How to redistribute pixels?

Posted is a picture of a cross section of a vein. I have transformed the red interior line into a circle and need to have the pixels on the black line surround the new circular red line, both red and black lines having the same centers. The transformed black line should not be a perfect circle. As you can see in the picture, the distance between the black and red lines differ and once the black line is transformed, it should maintain the same varying distance between it and the red line as before. My question is how can I do this?

Answer 1

It's not too hard if you think about it systematically. Just take it one step at a time. The key is to convert the lines into masks and use bwdist().

1. use poly2mask on the black line to get a mask
2. use poly2mask on the red line to get a smaller mask
3. subtract the two masks to get the "in between" region.
4. call bwdist() to get the Euclidean Distance Transform (EDT). This is the distance from the region to the nearest edge, so twice that is the thickness of the region at that point. So multiply the EDT by 2.
5. call bwmorph() to get the skeleton. This is the centerline of the region.
6. Multiply the skeleton by the EDT. This will give you the thickness of the region.
7. bwdist will also give you the pixel on the red line that is closest to the black line.
8. Now you know the radius and center of the red circle. Radius is just perimeter/(2*pi). So you just run around the black circle, and get the red pixel that is closest to it.
9. You know the position of each red pixel - it's index on the circle. So you plot a red point there, and then plot a black point at that red radius plus 2*EDT for that black pixel.

Thank you for your input, I have looked it over and I have some questions:

4) why would 2*EDT be the region thickness? 6) I thought 4) gave the region thickness? 8) would the smallest distance between the red and black pixels be where the black pixel is directly above or orthogonal to a red pixel?

Since I know the position of each pixel on the red and black lines, how can I go pixel by pixel on a line and get the distance between a pixel on that line and a pixel on the other line? After the red and black lines are transformed, the area between the red and black lines pre-transformation and post-transformation should be the same.

(4) The region thickness is twice the EDT value at the center. Let's say you had a region 20 pixels wide. Well, at the center it will be 10 pixels to either side because if you're at the center line it has to go only half the thickness to get outside the region. So you need to double the value at the center to get the full thickness.

6) You multiply by the skeleton to give you the value along the centerline of the region. The EDT you get in (4) is a thick image. It's like a wide swatch the same size as the region in between your two curves. And the values of the EDT go from 0 or 1 at the edge (where a pixel inside the swath does not have far to go to get outside) all the way to half the thickness at the centerline. If you're 3 pixels away from the background, the EDT value is 3, if you're 4 pixels away it's 5, if you're 5 pixels away it's 5, and so on until you're the radius (half thickness) away from the edge. If you proceed toward the other side, it starts going down again until it again reaches 0 at the other edge (boundary).

Think of it like this. You're taking an inflatable ball and running it along the region between the two lines. The ball is always inflated as much as it can go and still be a circle. Sometimes, when the thickness is large, you can fit a large ball, and sometimes, when the thickness is small, you'll have a small ball. The EDT along the centerline (skeleton, spine) is the radius of the ball at that point.

Does that explain it better?

Thanks for the clarification, I understand what the EDT and skeleton are now. So far I have: #1 created the masks #2 gotten the "in between" region and made it hollow #3 did a EDT calc between the two boundary lines of the region #4 found the skeleton of the hollow region #5 removed the branches from the skeleton (don't know purpose of branches) #6 found the perimeter of the region that will be transformed in a circle #7 used that perimeter to find the diameter for the transformed circle

When I did the EDT it gave me a giant "logic" matrix. So now how can I use MATLAB to go through all of the pixels on a boundary of the matrix and find the distance to the nearest pixel on the opposing line?

I appreciate your help.

Or is there a way of taking a pixel on a line, lets say the black line, and then find the shortest path to the red line and storing that path as a vector?

Answer 2

So now how can I use MATLAB to go through all of the pixels on a boundary of the matrix and find the distance to the nearest pixel on the opposing line?

