Additional Lines

The condition on the crossratio used in the above section is only a necessary con-dition to identify the structure we are looking for. Usually a number of adcon-ditional line-sets with similar crossratios exist in any given image. In MOVIS, I therefore decided that finding two stripes (four lines) with given crossratio is not sufficient evidence for a_::::::zebra_::::::::::crossing_:zebra-crossing (the same argument could be made for any other repeating structure). Instead, a minimum of three stripes (six lines) is required.

Luckily it is relatively easy to identify additional _::::line:::::::::::segments_:line-segments by using an adaption of the canonical-frame approach described in Section 4.5.1.2.

Within the canonical frame, the locations of all other lines potentially belonging to the structure in question are known. These can then be backprojected into the image to get the approximate position of additional stripes in the image. If corresponding stripes are found, these are then added to the set of four lines to form a hypothesis.

In addition, this also means that the location of the stripes’ vanishing line is known (the backprojection::::::::::::::::_:back-projection of a line at infinity), in Section 5.4 this will be used for verification — the backprojected

::::::::::

vanishing

:::::line

:vanishing-line should coincide with the horizon of the image. Figure 5.10 shows an example where the position of a minimum of three lines within the image is sufficient to predict the

5.3.3 Additional Lines 125

PSfrag replacements

x,` x<sup>0</sup>,`⁰ x⁰⁰,`⁰⁰

T

T₁ T₂

Figure 5.11: Decomposition of T into two transformations T = T₂T₁, using an intermediate canonical frame representation.

position of an infinite number of additional lines. In the following an alternative approach to the ones described in Section 4.5.1.2 is given.

The most accurate way to achieve the

::::::::::::::::

backprojection

:back-projection is to find a (5 degrees of freedom) transformation T^T from a canonical frame (of, say, horizontal lines of known position) into the image that minimises the distance between the proposed and the measured lines. Once this transformation is found, it is then easy to predict other lines by calculating T^T`<sup>00</sup><sub>i</sub>, where `⁰⁰_i = (0,1, c⁰⁰_i)^T is one of the lines in the canonical frame (see Figure 5.11). By the same idea, the vanishing_::::::::::

::::line_:vanishing-line can be found by calculating T^T(0,0,1)^T. As for many of the problems which we encountered in Section 4 there is again no closed-form solution to the problem of finding T^T.

A somewhat similar but much faster approach finds the inverse transformationT^−T such that the distance between the proposed and measured lines becomes minimal within the canonical frame (instead of the image). A very efficient approximation for this transformation exists under the assumption of small errors. It is then possible to decomposeTinto two matricesT₁and T₂ for which we can solve separately. T^−T₁ transforms the lines into an intermediate canonical frame in which all lines are (as near as possible) horizontal. T^−T₂ is the transformation into the final canonical frame, in which the individual lines will end up in definite positions (compare Figure 5.11).

It is immediately clear from the above that the matrix in Equation (4.28) could be used as T₁, as could be any other transformation uniquely defined by the

::::::::::

vanishing

::::::

pointvanishing-point; a nicer example is the matrix

T₁ =

126 Grouping

Figure 5.12: Monte-Carlo simulation of vanishing-line calculation using the canonical-frame algorithm for three typical constellations. Notice that small errors in the vanishing-point coincide with big errors in the orientation of the vanishing-line and vice-versa; compare also Figure 4.9, where the same lines were used to calculate the vanishing-point.

if x=y= 0. All that remains is to find the 3 degrees of freedom transformation T₂ =



 1 0 0 0 1 t_y 0 py s



 (5.3)

for which a closed form solution exists.

The decomposition of T into T₁ and T₂ is strictly speaking only possible if either all lines `⁰_i in the intermediate canonical frame are exactly horizontal, or if py ≡0, since T₂ with p_y 6= 0 will change the angle of all non-horizontal lines. However, if the assumption that all lines were originally parallel is true, and if the vanishing_::::::::::

::::::

point_:vanishing-point used to determineT₁ was calculated using one of the methods described in Section 4.4, then we can also guarantee that the lines in the interme-diate frame are as horizontal as possible — any deviation must be an error in the measurements, which should be corrected — and the change in the angle will be small (and can in fact be ignored). The results of a Monte-Carlo simulation in Fig-ure 5.12 show that the above approximation works quite well, although it is clear that the small-error assumption is not valid anymore for the resulting lines.

It is quite instructive to have a closer look at the matrix T=T₂T₁. It was already mentioned that it has 5 degrees of freedom. These determine uniquely (up to scale) the last two rows — the first row can be chosen arbitrarily as long as the matrix does not become singular. We also see, when

::::::::::::::::

backprojecting

:back-projecting the line at infinity T^T(0,0,1)^T, that the third row is nothing but the vanishing line in the image, fixing 2 degrees of freedom. By the same argument we see that the second row is the

:::::::::::::::::backprojection

:back-projection of the horizontal line through the origin (0,1,0)^T, leaving 1 degree of freedom to be fixed. What is the remaining degree of freedom used for? It is easy to see that any line passing through the vanishing point in the image (and therefore horizontal in the canonical frame) can be constructed as a linear combination of the second and third row by calculating T^T(0, b,1−b)^T. The last degree of freedom fixes where in the image a line with given

5.3.4 Merging Hypotheses 127