[Rtk-users] Model-based image reconstruction based on the RTK modules

Thu Nov 3 02:23:25 EDT 2016

Hello Vahid,

Thank you for this insight on Newton's methods. Yes, the output of the 
backprojection filter, in SART, is the gradient of the cost function. 
You may have noticed that the pipeline performs a division of the 
forward projection by something coming from "RayBoxIntersectionFilter". 
This is to normalize the forward projection, so that R^T R f ~= blurry 
f. If you don't do it, you'll have R^T R f ~= alpha * blurry f, with 
alpha that can reach 200 or so, and your algorithm will quickly diverge.

You could try to extract the gradient from the conjugate gradient filter 
as well, but it is significantly more tricky: first, the CG filter was 
implemented from the following algorithm, taken from wikipedia (which 
embeds the normalization I was mentioning earlier):

{\begin{aligned}&\mathbf {r} _{0}:=\mathbf {b} -\mathbf {Ax} 
_{0}\\&\mathbf {p} _{0}:=\mathbf {r} 
_{0}\\&k:=0\\&{\hbox{repeat}}\\&\qquad \alpha _{k}:={\frac {\mathbf {r} 
_{k}^{\mathsf {T}}\mathbf {r} _{k}}{\mathbf {p} _{k}^{\mathsf 
{T}}\mathbf {Ap} _{k}}}\\&\qquad \mathbf {x} _{k+1}:=\mathbf {x} 
_{k}+\alpha _{k}\mathbf {p} _{k}\\&\qquad \mathbf {r} _{k+1}:=\mathbf 
{r} _{k}-\alpha _{k}\mathbf {Ap} _{k}\\&\qquad {\hbox{if 
}}r_{k+1}{\hbox{ is sufficiently small then exit loop}}\\&\qquad \beta 
_{k}:={\frac {\mathbf {r} _{k+1}^{\mathsf {T}}\mathbf {r} 
_{k+1}}{\mathbf {r} _{k}^{\mathsf {T}}\mathbf {r} _{k}}}\\&\qquad 
\mathbf {p} _{k+1}:=\mathbf {r} _{k+1}+\beta _{k}\mathbf {p} 
_{k}\\&\qquad k:=k+1\\&{\hbox{end repeat}}\\&{\hbox{The result is 
}}\mathbf {x} _{k+1}\end{aligned}}

In this algorithm, it is not clear to me what exactly is the gradient of 
the cost function. I would say it is something like "- r_k", but I'm not 
sure. Second, as you see, the CG filter needs an operator A, which may 
differ from one problem to another, so this operator is implemented in a 
separate filter, which in your case would be 
rtkReconstructionConjugateGradientOperator, with the laplacian 
regularization parameter gamma set to 0.

Note that we never actually store the system matrix R. Instead, the 
interpolation coefficient it contains are re-computed on the fly 
everytime we forward project. And the same holds for backprojection, i.e 
the matrix R^T.

Best,

Cyril

On 11/03/2016 03:38 AM, vahid ettehadi wrote:
> 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 
> <cyril.mory at creatis.insa-lyon.fr> 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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