Inferring real-world coordinates from pixel positions

Hi all. I have an image with a few control points (CPs), for which I know both the pixel coordinates and their corresponding real-world UTM coordinates, including elevation. I’d like to use these control points to estimate the UTM coordinates of another feature in the image, for which I only have the pixel coordinates. I’ve explored some image rectification functions, but I’m not sure they are the right tool for this task. Does anyone have suggestions? I've attached the image — the control points are marked in green, and the feature I'm trying to locate is indicated by the red line. The pixel positions (u, v) and the corresponding UTM coordinates (x,y) are reported below. Thanks !

u = [1333 1265 1227 1481 937 842 667];

v = [1066 910 822 499 558 539 532];

x = [ 498899 498895 498892 498854 498734 498682 498584];

y = [4915388 4915389 4915391 4915430 4915441 4915442 4915425];

z = [6.268 6.270 6.264 11.512 4.764 6.698 6.705];

Answer 1

[William Rose]

@Francesco,

The projection equation from world (x,y,z) to image (u,v) is

We can estimate A from the control points:

% Control points in image coords and world coords
u = [1333 1265 1227 1481 937 842 667];
v = [1066  910  822  499 558 539 532];
x = [ 498899  498895  498892  498854  498734  498682  498584];
y = [4915388 4915389 4915391 4915430 4915441 4915442 4915425];
z = [6.268 6.270 6.264 11.512 4.764 6.698 6.705];
mx=mean(x); my=mean(y);     % mean x,y values
xzm=x-mx; yzm=y-my;         % x,y with mean removed (zm=zero mean)
% Camera projection matrix 
imagePoints=[u',v'];
worldPointszm=[xzm',yzm',z'];
A=estimateCameraProjection(imagePoints,worldPointszm);  % camProjection

I removed the mean values from x and y before computing A, to improve numeric accuracy. We can add the mean x,y values back later.

Each red point in the image is at sea level, plus or minus the tide. Let us assume that we know the z value (z=zc) for an unknown world point, and we want to find the x,y values for the unknown world point. Then the equation above becomes

Rearrange equation above:

The last equation above is what we need to convert (u,v) to (x,y), when z=zc is known. If the vector obtained by doing the left hand side above does not have the 3rd element=1, then divide the vector by the third element to get a normalized vector.

I estimated pixel values (u,v) for three points along the red shoreline, and I assume zc=0 along the shoreline. I used the formula above to compute the world coordinates (x,y) of the image points. Script attached. I plotted the green points and the red shoreline points in world coordinates in 3D. The screenshot below shows a view of the points from directly overhead.

[William Rose]

The image is from Spiaggia Sturla, Genova, Italia. Specify UTM zone 32 to convert the UTM easting, northing values to longitude, latitude.

[Francesco]

thank you so much William, it worked fine !

[Matt J]

@Francesco, if so you should Accept-click the answer.

[William Rose]

@Francesco, you're welcome. @Matt J, thank you.

Answer 2

[Matt J]

You need a stereo camera,

https://www.mathworks.com/discovery/stereo-vision.html