[Insight-users] Question about levenberg-marguardt optimization
for the normalized correlation metric
Luis Ibanez
luis.ibanez@kitware.com
Wed May 12 22:17:17 EDT 2004
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Hi Carolyn, Zhaoqiang,
The Levenberg-Marquardt optimization method does not requires
second derivatives. It only needs the Jacobian that relates
changes in the array of parameters to be optimized with respect
to the array of parameters that control the errors.
For example, if we have two sets of points and want to minimize
the sum of square distances between them by applying a rigid
transform to one of the sets, we will be able to relate the
variations in point positions to the variation of the rigid
transform parameters. Such equation willl look like the one
in the attached image : EquationJacobian.gif.
Where the left vector is "E" = the array of errors to be miniimized.
The matrix is the Jacobian = "J", and the array of transform paramters
is "D". The equation in the image represents then:
E = J . D
Solving this problem using the Newton's method will do:
D = ( J^T . J )^-1 . J^T . E
(that looks nicer in the attached image: EquationNewton.gif
Solving the same problem using a Gradient Descent algorithm
will use the equation:
D = 1/lambda . J^T . E
(that looks nicer in the attached image: EquationStepest.gif
Finally, using the Levenberg-Marquardt algorithm one iteration
of the solution will compute:
(J^T . J + lambda . I ) . D = J^T . E
(that looks nicer in the attached image: EquationLevenbergMarquard.gif
As you can see, there is no need for second derivatives. The problem
is solved iteratively by using a linearized form of the non-linear
relationship between the parameters in D and the parameters in E.
Note that both D and E are actually differentials. That is, D is the
array of differentials from the transform parameters, while E is the
array of differentials from the errors.
As far as the software implementation goes, ITK is simply wrapping
the Levenberg-Marquardt algorithm from VXL/VNL, that in its turn
is calling the function lmderl_() from linpack.
-----
NormalizedCorrelation is not well suited to be used along with
the Levenberg-Marquardt method. In fact, no metric that returns
a single value is well suited for this optimizer.
That is, none of the Image metrics deriving from
SingleValueCostFunction:
http://www.itk.org/Insight/Doxygen/html/classitk_1_1SingleValuedCostFunction.html
You should use it only with metric deriving from
MultipleValueCostFunction:
http://www.itk.org/Insight/Doxygen/html/classitk_1_1MultipleValuedCostFunction.html
An intersting application of the Levenberg-Marquardt optimizer
can be found in the context of iterative closest point (ICP),
where the values to be minizimed are an array of distances.
You might want to look at the two ICP examples under:
Insight/Examples/Registration/
IterativeClosestPoint1.cxx
IterativeClosestPoint2.cxx
Both of them use this optimizer.
Regards,
Luis
------------------------
Carolyn Johnston wrote:
> (Forwarded question from a colleague -- C)
>
> When optimizing the mean square error metric with levenberg-marquart
> approach, the second derivative matrix (Hessian matrix) of chisq metric
> function can be approximated to the product of the first derivative of
> chisq metric w.r.t. to parameters, after ignoring the second derivatives.
> Is this approach still valid for normalized correlation metric?
>
>> From our derivation, the second derivative matrix of this metric is quite
>
> different from the above approximation, i.e. the product of two first
> derivatives. We looked through itk source codes and only see the first
> derivative is calculated in the
> itkNormalizedCorrelationImageToImageMetric.txx file, but no second
> derivative calculation is calculated. We didn't find itk source codes
> that generate the Hessen matrix for levenberg-marquardt optimizer. Are
> we missing something here?
>
> Thanks, Zhaoqiang Bi
>
>
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