As in the previous case, we assume that we have the coordinates at a particular point, and recover the surface shape along a strip extending from that point. Provided we know p and q at the given point, then as we extend our solution from to the changes in p and q can be computed by

where i.e. , and .

Figure 7: Differential surface geometry and the characteristic strip

This can be written more compactly as

The Hessian provides information on surface curvature; the trace of the Hessian is the Laplacian of the depth, which is twice the mean curvature (discussed elsewhere).

Next, we recall the image irradiance equation 4. Differentiate each side of with respect to x using the chain rule,

Similarly, differentiate with respect to y,

or, using the Hessian matrix,

Now, we have a relationship between the image gradient and the gradient in the reflectance map . Equations 35 and 36 have three unknowns and . In general this cannot be solved. However, we can continue the solution in a specific direction, as we did in the linear case. Let

From Equations 34 and 38.

where is a small quantity. Examining Figure 8, as we extend the strip, so we move to a new point in and spaces. The change in can be computed provided the step taken in image space, , shown by the dotted arrow is parallel to the direction of steepest descent in gradient space, shown by the solid arrow. In turn, the direction of change in gradient space (dotted) arrow is parallel to the image gradient (solid arrow).

As in the linear case, we require starting values for the strips; these may be occluding contours, or more strictly parts of the surface adjacent to the occluding contours, or alternatively the ``singular points'' which correspond to maxima in the reflectance map.

 
Start the solution at a known point 
REPEAT
	Take a small step  along a characteristic strip  from 
		to . This is parallel to the direction of 
		steepest ascent or descent in the reflectance map 
		( the dotted arrow in  	space)
	 Compute changes in p and q from the changes in 
	 Define new values for p and q
	 Compute new values for R,  and 
	 Compute the new value for z
	 Reset start point to new value of 
UNTIL ( a discontinuity is reached in the image function)

As an illustration of the method, Figure 9 shows needle diagrams of simple spheres, computed from the shading pattern and the extremal contour which surrounds the sphere's surface. In fact this result was obtained by a minimisation method, but it serevs as an illustration. The difference between the two pictures is in the raytracer; the one on the left uses a more crudely factetted approximation and the needle map is further for a true sphere. Another illustration is given in Figure 10 from [Horn, 1986]. This uses the tip of the nose, or more accurately a quasi-circular strip surrounding the nose as the set of starting points for the growth of the characteristic strips. The result is a relief map of part of the face.