[Insight-users] Computing the gradient in the Metric?

[motes motes]

Thanks for the hint I have read those pages but I still have two
questions to the below expression:

          sum += 2.0 * diff * jacobian( dim, par ) * gradient[dim];

1) In the above expression gradient[dim] corresponds to the gradient
for the current pixel in the moving image? On page 249 in the "Insight
Into Images" this is the first term in (10.6). But why are the
gradient of the moving image used to compute the derivative of the
metric? I know this is a very basic question but I still cannot see
the connection.

2) When I debug the code the jacobian actually just contains the
weights (computed using the transform kernel) for each node in the
support region. But are the jacobian not supposed to contain the first
order partial derivatives? It seems a bit confusing that the jacobian
is used as a container for the kernel transform weights.





On Sun, Sep 6, 2009 at 1:01 PM, Neuner Markus<neuner.markus at gmx.net> wrote:
> Hi,
>
> To understand why the jacobian is used i would suggest to read pages 249-251
> in the Book "Insight Into Images" by Terry S. Yoo. Ther eis described how
> and why the jacobian is used.
>
> Greets
>
> motes motes wrote:
>>
>> In itkMeanSquaresImageToImageMetric.txx the metric value and the
>> gradient is computed. I pretty much understand how the metric value is
>> computed but am a bif confused on how the gradient is computed:
>>
>>
>>
>>
>>      const RealType movingValue  = this->m_Interpolator->Evaluate(
>> transformedPoint );
>>      const TransformJacobianType & jacobian =
>> this->m_Transform->GetJacobian( inputPoint );
>>      const RealType fixedValue     = ti.Value();
>>      this->m_NumberOfPixelsCounted++;
>>      const RealType diff = movingValue - fixedValue;
>>      measure += diff * diff;
>>
>>      for(unsigned int par=0; par<ParametersDimension; par++)
>>        {
>>        RealType sum = NumericTraits< RealType >::Zero;
>>        for(unsigned int dim=0; dim<ImageDimension; dim++)
>>          {
>>          sum += 2.0 * diff * jacobian( dim, par ) * gradient[dim];
>>          }
>>        derivative[par] += sum;
>>        }
>>      }
>>
>> It basically comes down to this line:
>>
>>
>>
>>          sum += 2.0 * diff * jacobian( dim, par ) * gradient[dim];
>>
>> why is the jacobian multiplied with the current gradient?
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