<html><head></head><body><div style="color:#000; background-color:#fff; font-family:HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, sans-serif;font-size:14px"><div id="yui_3_16_0_ym19_1_1478133746044_18490"><span id="yui_3_16_0_ym19_1_1478133746044_18491">Hello Simon and Cyril,</span></div><div id="yui_3_16_0_ym19_1_1478133746044_18492"><span id="yui_3_16_0_ym19_1_1478133746044_18493">Thanks for the reply.</span></div><div dir="ltr" id="yui_3_16_0_ym19_1_1478133746044_18494"><span id="yui_3_16_0_ym19_1_1478133746044_18495">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 </span>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.</div><div dir="ltr" id="yui_3_16_0_ym19_1_1478133746044_18494">Thanks again for the reply.</div><div dir="ltr" id="yui_3_16_0_ym19_1_1478133746044_18494"><br></div><div dir="ltr" id="yui_3_16_0_ym19_1_1478133746044_18494">@Cyril:</div><div dir="ltr" id="yui_3_16_0_ym19_1_1478133746044_18494">Please correct me if I am wrong. you mean the output of backProjectionFilter is the gradient of defined cost function?</div><div dir="ltr" id="yui_3_16_0_ym19_1_1478133746044_18494"><br></div><div dir="ltr" id="yui_3_16_0_ym19_1_1478133746044_18494">Regards,</div><div dir="ltr" id="yui_3_16_0_ym19_1_1478133746044_18494">Vahid</div> <div class="qtdSeparateBR"><br><br></div><div class="yahoo_quoted" style="display: block;"> <div style="font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, sans-serif; font-size: 14px;"> <div style="font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, sans-serif; font-size: 16px;"> <div dir="ltr"><font size="2" face="Arial"> On Wednesday, November 2, 2016 2:53 AM, Cyril Mory <cyril.mory@creatis.insa-lyon.fr> wrote:<br></font></div> <br><br> <div class="y_msg_container"><div id="yiv2652688545"><div>
<div>Hi Vahid,</div>
<div>Welcome to RTK :)</div>
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. <br clear="none">
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)<br clear="none">
<br clear="none">
digraph SARTConeBeamReconstructionFilter {<br clear="none">
<br clear="none">
Input0 [ label="Input 0 (Volume)"];<br clear="none">
Input0 [shape=Mdiamond];<br clear="none">
Input1 [label="Input 1 (Projections)"];<br clear="none">
Input1 [shape=Mdiamond];<br clear="none">
<br clear="none">
node [shape=box];<br clear="none">
ForwardProject [ label="rtk::ForwardProjectionImageFilter" URL="\ref
rtk::ForwardProjectionImageFilter"];<br clear="none">
Extract [ label="itk::ExtractImageFilter" URL="\ref
itk::ExtractImageFilter"];<br clear="none">
MultiplyByZero [ label="itk::MultiplyImageFilter (by zero)"
URL="\ref itk::MultiplyImageFilter"];<br clear="none">
AfterExtract [label="", fixedsize="false", width=0, height=0,
shape=none];<br clear="none">
Subtract [ label="itk::SubtractImageFilter" URL="\ref
itk::SubtractImageFilter"];<br clear="none">
MultiplyByLambda [ label="itk::MultiplyImageFilter (by lambda)"
URL="\ref itk::MultiplyImageFilter"];<br clear="none">
Divide [ label="itk::DivideOrZeroOutImageFilter" URL="\ref
itk::DivideOrZeroOutImageFilter"];<br clear="none">
GatingWeight [ label="itk::MultiplyImageFilter (by gating weight)"
URL="\ref itk::MultiplyImageFilter", style=dashed];<br clear="none">
Displaced [ label="rtk::DisplacedDetectorImageFilter" URL="\ref
rtk::DisplacedDetectorImageFilter"];<br clear="none">
ConstantProjectionStack [ label="rtk::ConstantImageSource" URL="\ref
rtk::ConstantImageSource"];<br clear="none">
ExtractConstantProjection [ label="itk::ExtractImageFilter"
URL="\ref itk::ExtractImageFilter"];<br clear="none">
RayBox [ label="rtk::RayBoxIntersectionImageFilter" URL="\ref
