[Insight-users] Gradient-based optimization and Jacobian-based partial derivative estimation in 2D/3D scenarios

[Lars Friedrich Lars]

Hello,

the current implementation of InsightApplication 'IntensityBased2D3DRegistration' uses the RegularStepGradientDescentOptimizer and the GradientDifferenceImageToImageMetric. This optimizer directly uses the gradient information of the cost function retrieved via metric->GetDerivative() for cost function minimization. In the case of GradientDifferenceImageToImageMetric a 'derivative-delta' is utilized to generate positions around the current transformation position in order to generate an estimation of the partial derivatives of the transformation parameters.

However, if you replaced the metric type in this example with the MeanSquaresImageToImageMetric, the partial derivatives are estimated in a fundamentally different way. Internally this metric uses the Jacobian of the transformation amongst others. The ITK 2D/3D-registration implementation treats the 3D volume as moving image and the (2D) fixed image as single-sliced 3D fixed image. The transformation is the transformation of the volume and maps the fixed image points 'somewhere'. However, the transformed fixed image points are used to compute the Jacobian and are then used for partial derivative estimation.

Is this kind of derivative estimation valid in situations where one uses ITK's 2D/3D-registration approach together with optimizers that require direct gradient information? There are many ITK metrics that internally use the Jacobian to estimate the gradient (MeanSquaresImageToImageMetric, MutualInformationImageToImageMetric, MattesMutualInformationImageToImageMetric ...). Can these metric types only be used together with optimizer that do not require direct gradient information?

The more I think about this, the more I get confused. Has someone thought about this or does someone have answers to these questions?

Thank you!


best regards,

lars

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