Image Processing Reference
In-Depth Information
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Fig. 10.32. The same as in Fig. 10.31 but the graphs on the left represent angle estimations
on the line joining point 5 to the center (green) and on the line joining point 6 to the center
(blue) marked in Fig. 10.15. The graphs on the right represent the corresponding |I 20 | (solid)
and I 11 (dashed), respectively. The measurements on the noisy line are at the bottom.
through all discrete φ values of the grid A established in step 3. For each such
φ k , one b is obtained, which is rounded off towards the closest discrete b value
of the grid cells. The final A ( φ m ,b m ) obtained after the voting procedure is the
Hough transform for lines.
5. A long line causes a high peak in the ( φ, b ) plane, because every long line in
the ( x, y )-plane generates a point in the ( φ, b ) plane. Accordingly, the position
of each such peak yields a specific ( φ, b ) parameter that represents an infinitely
long line in the ( x, y )-plane.
The Hough transform procedure above can be simplified further by voting only for
one parameter cell ( φ, b ) per edge point ( x, y ) in step 4, the most likely direction
φ k
θ ( x, y ) and its corresponding b value closest to a grid node. The robustness can
be improved by smoothing the accumulator before finding the peaks. Accordingly,
the votes cast for the line ( φ j ,b j ) are
A ( φ j ,b j )=
l
δ [ θ ( x j + x l ,y j + y l )
θ ( x j ,y j )]
(10.95)
where ( x l ,y l ) are the edge points along the line given φ j ,b j having the direction
θ ( x j ,y j ). The sum represents a filtering that counts the occurrence of the edges
encountered along the line. The occurrence is w.r.t. edges and edge directions, i.e.,
if there is a directional conflict the value of the cast vote is zero.
Nonuniqueness of Line Directions
In step 1 we have a model of a line for every fixed pair of parameters of ( ϕ, b ).
However, the model parameters and the lines do not uniquely correspond to each
other because by substituting ( φ + π,
b ) in Eq. (10.91), one sees that this parameter
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