If the rows of A are the pixel coordinates on the black line and B the red line, then you can use

http://www.mathworks.com/matlabcentral/fileexchange/18937-ipdm-inter-point-distance-matrix

as follows

s=ipdm(A,B,'Subset', 'NearestNeighbor','Result','Structure');

Then s.columnindex will be the indices in B of the closest points on the red line. The corresponding distances will be in s.distance.

When run with this specific syntax, the column index is what will be most useful. It tells you that

 B(short_dist.columnindex(i), :)

is the closest point on the red line to A(i,:). And if you want the vector between the two points, just do

 B(short_dist.columnindex(i), :) - A(i,:)

for all i=1...1856. Or, in vectorized form, simply

 B(short_dist.columnindex, :) - A

I implemented your idea into my code and it almost works. It has some difficulty calculating the new red line around the right side of the wall where it is folded up. Here is how I am calculating the transform so far:

1. use the ipdm function to calculate the vectors between the red and black lines pre-transformation using your previous comment
2. calculate the perimeter of the black line [units=pixels]
3. convert that perimeter into a radius
4. multiply that radius by the pixel resolution to convert units into microns
5. using that radius, draw a circle that is centered at the origin (has the same number of points that the black line had in pixels)
6. add the vectors from the ipdm function to the circle to get the red line.

As you will see in the picture, the boundaries are not correct. Any ideas on what when wrong?

I' a bit confused. In your originally posted question, you say the red line is transformed into a perfect circle and the black line transforms into a non-circle derived from the red line. Here, however, you seem to have done the opposite.

I suspect you will get what you're looking for if you just interchange the roles of A and B in all steps, including the use of IPDM, i.e., you should be finding the distances/vectors of red-to-black instead of black-to-red.

Tony DeStephens commented:

Yes, originally I wanted to transform the red line into a perfect circle, now I want to instead transform the black line into a perfect circle and the red line which is a non-circle gets derived from the black line. I apologize for the confusion.

So if I now wanted to instead keep the black line a circle I would just re-arrange the roles of A and B? Something like this:

s=ipdm(B,A,'Subset', 'NearestNeighbor','Result','Structure');

vector = A(short_dist.columnindex(i), :) - B(i,:);

A_new = equations for cirlce;

B_new = A_new + vector;

Is this correct?

So if I now wanted to instead keep the black line a circle I would just re-arrange the roles of A and B?

No. I only suggested interchanging A and B because it sounded like you had the red line and black line mixed up. If you're now interested in transforming the black line into the circle, that doesn't apply.

6.add the vectors from the ipdm function to the circle to get the red line.

I think this step is the real problem. It sounds like you want to be stepping towards the center of the black circle by the length of the IPDM vectors. In particular, let v be the vector given by ipdm, let A_new be the corresponding point on the black circle, and let C be the position of the center of the black circle. Then I think you want,

 B_new = A_new + norm(v)*(C-A_new)/norm(C-A_new)

I had v = B(short_dist.columnindex, :) - A

No, you need to be looping over all A_new individually.

Incidentally, you should be replying to me in comments so that people know whose answer you're addressing.

Your code worked great, thanks for the help!

To check that the transform was done properly, I am calculating the area between the two lines both before and after the transformation and comparing them. I am using the poly2mask function to create a binary image from the post transform coordinates, but only 1/4 of the image is showing up. I am using the following commands where BW is the binary image of the ROI pre-transformation:

BW_A_new = poly2mask(A_new_xunit, A_new_yunit, length(BW(:,1)), length(BW(1,:)));

BW_B_new = poly2mask(B_new_xunit, B_new_yunit, length(BW(:,1)), length(BW(1,:)));

in_between_region_post = BW_A_new - BW_B_new;

figure (4), imshow(in_between_region_post)

Am I calling poly2mask incorrectly? How do I get the whole image to so up?

Also, I am using the imfreehand function to draw the lines on the picture, but I am going to have to draw a 3rd line that outlines the boundary of the inside diameter (which is just inside the red line). In order to do this I will have to be constantly stopping, zooming in and moving the image. So the question is how can I stop and go while outlining an image using imfreehand?