rtk::RayBoxIntersectionImageFilter"];<br clear="none">
ConstantVolume [ label="rtk::ConstantImageSource" URL="\ref
rtk::ConstantImageSource"];<br clear="none">
BackProjection [ label="rtk::BackProjectionImageFilter" URL="\ref
rtk::BackProjectionImageFilter"];<br clear="none">
OutofInput0 [label="", fixedsize="false", width=0, height=0,
shape=none];<br clear="none">
OutofBP [label="", fixedsize="false", width=0, height=0,
shape=none];<br clear="none">
BeforeBP [label="", fixedsize="false", width=0, height=0,
shape=none];<br clear="none">
BeforeAdd [label="", fixedsize="false", width=0, height=0,
shape=none];<br clear="none">
Input0 -> OutofInput0 [arrowhead=none];<br clear="none">
OutofInput0 -> ForwardProject;<br clear="none">
ConstantVolume -> BeforeBP [arrowhead=none];<br clear="none">
BeforeBP -> BackProjection;<br clear="none">
Extract -> AfterExtract[arrowhead=none];<br clear="none">
AfterExtract -> MultiplyByZero;<br clear="none">
AfterExtract -> Subtract;<br clear="none">
MultiplyByZero -> ForwardProject;<br clear="none">
Input1 -> Extract;<br clear="none">
ForwardProject -> Subtract;<br clear="none">
Subtract -> MultiplyByLambda;<br clear="none">
MultiplyByLambda -> Divide;<br clear="none">
Divide -> GatingWeight;<br clear="none">
GatingWeight -> Displaced;<br clear="none">
ConstantProjectionStack -> ExtractConstantProjection;<br clear="none">
ExtractConstantProjection -> RayBox;<br clear="none">
RayBox -> Divide;<br clear="none">
Displaced -> BackProjection;<br clear="none">
BackProjection -> OutofBP [arrowhead=none];<br clear="none">
}<br clear="none">
<br clear="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. <br clear="none">
<br clear="none">
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
<a rel="nofollow" shape="rect" class="yiv2652688545moz-txt-link-freetext" target="_blank" href="https://itk.org/Doxygen/html/classitk_1_1CostFunctionTemplate.html">https://itk.org/Doxygen/html/classitk_1_1CostFunctionTemplate.html</a>
if you want to try this approach.<br clear="none">
<br clear="none">
Best regards,<br clear="none">
Cyril<br clear="none">
<br clear="none">
<div class="yiv2652688545yqt2869888754" id="yiv2652688545yqt14261"><div class="yiv2652688545moz-cite-prefix">On 11/01/2016 05:50 PM, vahid ettehadi
via Rtk-users wrote:<br clear="none">
</div>
<blockquote type="cite">
<div style="color:#000;background-color:#fff;font-family:HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, sans-serif;font-size:14px;">
<div id="yiv2652688545yui_3_16_0_ym19_1_1478018323282_15982">Hello RTK users
and developers,</div>
<div id="yiv2652688545yui_3_16_0_ym19_1_1478018323282_15982"><br clear="none">
</div>
<div id="yiv2652688545yui_3_16_0_ym19_1_1478018323282_15982">I already
implemented the RTK and reconstructed some images with the FDK
algorithm implemented in RTK. It works well. Thanks to RTK
developers.<br clear="none">
</div>
<div dir="ltr" id="yiv2652688545yui_3_16_0_ym19_1_1478018323282_17709">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. </div>
<div dir="ltr" id="yiv2652688545yui_3_16_0_ym19_1_1478018323282_17709"><br clear="none">
</div>
<div dir="ltr" id="yiv2652688545yui_3_16_0_ym19_1_1478018323282_17709">Best
Regards,</div>
<div dir="ltr" id="yiv2652688545yui_3_16_0_ym19_1_1478018323282_17709">Vahid</div>
<div dir="ltr" id="yiv2652688545yui_3_16_0_ym19_1_1478018323282_17710"><br clear="none" id="yiv2652688545yui_3_16_0_ym19_1_1478018323282_17711">
</div>
</div>
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<fieldset class="yiv2652688545mimeAttachmentHeader"></fieldset>
<br clear="none">
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