Page - 108 - in Proceedings - OAGM & ARW Joint Workshop 2016 on "Computer Vision and Robotics“

transformationsasexplainedabove: Π1(x,y) = Π(x,y) (2) Π2(x,y) = { Π(W,y)−Π(x,y) ifx 6=W, Π(W,y) ifx=W. Π3(x,y) = { Π(x,H)−Π(x,y) ify 6=H, Π(x,H) ify=H. Π4(x,y) =          A(x,y) ifx 6=W,y 6=H, Π(x,H)−Π(x,y) ifx=W,y 6=H, Π(W,y)−Π(x,y) ifx 6=W,y=H, Π(x,y) ifx=W,y=H. A(x,y) = Π(W,H)+Π(x,y)−Π(W,y)2−Π(x,H) (3) W indicates the last validx index of a row andH the last validy index of a column. Care has to be taken, if any index lies on the edge: Here, some components simply refer to the same area and are equal. Now,wemakeuseoftransformationswhichthesummationtermsareinvariant to: weareinterestedin the difference between maximum and minimum of the summation. Thus, adding a constant factor to allelementswithin the integral imagewillnotaffect thedifferencebetweenmaximumandminimum. When it comes to (3), the term Π(W,H) clearly is constant, neither depending on index variablex nor on y. As a result, it can be omitted for the discrepancy norm calculation. Note that this needs to be compensated in the others cases of Π4, too. Taking the constraint for the indexes into account, Equations (4) and (5) areobtained: Π˜4(x,y) =          A˜(x,y) ifx 6=W,y 6=H, −Π(W,y) ifx=W,y 6=H, −Π(x,H) ifx 6=W,y=H, 0 ifx=W,y=H. , (4) A˜(x,y) = Π(x,y)−Π(W,y)−Π(x,H). (5) 2.1. Reducing the compareoperations So far, the discrepancy norm in 2D has been reduced to computing a single integral image and ex- pressing the other forms based on this single one. It still requires 8 compare operations per pixel: one for the minimum, one for the maximum and this has to be done four times for the different com- ponents. Some of these operations are redundant if we concentrate on a specific row, meaning y is constant. Applying this method to Π2 of Equation (2), we obtain equations (6) and (7) which differ only by a constant and a sign. The negative sign will swap minimum and maximum. As a result, the second component can be deduced with simple operations that are only necessary at the end of each 108