I think it's because you're using x,y coordinates with an origin somewhere in the middle of the image. They need to be in matrix coordinates, if I'm not mistaken.

I'm a bit skeptical though that the area between the curves will be preserved (I could be wrong, though). Was that the goal all along? You should have said so in your original post.

They need to be in matrix coordinates

how do I make a transformation from Cartesian coords to matrix coords?

I'm a bit skeptical though that the area between the curves will be preserved (I could be wrong, though). Was that the goal all along? You should have said so in your original post

If the distances between the pixels are the same before and after the transform then my thought was that the area change would not change. I mentioned it in my original post when I said "it should maintain the same varying distance between it(the black line) and the red line as before"

how do I make a transformation from Cartesian coords to matrix coords?

You first convert Cartesian coordinates to pixel length units (instead of millimeters or whatever). Then you translate to a new origin. The matrix coordinate origin is in the top left corner.

If the distances between the pixels are the same before and after the transform then my thought was that the area change would not change.

Don't think so. Consider the simple case of 2 concentric circles with radii a>b. The area between them is

 InitialArea = pi*(a^2-b^2)

Now transform the circles simply by increasing both radii by the same amount, x, so that the distance between them is preserved at a-b. The new area is

 NewArea = pi*((a+x)^2-(b+x)^2)
         = InitialArea+2*pi*(a-b)*x

Clearly the distance between the curves is invariant to x but the area between them is not.

You first convert Cartesian coordinates to pixel length units (instead of millimeters or whatever). Then you translate to a new origin. The matrix coordinate origin is in the top left corner.

In the binary image that I posted, the units were in pixel length. I tried to move the ROI origin into the middle of the figure by adding about half of the figure width (unit pixels) and subtracting half the figure height (unit pixels). When I did this and used imshow to bring up the image nothing showed up. The figure was completely black and the ROI was gone.

Clearly the distance between the curves is invariant to x but the area between them is not

I see what you are saying, but what if the perimeter of the black line was preserved, so that it was the same for the pre and post transform as well as preserving the distances between the red and black line? Would the pre and post transform areas be the same then?

Also, how can I stop and go while outlining an image using imfreehand? Currently, once I start the drawing I can't stop until its complete, but I want to be able to stop so that I can zoom in and then resume outlining

When I did this and used imshow to bring up the image nothing showed up. The figure was completely black and the ROI was gone.

We can't see your calculations, so we can't see the relationship between your current coordinates and the one that poly2mask works with. By experimenting with with poly2mask and reading the examples in its documentation, you should be able to understand the coordinates it uses.

I see what you are saying, but what if the perimeter of the black line was preserved, so that it was the same for the pre and post transform as well as preserving the distances between the red and black line?

It might be true if the closest point from points on the A curve to the B curve is a 1-1 mapping. Not sure how to confirm that with rigorous calculus, though. It doesn't seem to be true when the mapping is not 1-1 however.

Suppose for example your red curve was 1 side of a unit square and your black curve were the remaining 3 sides of the square. In this case, the tips of the red edge are the nearest points to every point on the black edge intersecting them, so the nearest-point mapping is not 1-1. Now think about how the initial area (equal to 1) changes if you transform the black curve by straightening it into a line segment with the same initial total length = 3. If you construct a new red border based on distance along rays perpendicular from this new black line, you will end up with a trapezoid of area 2.

Also, how can I stop and go while outlining an image using imfreehand?

Don't think you can do that, but you can call imfreehand with 'Closed'=false. Then it will just draw an open curve. You can then zoom and call imfreehand again and draw a new segment of the curve, etc... With just a little bit of coding, you can accumulate all the points on the different curve segments.

B_new = A_new + norm(v) x (C-A_new)/norm(C-A_new)

Can you explain how you got this equation that calculates the new red line?

call imfreehand with 'Closed'=false

I have now implemented this to my code, but is there a way that I can have a newly drawn open line to automatically connect the previously drawn open line? Similar to how when 'closed'=true automatically closes when the mouse button is let go? That way when I finish drawing my ROI it will be closed.

