Reconstruction and Approximation - Vision with Direction - page 111

in Eq. (8.15), with Qr , where Q is a rotation matrix :
f ( Qr )=
l

f ( r l ) μ ( Qr −

r l ) (8.22)

As an example of rotation matrices, we mention
Q = cos θ sin θ
−

(8.23)

sin θ cos θ

which is the matrix used to rotate 2D images. The new coordinates r can be sampled
at a desired rate yielding r k . If the image f is band-limited so is f ( Qr ) because a
rotation merely rotates the spectrum:
f ( Qr k )=
l

f ( r l ) μ ( Qr k −

r l ) (8.24)

Inspecting this equation shows that rotation is achieved by a scalar product be-
tween the image values on the original grid and the interpolation function rotated
via Qr k to align the axes of the new coordinates r , but sampled at the original
grid points. The result of the scalar product is placed in the new grid at the location
r k . In other words, here too, the rotation operation is absorbed by the interpolation
function. The interpolator is aligned to the new grid coordinates, but it is sampled
at the old grid points to generate the scalar product coefficients. A scalar product
between the latter and the function samples on the old grid delivers the rotated func-
tion value on the new grid. The sampling rate on the new grid must be appropriately
chosen so it does not lose some spectral components. This is because a 2D image f
can be faithfully sampled on a rectangular grid only if F is confined to the square
[

π, π ] for an N D
image), according to the Nyquist theorem. Rotating f , and thereby F , will require a
denser sampling rate on the new grid if all spectral components are to be retained.
In Fig. 8.9 we illustrate the frequency behavior of two example rotations. Using
the original density will result in a rotation aliasing caused by certain components,
see the “magenta” zones, which may not be acceptable to some applic a tions. For ex-

− π, π ] × [ − π, π ], (and to the hypercube [ − π, π ] × [ − π, π ] × ... [ −

ample, a rotation with θ = π/ 4 may require a sampling grid that is √ 2 times denser
than the old grid if all frequency components are to be ret ai ned. For square-shaped

2D images, a frequency band magnification with a factor √ 2 is the largest factor for
any rotation. Alternatively, the area between the blue circle and the boundaries of
the red square, representing the frequencies that risk causing aliasing, can be simply
suppressed by a lowpass filtering as this is an area which could be accepted as noise
by the application at hand. Such a filtering can be incorporated directly into the rota-
tion operator and amounts to having a larger kernel to reconstruct f from its samples.
Similarly, in N D images having the sam e si ze in all coordinate axes, the maximum

density magnification due to rotation is √ N .
Affine warping of an image. Evidently, the same signal processing principles
and reasoning used to discretize rotation apply to discretizing affine coordinate trans-
formations , also known as affine warping . The old coordinates are replaced by