Direction in Curvilinear Coordinates - Vision with Direction - page 217

φ j

R j

Fig. 11.4. A collection of edges describing an object to be identified is shown in green .
An edge with the direction φ j and the distance R j to the reference point, marked with a × ,
are shown in magenta . The latter information is used by GHT to encode the position of the
prototype. All edge matches occuring in the target and the image result in a vote cast to the
corresponding reference point in the image

presence of target curves. It consists of a table look-up indexed by the direction of
the target curve and a voting procedure, [12, 111]. The table entries are discrete edge
directions of the edges of the prototype. To each tabulated direction corresponds a list
of edge positions, expressed relative to a unique reference point (origin) of the pro-
totype. There is a direction associated with each contour edge, both in the image and
in the prototype. Typically the contour points and their directions are obtained after
applying a threshold to the gradient-filtered image and the quantized gradient direc-
tions. This is because edges participating in the GHT poll are binary, i.e., they either
exist with a unique direction, or they do not. Given the table, one can easily construct
the contours of the prototype, or inversely construct the table, given the contours of
the prototype. The contours of a transistor, an example prototype, its reference point,
an edge, and its ( φ j ,R j ) parameters used in a GHT table are shown in Fig. 11.4.
Accordingly, the GHT direction table is equivalent to a 2D template containing the
contour points and their directions. The GHT procedure consists of a 2D nonlinear
filtering identical to sliding a template over the contour-direction image and counting
the direction matches per template position. Thus, the vote accumulator, A ,istwo-
dimensional and has usually the same resolution as the image itself in this process.
However, it may be necessary to estimate more than two parameters, e.g., transla-
tion, rotation, and scale. This can be handled by the 2D accumulators too, provided
that one can fix the other parameters by use of a numerical optimization procedure,
such as gradient descent. The peaks in the accumulator A will indicate the positions
where the template presence is most likely. The accumulator votes are given by