Or would I be better off just trying to have the start of the new open line overlap the end of the previous open line so that the finished ROI is closed?

If I drew a straight line that went from the red line to the black line, what would the equation be that would transform this line onto the new red and black lines

so far I found the vector between each point of the line (goes from A to B) and the closest point on the red line.

vector = ipdm(line,B,'Subset','nearest','result','struct')

Then maybe add that vector to the transformed point on the red line?

Can you explain how you got this equation that calculates the new red line?

The length of the new difference vector (B_new - A_new) will be norm(v)=norm(B_old-A_old), preserving the previous distance. The direction of the new difference vector will be (C-A_new)/norm(C-A_new) which points radially inward from the boundary of the new black circle.

I have now implemented this to my code, but is there a way that I can have a newly drawn open line to automatically connect the previously drawn open line?

Not that I know of, but you could consider using impoly instead of imfreehand. With impoly, you can freely zoom in, add new vertices, and bend the boundary of the polygon as finely as you like.

Then maybe add that vector to the transformed point on the red line?

Thought you tried that before and it wasn't what you wanted.

Is there a way to use the ipdm code to go from a pixel on the black line to a pixel on the red line that is normal to it? Would this be a better method in preserving the distance between the two lines than finding the nearest neighbor? Why or why not? The one of the goals for this transfiguration is to have a 1-1 mapping.

Would this be a better method in preserving the distance between the two lines than finding the nearest neighbor

No one but you knows what "better" means (because it depends on what you're trying to achieve with all this).

one of the goals for this transfiguration is to have a 1-1 mapping.

It would really be better if you described all of the goals of the transformation. As you can see, leaving design requirements unspoken leads to proposals that don't satisfy them.

It would really be better if you described all of the goals of the transformation

The goal is to make the black line circular and remap the red line such that the distance between the red and black lines are preserved, which would mean that 1-1 mapping is achieved.

You had mentioned before that your idea with finding the nearest neighbor, preserving that distance and then plotting the new red line radially inward from the new black line might not be 1-1.

So I was thinking that instead of going pixel by pixel on the black line and finding the nearest pixel neighbor on the red line, instead find the red line pixel that is orthogonal to the black line pixel and tracking that distance instead.

So the only thing that would change would be how I used the ipdm function, this is if it had the capability to find the red line pixel that is orthogonal to a given black line pixel.

So I was thinking that instead of going pixel by pixel on the black line and finding the nearest pixel neighbor on the red line, instead find the red line pixel that is orthogonal to the black line pixel and tracking that distance instead.

That wouldn't guarantee a 1-1 mapping, I'm afraid. If either curve bends sharply enough then a perpendicular emanating from certain points on the outer black curve may not intersect the interior red curve at all (imagine concentric squares). Conversely, a perpendicular radiating from the interior curve to the outer curve would skip over points at the corners.

I think you need to be analyzing this more generally. Suppose we parametrize the given red and black curves, (e.g., using splines) as

r_black(t) = [x_black(t), y_black(t)]

r_red(t) = [x_red(t), y_red(t)]

Any such parametrization defines a 1-1 mapping between r_red and r_black as long as each curve is swept out completely as t increases monotonically from 0 to 1.

Similarly write down a parametric formula for the target black circle

r_circ(t) = [x_circ(t), y_circ(t)]

Then there is an infinite set of distance-preserving new red cuves r(t) given by

    r(t) = r_circ(t)+ norm(r_red(t) - r_black(t))* v(t)

where v(t) is any vector satisfying norm(v(t))=1 at all t. The set is infinite because there are infinite possibilities for v(t) and also because there are an infinite variety of ways to parametrize r_red, r_black, and r_circ as functions of t.

The thing to do therefore is to specify what properties that you want r(t) to have in addition to distance preservation (all choices satisfy that) so that the choices for some of these things can be narrowed down.

Any such parametrization defines a 1-1 mapping between r_red and r_black as long as each curve is swept out completely as t increases monotonically from 0 to 1.

