts:

( I − R ) 2 = ( R ( p , q ) − I ( x , y )) 2 (5.51) ,

( R x − I x ) 2

+ ( R y − I y ) 2 = ( R p ( p , q ) p x + R q ( p , q ) q x − I x ( x , y )) 2

(5.52)

+ ( R p ( p , q ) p y + R q ( p , q ) q y − I y ( x , y )) 2

,

µ (( Z x − p ) 2 + ( Z y − q ) 2 ) . (5.53)

Using the technique of calculus of variations in Section 5.2.3 to mini-
mize the energy function (5.50) is equivalent to solving the following Euler
equation:

F p − ∂

∂ x ∂ F ∂ p x − ∂ ∂ y ∂ F ∂ p y = 0 ,

(5.54)

∂ x ∂ F ∂ y ∂ F

F q − ∂

∂ q x − ∂

∂ q y = 0 ,

∂ x ∂ F ∂ y ∂ F

F Z − ∂

∂ Z x − ∂ ∂ Z y = 0 .

By taking the first-order terms in the Taylor series of the reflectance map,
Zheng-Chellappa [70] simplified the Euler equation. For example, F p can be
approximated by the following equation:

F p ≈ 2[ R − I ( x , y )] R p + µ ( p − Z x ) . (5.55)

From Eq. (5.55), we observe that the higher order derivatives,
R pp , R pq , R qp , and R qq , are omitted because we only take the first-order Tay-
lor expansion. Similarly, we can get F q and F Z and all the other variables in Eq.
(5.54). Finally, we get the following iterative formula (the current values of p ,