[Insight-developers] Demons voodoo
Miller, James V (Research)
millerjv at crd.ge.com
Thu, 8 Apr 2004 11:11:33 -0400
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Lydia,
We were having some problems reconciling the output of the demons
registration filter with our understanding of the demons paper. I think I
have things worked out but it would require changing the code/behavior. I
have made the modfications locally and the effect of the results are in the
attached image. All of this is very complicated, having to keep track of
what the warps fields actually do, so I'll break my discussion up into
stages and use the "brain index".
(1 brain needed to read this section): Demon's paper
According to the demon's paper (page 15 and equation 4 on page 17), the goal
of the demons algorithm is to "make an image g appear more like an image f".
Equation 4 is essentially
v = (g -f) grad(f) / (gradmag(f)^2 + (g - f))
I interpret this to mean the vector v is the displacement that is needed to
move a position in image g such that it will lie on a pixel in image f that
has the same intensity as the pixel at the original position in g (this last
bit is just the basic premise of the demons algorithm).
In other words, the displacement field calculated will tell you for each
pixel in g, where the corresponding pixel in f is.
(2 brains needed to read this section): ITK implementation
The demons algorithm is implemented using the finite difference engine,
where the output is the vector field of displacements. The finite difference
engine iterates over the output vector image, passing the index of vector
pixel to the DemonsRegistrationFunction. In pseudo-code, this function
computes the following:
Calculate:
f(pi)
grad(f(pi))
g(pi + vi)
dvi = (f - g) grad(f) / (gradmag(f)^2 + (f - g))
Mapping the code to the equations above, f is the fixed image and g is the
moving image. Note that this is calculating a vector at a point pi that is
associated with image f. This seems to be calculating a vector field that
is keeping track of "where a pixel in f goes" as f is warped to g.
(2 brains needed to read this section): Proposed ITK implementation
Given the equation in the demons paper, I interpret the vector field to map
a position in image g to where in image f the corresponding pixel lies.
Therefore the output of the filter should be a vector field where the pixel
indices correspond to pixel indices in the moving image g and the vector
values at these indices say where in f to sample the values. I propose,
therefore, the code should be
Calculate:
g(pi)
f(pi + v)
grad(f(pi+v))
dvi = (g - f) grad(f) / (gradmag(f)^2 + (g-f))
Again, the fixed image is f and the moving image is g. The output vector
field is index aligned with g and each vector gives a displacement to a
corresponding pixel in f. In essence, the vector field is keeping track of
"where a pixel in g came from" as g is warped to f.
(3 brains needed to read this section): Warping using the demons
displacement field
ITK's WarpImageFilter takes a displacement field that is index aligned with
the output of the WarpImageFilter and using the vector at each pixel to
determine where in the input image to probe to produce the output.
In the current implementation, we pass the moving image (g) and the demons
displacement filed (d) to the WarpImageFilter to produce an "estimate" of
the fixed image.
In my proposal, the deformation field is in the other direction, so we would
need to pass the fixed image (f) and the demons displacement field (g) to
the WarpImageFilter to produce an "estimate" of the moving image.
Summary/Impact
My main concern with the current implementation is that the vector field
computed does not match the mapping (g->f) of the original demons paper. So
people familar using the paper as a reference to how the filter should
behave can be confused (we have already proven that). The filter is
currently calculating vectors of where a pixel in f should go and opposed to
where a pixel in g came from. I don't think these are quite equivalent. As
an example, see the attached png image. I took a circle and warped it to a
square using the current ITK implementation and my proposed modifications.
Because the sense of the vector fields is different in the two
implementations, which image is considered the fixed or moving is different.
However, you can accomplish the end task with both implementations. The
first row of the figure shows the current implementation. As the circle
deforms into a square it actually changes topology twice (4 small hole
appear in the corners of the square and are then filled in). The second row
of the figure shows my alterations which seems more "regularized". However,
the new implementation requires more iterations to reach the destination.
So the new implementation needs more iterations. It also requires that the
grad(f(pi+vi)) be calculated. Currently, I am using the same central
difference image function as before but now I pass in a physical coordinate
instead of an index. This image function will simply evaluate the gradient
at the nearest pixel. So overall, the proposed implementation requires more
computations. However, it looks to me like the deformation path is more
stable and the computed deformation field seems to agree (with me anyway)
with the original Demons paper.
But perhaps more disruptively, the "sense" of the deformation field (moving
to fixed as opposed to fixed to moving) is different.