So if the red and black lines were drawn as monotonic splines, then 1-1 mapping will be achieved?

specify what properties that you want r(t) to have in addition to distance preservation

Distance preservation is the main property that I want. Other than that I want the region between the lines pre-transformation to be the same as post-transformation.

So if the red and black lines were drawn as monotonic splines, then 1-1 mapping will be achieved?

I don't know what you mean by a "monotonic spline". If you have 2 space curves r1(t) and r2(t) which are each 1-1 functions of t, then likewise there is a 1-1 relationship between the points on the curves, given by

r2(t(r1))

In other words, a given coordinate r1 maps to some unique t which maps to a unique r2 and vice versa.

Other than that I want the region between the lines pre-transformation to be the same as post-transformation.

What does it mean for 2 regions to be "the same"? Obviously the regions will have a different shape post-transformation. You mean their areas should be the same?

If you have 2 space curves r1(t) and r2(t) which are each 1-1 functions of t

So right now the red and black lines are hand drawn, but your saying that they need to be defined by parametric equations each being a function of t?

You mean their areas should be the same?

Yes, I want the areas to be the same. sorry for the confusion.

So right now the red and black lines are hand drawn, but your saying that they need to be defined by parametric equations each being a function of t?

Yes, but for example, if you used impoly instead of imfreehand, you know that the curve is piecewise linear. It would be very easy to write down a piecewise linear r(t) once the vertices are obtained. Similarly, you could use the SPLINE command to get a cubic spline fit of the vertices

Yes, I want the areas to be the same. sorry for the confusion.

This is starting to sound doubtfully difficult. In order for the transformation to be both length and area preserving, it would have to probably consist of piece-wise rigid rotation/translations that would map small sections of the initial black curve to the final black circle. If you applied such a transformation, I can see the red curve getting transformed in ways you don't like, e.g., bending in ways that make it self-intersecting...

What exactly is the practical purpose of all this again? If we succeed at finding this transformation, what does it then allow you to do?

you could use the SPLINE command to get a cubic spline fit of the vertices

Do the red and black lines need to have the same number of vertices and should the vertices for each line have the same polar coordinates with respect to the centroid of the figure? So in other words, would each of the vertices on the red and black lines line up radially with the centriod?

map small sections of the initial black curve to the final black circle

Yes, if the sections were small enough I think that this might work. However, how would you split the region between the two lines into smaller sections?

What exactly is the practical purpose of all this again?

I basically want to track the radial position of the cells in between those two lines after the transformation.

I basically want to track the radial position of the cells in between those two lines after the transformation.

So basically, the whole idea all along was to transform the 2 curves into polar curves r(theta) where r is a 1-1 function of theta? Does the transformed black curve even have to be a circle?

So basically, the whole idea all along was to transform the 2 curves into polar curves r(theta) where r is a 1-1 function of theta?

Yes, that seems like it would be the best way to get the transformation that I want.

Does the transformed black curve even have to be a circle?

Yes

Does the transformed black curve even have to be a circle? Yes

I was looking for a little more elaboration here. It doesn't sound like it needs to be a circle, if all you want to be able to do (post-transformation) is to draw a radial line through each cell and see where that line intersects the red and black curves.

Aside from that, I'd like to step back for a moment and try to reverse engineer things. Suppose we start with the transformed red and black curves that you're looking for. Suppose I draw a radial line to the black circle and observe the chord running between its intersection with the red and black curves. Now, suppose that I invert the transformation, taking us back to the original red and black curves, but I also apply that inverse transformation to the chord (giving me a "pre-chord") and observe how it changes shape.

My question(s): what hard requirements do you have on that pre-chord? You've said you want the pre-chord and the chord to have the same length. However, does the pre-chord still have to be a straight line? Should the pre-chord intersect the pre-red and pre-black curves at only its 2 end points or is it allowed to cross them at other places as well?

Obviously, the more conditions you have on how chords/pre-chords map to one another, the harder it will be to find the desired transformation (if one even exists), so you should set only the requirements that you really and truly need. You should also explain why you think you need those conditions, so I can challenge that reasoning (some of the things you want may turn out to be impossible) and maybe find more feasible alternatives.

