Hello Simon and Cyril,
Thanks for the reply.
You are right Simon. I did not notice it too in the literature. The
main problem as you said is the storage. Actually I developed the
conjugate gradient (CG), quasi-Newton and Newton optimization methods
for optical tomography and I intended to apply them to the CT
reconstruction as well. I implemented the Newton's methods
(Gauss-Newton and Levenberg-Marquardt) in a
Jacobian-Free-Newton-Krylov approaches to avoid the matrix
multiplication of Jacobians (sensitivity). It means we only need to
store the Jacobian matrix for the these methods (the matrix R that
Cyril was mentioned), that is still a big matrix for practical
problems in CT reconstruction. For the quasi-Newton I adapted an
L-BFGS algorithm that only need the 3 or 8 last iterations of the
gradient vector to calculate the Hessian matrix. In my case, the
L-BFGS and Newton's methods was much faster than the CG as you know
because of using the second order derivative (hessian matrix). I saw
in your last paper you implement the conjugate gradient method, so I
thought it might be easy to extract the gradient vector from CG
modules and solve the cost function within the quasi-Newton/Newton
methods. I will look at the codes to see what I can do.
Thanks again for the reply.
@Cyril:
Please correct me if I am wrong. you mean the output of
backProjectionFilter is the gradient of defined cost function?
Regards,
Vahid
On Wednesday, November 2, 2016 2:53 AM, Cyril Mory
<[email protected]>
<mailto:[email protected]> wrote:
Hi Vahid,
Welcome to RTK :)
Indeed, there are several iterative methods already implemented in
RTK, but none of the filters allows you to easily extract the
gradient of the least squares function there are minimizing.
If you need to minimize the classical non-regularized tomographic
cost function, ie || R f - p ||², with R the forward projection
operator, f the volume you are looking for, and p the measured
projections, my best advice would be to copy some part of the
pipeline of rtkSARTConeBeamReconstructionFilter to get the job done,
ie the following part (copy-paste this into webgraphviz.com)
digraph SARTConeBeamReconstructionFilter {
Input0 [ label="Input 0 (Volume)"];
Input0 [shape=Mdiamond];
Input1 [label="Input 1 (Projections)"];
Input1 [shape=Mdiamond];
node [shape=box];
ForwardProject [ label="rtk::ForwardProjectionImageFilter" URL="\ref
rtk::ForwardProjectionImageFilter"];
Extract [ label="itk::ExtractImageFilter" URL="\ref
itk::ExtractImageFilter"];
MultiplyByZero [ label="itk::MultiplyImageFilter (by zero)" URL="\ref
itk::MultiplyImageFilter"];
AfterExtract [label="", fixedsize="false", width=0, height=0,
shape=none];
Subtract [ label="itk::SubtractImageFilter" URL="\ref
itk::SubtractImageFilter"];
MultiplyByLambda [ label="itk::MultiplyImageFilter (by lambda)"
URL="\ref itk::MultiplyImageFilter"];
Divide [ label="itk::DivideOrZeroOutImageFilter" URL="\ref
itk::DivideOrZeroOutImageFilter"];
GatingWeight [ label="itk::MultiplyImageFilter (by gating weight)"
URL="\ref itk::MultiplyImageFilter", style=dashed];
Displaced [ label="rtk::DisplacedDetectorImageFilter" URL="\ref
rtk::DisplacedDetectorImageFilter"];
ConstantProjectionStack [ label="rtk::ConstantImageSource" URL="\ref
rtk::ConstantImageSource"];
ExtractConstantProjection [ label="itk::ExtractImageFilter" URL="\ref
itk::ExtractImageFilter"];
RayBox [ label="rtk::RayBoxIntersectionImageFilter" URL="\ref
rtk::RayBoxIntersectionImageFilter"];
ConstantVolume [ label="rtk::ConstantImageSource" URL="\ref
rtk::ConstantImageSource"];
BackProjection [ label="rtk::BackProjectionImageFilter" URL="\ref
rtk::BackProjectionImageFilter"];
OutofInput0 [label="", fixedsize="false", width=0, height=0, shape=none];
OutofBP [label="", fixedsize="false", width=0, height=0, shape=none];
BeforeBP [label="", fixedsize="false", width=0, height=0, shape=none];
BeforeAdd [label="", fixedsize="false", width=0, height=0, shape=none];
Input0 -> OutofInput0 [arrowhead=none];
OutofInput0 -> ForwardProject;
ConstantVolume -> BeforeBP [arrowhead=none];
BeforeBP -> BackProjection;
Extract -> AfterExtract[arrowhead=none];
AfterExtract -> MultiplyByZero;
AfterExtract -> Subtract;
MultiplyByZero -> ForwardProject;
Input1 -> Extract;
ForwardProject -> Subtract;
Subtract -> MultiplyByLambda;
MultiplyByLambda -> Divide;
Divide -> GatingWeight;
GatingWeight -> Displaced;
ConstantProjectionStack -> ExtractConstantProjection;
ExtractConstantProjection -> RayBox;
RayBox -> Divide;
Displaced -> BackProjection;
BackProjection -> OutofBP [arrowhead=none];
}
As you can see, it is a very large part of the SART reconstruction
filter, so yoiu might be better off just copying the whole
SARTConeBeamReconstructionFilter and modifying it.
Of course, you could also look into ITK's cost function class, and
see if one of the classes inherited from it suits your needs,
implement your cost function this way, and use ITK's off-the-shelf
solvers to minimize it. See the inheritance diagram in
https://itk.org/Doxygen/html/classitk_1_1CostFunctionTemplate.html if
you want to try this approach.
Best regards,
Cyril
On 11/01/2016 05:50 PM, vahid ettehadi via Rtk-users wrote:
Hello RTK users and developers,
I already implemented the RTK and reconstructed some images with the
FDK algorithm implemented in RTK. It works well. Thanks to RTK
developers.
Now, I am trying to develop a model-based image reconstruction for
our cone-beam micro-CT. I see already that some iterative algorithms
like ART and its modifications and conjugate-gradient (CG) method
are implemented in the RTK. I want to develop a model-based
reconstruction through the Newton/quasi-Newton optimizations
methods. I was wondering is it possible to extract the gradient of
least square function from implemented algorithms like CG module?
Any recommendation will be appreciated.
Best Regards,
Vahid
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