Got a book “Mathematical Methods in Image Reconstruction” by Frank Natterer and Frank Wübbeling.  It brings about my old thought.

Say here is a picture of a box, or something rectangular.  And let us assume that we roughly know position and attitude of the box before examining the picture.  When we want to measure the position more accurately, sometimes we take the following procedure:

1. take orthogonal projection of (a certain interesting region of) the image to have 1 dimensional signal (integrate along parallel lines, in other words),
2. apply derivative filter (edge detector) to the 1D signal,
3. find peaks in the filtered signal.

Then we will know the position of 2 sides of the box.

But what if we do not know a priori the attitude of the box.  We can not decide in which direction we should project the image.

One thing we can do for this case, if we still stick to the projection idea, is to project the image in many directions.  And pick results that give the strongest edge filter response.  Range of projection angle might be [0, 180) degrees if we know nothing about the pose of the box.

If we describe the procedure (up to projection) in mathematical way, we will reach 2D Radon transform.

In computer vision’s realm, there is a well-known technique for geometric feature extraction called Hough transform.  In Hough transform for line detection, a most ineteresting step is to plot set of points in (r-$\theta$) plane for each feature point in the original image (x-y) plane.  And I think this plotting step is essentially the same as the Radon transform.

I learned Hough transform when I was with computer vision company.  But I did not know of Radon transform until I learned computer tomography things.

Some descriptions of Hough transform I met do not mention Radon transform, which is unfortunate because

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4 Responses to Radon, Hough, projection to 1D

1. voodoo child says:

And one should also remember not to place the box under an elephant.

• azumih says:

mmm. Okay, please drop me some more comments again. I will finish to say why it is a pity soon.

2. voodoo child says:

One should of course remember that pink elephants are acceptable because they are extremely light.

They are also very happy and positively delight in the Hough transform.

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