I was looking for a little more elaboration here

When the vein has blood is flowing through it, the black line, which represents a boundary in the vein wall, is circular. So one of the requirements for this transformation is that the black line is circular and then the red line is remapped around the black line, while trying to preserve the pre-transform distance between the two lines. 1-1 mapping is desired in order to have an accurate transformation and I do not want to lose any area between the two curves since a cell might be located in the area that is lost.

does the pre-chord still have to be a straight line?

What are the implications if it is or is not? I would think that if the pre-chord line were straight and the post-chord line was not straight, or vice versa, then there would be a some sort of rotation that occurred. So based on that assumption, I would say that they should be straight.

Should the pre-chord intersect the pre-red and pre-black curves at only its 2 end points or is it allowed to cross them at other places as well?

I would assume that as long as the pre-chord intersects the same points on the post-red and post-black curves as it did on the pre-red and pre-black curves and that the length of the curve between those two points pre and post transform are the same, then it is OK. If this were not true then I would think that the distance is not preserved.

When the vein has blood is flowing through it, the black line, which represents a boundary in the vein wall, is circular

Do you know for a fact that the "distance" between the red and black curves remains the same in physical reality? If you know that for a fact, you ought to have some basis, based on biology, of defining that distance as well as the point on the red curve that a given point on the black curve is partnered with. We've been lacking those definitions, so far. Obviously, the nearest-point mapping from the black curve to the red curve does not define the partnering. We tried that...

To my inexpert eye, I can see that the total area between the 2 curves would remain the same assuming the material sandwiched in between the 2 curves (cells?) has a fixed density, but otherwise no other biomechanical laws about how the red boundary should transform are apparent to me. We need to know those laws.

Do you know for a fact that the "distance" between the red and black curves remains the same in physical reality?

For now, compressibility effects are assumed to be non existent. All I want for this first transformation is to preserve the distance and area between the two curves, make the black line circular, and then remap the red curve with respect to the black curve. The current transform that we are talking about will not take pressure or any other biomechanical effects into account. After this initial transformation there will be others that need to be addressed that will take these aforementioned effects into account, but I just needed help figuring out how to program this first transform so that I can get started.

Obviously, the nearest-point mapping from the black curve to the red curve does not define the partnering.

This method was close, with a percent difference between the pre and post transform areas being from <1% - 8%. So while distance between the two curves is preserved, area is not.

We need to know those laws

Yes, but only for later transformations. Right now, for the first of many transforms, these laws are not taken into account. To preserve area, the idea of splitting the region between the two curves into smaller sections and finding the distances between the two curves in these smaller regions is a good one. How would you go about programming it though? Is there anything in the IPDM code that can be used to find these distances?

Ideas for the transformation

I was thinking about splitting up the region between the two lines into a number of smaller "trapezoidal like" sections. The top and bottom of the trapezoid will be the red and black curves with a predetermined length, where the black curve is the longer length of the trapezoid. The sides of the trapezoid run laterally between the red and black lines and should be straight. After the transformation, the black curve will be a circular arc and the red line will adjust according to the pre-transform distances in order to preserve distance. In order to preserve area, the lateral lines of the trapezoidal will adjust the angles at which they were orientated, but still remain straight, to preserve area.

Yes, but only for later transformations. Right now, for the first of many transforms, these laws are not taken into account.

I think you might be missing my point. I'm suggesting that not taking at least some of those laws into account is making things harder than including them. Even if you just want to preserve area and "distance" (whatever that is) between the two boundaries, the reason those things are preserved in real life is the complex result of fluid mechanical phenomena like applied pressure and non-compressibility. Writing down the equations for those gives one a better chance of deriving the proper transformation.

I think we're miles away from accomplishing this with an ad hoc geometric transformation. Rigidly rotating the black boundary piece by piece to align with the desired circle doesn't seem to me like it's going to work. If you were to do that in real life, it could stretch (or tear) the red boundary segment opposite it and that would change area.