Jim Miller
_____________________________________
Visualization & Computer Vision
GE Research
Bldg. KW, Room C218B
P.O. Box 8, Schenectady NY 12301
millerjv at research.ge.com <mailto:millerjv at research.ge.com>
james.miller at research.ge.com
(518) 387-4005, Dial Comm: 8*833-4005,
Cell: (518) 505-7065, Fax: (518) 387-6981
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<DIV><SPAN class=325063613-08042004><FONT size=2>(Original was bounced by the
message size filter on the list.)</FONT></SPAN></DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2></FONT></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2>Lydia, </FONT></SPAN></DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2></FONT></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2>We were having some problems
reconciling the output of the demons registration filter with our understanding
of the demons paper. I think I have things worked out but it would require
changing the code/behavior. I have made the modfications locally and the
effect of the results are in the attached image. All of this is very
complicated, having to keep track of what the warps fields actually do, so I'll
break my discussion up into stages and use the "brain
index".</FONT></SPAN></DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2></FONT></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2><STRONG>(1 brain needed to read
this section): Demon's paper</STRONG></FONT></SPAN></DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2></FONT></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2>According to the demon's paper
(page 15 and equation 4 on page 17), the goal of the demons algorithm is to
"make an image g appear more like an image f". Equation 4 is
essentially</FONT></SPAN></DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2></FONT></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004> <FONT size=2>v = (g -f)
grad(f) / (gradmag(f)^2 + (g - f))</FONT></SPAN></DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2></FONT></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2>I interpret this to mean the
vector v is the displacement that is needed to move a position in
image g such that it will lie on a pixel in image f that has the same
intensity as the pixel at the original position in g (this last bit is
just the basic premise of the demons algorithm).</FONT></SPAN></DIV>
<DIV><SPAN class=325063613-08042004></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2>In other words, the
displacement field calculated will tell you for each pixel in g, where the
corresponding pixel in f is.</FONT></SPAN></DIV>
<DIV><SPAN class=325063613-08042004></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004><STRONG><FONT size=2>(2 brains needed to
read this section): ITK implementation</FONT> </STRONG></SPAN></DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2></FONT></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2>The demons algorithm is
implemented using the finite difference engine, where the output is the vector
field of displacements. The finite difference engine iterates over the output
vector image, passing the index of vector pixel to the
DemonsRegistrationFunction. In pseudo-code, this function computes the
following:</FONT></SPAN></DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2></FONT></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2>
Calculate:</FONT></SPAN></DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2>
f(pi)</FONT></SPAN></DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2>
grad(f(pi))</FONT></SPAN></DIV>
<DIV><SPAN class=325063613-08042004> <FONT size=2>g(pi +
vi)</FONT></SPAN></DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2></FONT></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004> <FONT size=2>dvi =</FONT>
<FONT size=2> (f - g) grad(f) / (gradmag(f)^2 + (f -
g))</FONT></SPAN></DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2></FONT></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2>Mapping the code to the
equations above, f is the fixed image and g is the moving image. Note that this
is calculating a vector at a point pi that is associated with image f.
This seems to be calculating a vector field that is keeping track of "where a
pixel in f goes" as f is warped to g.</FONT></SPAN></DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2></FONT></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2>
<DIV><SPAN class=325063613-08042004><STRONG><FONT size=2>(2 brains needed to
read this section): Proposed ITK
implementation</FONT> </STRONG></SPAN></DIV>
<DIV><SPAN class=325063613-08042004><STRONG></STRONG></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004>Given the equation in the demons paper, I
interpret the vector field to map a position in image g to where in image f the
corresponding pixel lies. Therefore the output of the filter should be a
vector field where the pixel indices correspond to pixel indices in the moving
image g and the vector values at these indices say where in f to sample the
values. I propose, therefore, the code should be</SPAN></DIV>
<DIV><SPAN class=325063613-08042004></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004> Calculate:</SPAN></DIV>
<DIV><SPAN class=325063613-08042004> </SPAN><SPAN
class=325063613-08042004>g(pi)</SPAN></DIV>
<DIV><SPAN class=325063613-08042004> f(pi + v)</SPAN></DIV>
<DIV><SPAN class=325063613-08042004> grad(f(pi+v))</SPAN></DIV>
