  We reveal that the Analytic Signal phase, and its gradient have a hitherto
unstudied discontinuity in $2-D $ and higher dimensions. The shortcoming can
result in severe artifacts whereas the problem does not exist in $1-D $
signals. Direct use of Gabor phase, or its gradient, in computer vision and
biometric recognition e.g., as done in influential studies
\cite{fleet90,wiskott1997face}, may produce undesired results that will go
unnoticed unless special images similar to ours reveal them. Instead of the
Analytic Signal phase, we suggest the use of Linear Symmetry phase, relying on
more than one set of Gabor filters, but with a negligible computational add-on,
as a remedy. Gradient magnitudes of this phase are continuous in contrast to
that of the analytic signal whereas continuity of the gradient direction of the
phase is guaranteed if Linear Symmetry Tensor replaces gradient vector. The
suggested phase has also a built-in automatic scale estimator, useful for
robust detection of patterns by multi-scale processing. We show crucial
concepts on synthesized fingerprint images, where ground truth regarding
instantaneous frequency, (scale \& direction), and phase are known with
favorable results. A comparison to a baseline alternative is also reported. To
that end, a novel multi-scale minutia model where location, direction, and
scale of minutia parameters are steerable, without the creation of
uncontrollable minutia is also presented. This is a useful tool, to reduce
development times of minutia detection methods with explainable behavior. A
revealed consequence is that minutia directions are not determined by the
linear phase alone, but also by each other and the influence must be corrected
to obtain steerability and accurate ground truths. Essential conclusions are
readily transferable to $N-D $, and unrelated applications, e.g. optical flow
or disparity estimation in stereo.
