PhD thesis - Institute for Computer Graphics and Vision - Graz ...

4.2. Plane Sweep Depth Estimation 47
4.2.2.2 Normalized Correlation Coefficient
The widely used (zero-mean) normalized correlation coefficient for window-based local
matching of two images X and Y is (where ¯ X and ¯ Y denote the means inside the rectan-
gular region W)
ZNCC =
�
i∈W (Xi − ¯ X) (Yi − ¯ Y )
�� i∈W (Xi − ¯ �
X) 2
i∈W (Yi − ¯ Y ) 2
which is invariant under (affine linear) changes of luminance between images, but relatively
costly to calculate. Using integral images the ZNCC can be calculated in constant time
regardless of the correlation window size [Tsai and Lin, 2003], since
ZNCC =
� XiYi − ( � Xi) ( � Yi) /N
��� X2 i − ( � Xi) 2 � �� /N Y 2
i − ( � Yi) 2 �
/N
.
From the above formula it can be seen that five integral images are requires to calculate
the ZNCC: the integral image for � Xi, � Yi, � X 2 i , � Y 2
i and finally � XiYi. The
precision requirement for the higher order sums is 8 + 8 + log 2 512 + log 2 512 = 34 bit for
512 × 512 source images. The 32 bit floating point format of current GPUs has a mantissa
of 23 bit and artefacts due to precision loss may occur. Figure 4.2 illustrates the reduced
precision by depicting a ZNCC error image generated in software on a CPU and another
one computed on the GPU. An increasing loss of precision can be seen towards the lower
right corner of the image. Since the integral image generations starts from the upper left
corner, the lower right portion has the highest precision requirements within the integral
image.
Note that the precision requirements for the simple sums � Xi and � Yi are 26 bit for
8 bit images with 512 × 512 pixels resolution. By subtracting the image mean in advance
from the source image two additional precision bits can be saved: one by halving the
magnitude of the source values and another one by exploiting the sign bit in the integral
image.
Instead of creating full integral images, which allows box filtering with arbitrary window
sizes, it is usually sufficient to sum the values with a given specific window, since we do
not vary the aggregation window size during similarity score computation. Accumulation
of larger windows can be performed using a similar recursive doubling scheme as used for
integral image generation. Consequently, the precision requirements on the target buffer
storing the aggregated values depend on the window size, and these are substantially lower
than the requirements for integral images.
,