I'm not talking about the Hough Circle Transform (where you replace the representing eqns, hence the parameters with a 3D set) but a Hough Line Transform.

My solution to this:

A hough line transform maps lines to a single point in (p,theta) space. This point contains the number of points lying on the line in (x,y) space. Conversely, a point in (x,y) space maps to a curve in (p,theta) space, this is because that point could belong to an infinite number of lines.

  Now,

Imagine a circle,

Now the tangent is perpendicular at the point of contact with the radius. Since the circle is a symmetrical object, this applies to every point on it.

This means essentially, that a circle is constructed of such infinite lines which are perpendicular to the radius, the point of contact with the tangent of which is mapped to the bin in (p = radius, theta). So essentially, the circle in (X,Y) space becomes a line in (p,theta) space.

  Am I correct in saying this?

Then could we apply a hough transform on the hough matrix itself to detect circles in the original (X,Y) space?

(I know academics can be ruthless but newb here so please be kind when tearing me a new one)

I'm not talking about the Hough Circle Transform (where you replace the representing eqns, hence the parameters with a 3D set) but a Hough Line Transform.

My solution to this:

A hough line transform maps lines to a single point in (p,theta) space. This point contains the number of points lying on the line in $(x,y)$ (cartesian) space. Conversely, a point $\mathbf{A}=(x,y) \in \mathbb{R}^2$ maps to a curve in polar $(r,\theta)$ space. This is because a point belongs to an infinite number of lines. Now, imagine a circle:

where the tangent $\mathbf{AB}$ is perpendicular to the line connecting the center and the point of contact ($\mathbf{NA}$). Let's call this line the radial line. Since the circle is a symmetrical object, this applies to every point on it.

This means essentially, that a circle is constructed of such infinite tangent lines which are perpendicular to the radial line. The tangents map to the bin in polar space: $\mathbf{A} \rightarrow (r, \theta)$. So essentially, the circle in $(x,y)$ space becomes a line in $(p,\theta)$ space. Am I correct in saying this?

Then could we apply a Hough transform on the Hough matrix itself to detect circles in the original $(X,Y)$ space?

(I know academics can be ruthless but newbee here so please be kind when tearing me a new one.)