<DIV><SPAN class=325063613-08042004></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004> dvi = (g - f) grad(f) /
(gradmag(f)^2 + (g-f))</SPAN></DIV>
<DIV><SPAN class=325063613-08042004></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004>Again, the fixed image is f and the moving
image is g. The output vector field is index aligned with g and each
vector gives a displacement to a corresponding pixel in f. In essence, the
vector field is keeping track of "where a pixel in g came from" as g is warped
to f.</SPAN></DIV>
<DIV><SPAN class=325063613-08042004></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004>
<DIV><SPAN class=325063613-08042004><STRONG><FONT size=2>(3 brains needed to
read this section): Warping using the demons displacement
field</FONT></STRONG></SPAN></DIV>
<DIV><SPAN class=325063613-08042004><STRONG></STRONG></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004>ITK's WarpImageFilter takes a displacement
field that is index aligned with the output of the WarpImageFilter and using the
vector at each pixel to determine where in the input image to probe to produce
the output.</SPAN></DIV>
<DIV><SPAN class=325063613-08042004></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004>In the current implementation, we pass the
moving image (g) and the demons displacement filed (d) to the WarpImageFilter to
produce an "estimate" of the fixed image.</SPAN></DIV>
<DIV><SPAN class=325063613-08042004></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004>In my proposal, the deformation field is in
the other direction, so we would need to pass the fixed image (f) and the demons
displacement field (g) to the WarpImageFilter to produce an "estimate" of the
moving image.</SPAN></DIV>
<DIV><SPAN class=325063613-08042004></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004><STRONG>Summary/Impact</STRONG></SPAN></DIV>
<DIV><SPAN class=325063613-08042004><STRONG></STRONG></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004>My main concern with the current
implementation is that the vector field computed does not match the mapping
(g->f) of the original demons paper. So people familar using the paper
as a reference to how the filter should behave can be confused (we have already
proven that). The filter is currently calculating vectors of where a pixel in f
should go and opposed to where a pixel in g came from. I don't think these are
quite equivalent. As an example, see the attached png image. I took
a circle and warped it to a square using the current ITK implementation and my
proposed modifications. Because the sense of the vector fields is different in
the two implementations, which image is considered the fixed or moving is
different. However, you can accomplish the end task with both
implementations. The first row of the figure shows the current
implementation. As the circle deforms into a square it actually changes
topology twice (4 small hole appear in the corners of the square and are then
filled in). The second row of the figure shows my alterations which seems
more "regularized". However, the new implementation requires more
iterations to reach the destination.</SPAN></DIV>
<DIV><SPAN class=325063613-08042004></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004>So the new implementation needs more
iterations. It also requires that the grad(f(pi+vi)) be calculated.
Currently, I am using the same central difference image function as before but
now I pass in a physical coordinate instead of an index. This image
function will simply evaluate the gradient at the nearest pixel. So overall, the
proposed implementation requires more computations. However, it looks to
me like the deformation path is more stable and the computed deformation field
seems to agree (with me anyway) with the original Demons paper. </SPAN></DIV>
<DIV><SPAN class=325063613-08042004></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004>But perhaps more disruptively, the "sense"
of the deformation field (moving to fixed as opposed to fixed to moving) is
different.</SPAN></DIV>
<DIV></SPAN></FONT></SPAN><SPAN class=325063613-08042004><FONT
size=2></FONT></SPAN> </DIV></DIV></DIV>
<DIV><SPAN class=325063613-08042004><FONT size=2></FONT></SPAN> </DIV>
<DIV><SPAN class=325063613-08042004><STRONG><FONT
size=2></FONT></STRONG></SPAN> </DIV>
<DIV>
<P style="MARGIN: 0in 0in 0pt"><B><SPAN
style="COLOR: navy; FONT-FAMILY: 'Comic Sans MS'">Jim Miller</SPAN></B>
<BR><B><I><SPAN
style="FONT-SIZE: 10pt; COLOR: red; FONT-FAMILY: Arial">_____________________________________</SPAN></I></B><BR><EM><SPAN
style="FONT-SIZE: 7.5pt; COLOR: black; FONT-FAMILY: Arial">Visualization &
Computer Vision</SPAN></EM><I><SPAN
style="FONT-SIZE: 7.5pt; COLOR: black; FONT-FAMILY: Arial"><BR><EM>GE
Research</EM><BR><EM>Bldg. KW, Room C218B</EM><BR><EM>P.O. Box 8, Schenectady NY
12301</EM><BR><BR></SPAN></I><EM><U><SPAN
style="FONT-SIZE: 7.5pt; COLOR: blue"><A
href="mailto:millerjv at research.ge.com">millerjv at research.ge.com</A></SPAN></U></EM></P>
<P style="MARGIN: 0in 0in 0pt"><EM><U><SPAN
style="FONT-SIZE: 7.5pt; COLOR: blue">james.miller at research.ge.com</SPAN></U></EM><BR><I><SPAN
style="FONT-SIZE: 7.5pt; COLOR: black; FONT-FAMILY: Arial">(518) 387-4005, Dial
Comm: 8*833-4005, </SPAN></I><BR><I><SPAN
style="FONT-SIZE: 7.5pt; COLOR: black; FONT-FAMILY: Arial">Cell: (518) 505-7065,
Fax: (518) 387-6981</SPAN></I> </P></DIV>
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