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No, the tilted integral image is not needed for the Viola Jones algorithm. In fact, the original Viola Jones Paper only uses upright (=untilted) Haar-like features.

In An Extended Set of Haar-Like Features for Rapid Object Detection, Rainer Lienhart , Jochen Maydt, ICIP 2002 the authors showed that extending the original feature set with those tilted Haar-like features improves performance, i.e. the false positive rate was reduced.

AFAIK the Haar-cascade detector in OpenCV uses both normal and tilted Haar-like features by default. I suppose other object classes benefit more from tilted features than faces.