Writing down the equations for those gives one a better chance of deriving the proper transformation

All I need is the unpressurized circular geometry, once I have this I will use an FEA program that will calculate how the geometry changes in the blood flow environment

If you were to do that in real life, it could stretch (or tear) the red boundary segment opposite it and that would change area

I understand how there would be any stretching if the perimeters of the red and black line remain constant. I still don't see why this wouldn't work. A method to preserve distance between the two lines has already been established using the IPDM code, so in order to conserve area why couldn't the area of each section be adjusted to match the area of the section before the transform?

I have a feeling you still imagine the region between the curves as consisting of trapezoidal sections. For trapezoidal partitions to be possible, the black curve would have to be approximately parallel to the nearby red curve everywhere. But I see parts of the curves in your picture where this is distinctly not the case.

When referring to stretching, I was talking about the earlier idea of rotating different sections of the curves, though in my mind, the sections were quadrilaterals, not specifically trapezoids. To bend the black curve into a circle, you would have to rotate a quadrilateral section about the point on the black curve where neighboring sections meet, but that would exert stress on the point on the red curve where the same two sections meet.

I'm afraid I've lost faith in IPDM as the way to define corresponding points on the red and black curves. As I said earlier, I'm not confident that it maps point A on the black curve 1-1 to a point B on the red curve. I'm also not sure that all such chords AB do not cross. If chords do cross, those chords cannot map to distinct chords after the black curve is bent into a circle, contrary to your goals. You're hoping, I think, that all the chords map to distinct radial lines under that transformation...

All I need is the unpressurized circular geometry,

The "unpressurized circular geometry" is new terminology. How does the black boundary become circular if not by the application of some sort of pressure? And I thought you said earlier that it was preciesly the bloodflow that makes it circular in the first place, so what do you mean by "how the geometry changes in the blood flow environment?

I have a feeling you still imagine the region between the curves as consisting of trapezoidal sections.

If trapezoidal sections would not work, then would another shape like some sort of quadrilateral work?

I've lost faith in IPDM as the way to define corresponding points on the red and black curves.

How could you tell if it wasn't mapping 1-1, or if the chords were indeed crossing? Would it help if the curves were altered so that they had the same number of points? If not what is a better way to track the distance between the points on the two curves?

How does the black boundary become circular if not by the application of some sort of pressure?

I am just forcing it to be circular prior to it being pressurized. I figured that it would just be easier to do the transform that way.

Can you see any way in which I can do the transform that I want using the given specifications?

Can you see any way in which I can do the transform that I want using the given specifications?

Well, here's another idea. You're already using imfreehand or similar to trace the boundary of the curves. It seems to me that it would be essentially the same manual effort to use IMLINE or similar to draw lots of closely spaced line segments connecting the black curve to the red curve (essentially the pre-chords that we've been talking about). You would do this according to your own visual idea of which points on the black and red curves correspond, as opposed to using IPDM.

Once you've done that, you map all the points on the black curve to the desired circle. You would probably do this so that the distances between neighboring black points are all the same post-transformation as they were pre-transformation. You then use the lengths of the pre-chords to construct post-chords of the same length, except that the post chords are all directed inward toward the center of the black circle. The inner endpoints of the post-chords define the points on the transformed red curve and you're done.

Assuming you pack the pre-chords densely enough, you should get a pretty good approx of the curve you're looking for. Assuming, further, that a transform preserving both chord length and area does exist, I would expect this manual approach to do a fair bit better than the 8% area agreement you got via IPDM. If you don't get good area preservation, it would suggest to me that the requirements of chord length preservation and area preservation are in conflict. In either case, I wonder if you should really be caring about preserving distance and area so precisely. It sounds to me like you're planning to use this anyway as nothing more than an ad hoc initial approximation to something that's going to be refined later using something more principled (FEA?).

Could I preserve area by multiplying the transformed curves by sqrt(A_orig/A_new)?

I don't know what it means to "multiply a curve" by something. If you mean to scale the chord lengths by that, then perhaps, but then chord-length won't be preserved.