Reconstruction and Approximation - Vision with Direction - page 108

f can be seen as a scalar-valued image defined on an N -dimensional space repre-
sented by r .If f has finite frequencies, i.e., F is zero outside of a limited volume,
∂f
∂x j will also have finite frequencies. This is because the latter is equivalent to iω j F
in the frequency domain, which will evidently be zero in the same places (volume)
as F is zero. Thus, the partial derivatives of the continuous function can be faithfully
represented by sampling these functions on the same grid that samples the original
image. To compute the partial derivatives, the operator

∂

∂x i is applied to Eq. (8.15),
and it is moved past the sum because partial derivation is a linear operator:

=
l

∂f ( r )
∂x i

f ( r l ) ∂μ ( r

− r l )

, i, j :1 ··· N (8.17)

∂x i

At this point, we must require that the interpolation function μ is differentiable.
Assuming this, the linear operator can be absorbed by the interpolator function, and
the resulting continuous function can be discretized too. Because ∂ ∂x has the same
finite frequencies as f , we can also sample it on the same grid r k , yielding

=
l

∂f ( r k )
∂x i

f ( r l ) ∂μ ( r k −

r l )

, i, j :1 ··· N (8.18)

∂x i

This is a discrete scalar product of two functions, the first of which is the original
discrete image. The second function is the discretized interpolation function after it
has absorbed the partial derivation operator. As can be seen from the right-hand side
of Eq. (8.18), the approximation can be realized by a convolution when the values of
∂f
∂x i

at all points of the grid have to be computed.

Arbitrary Translation or Shifting

Translating an image f ( r ) with Δ r is another frequently needed operation. Here,
Δ r does not have to be a multiple of the sampling period that is assumed to be 1.
For example the translation could be Δ r =(0 . 25 , 0 . 75) T

in 2D. In the continuous

domain, this is equivalent to a coordinate transformation,

r = r + Δ r (8.19)

so that the translated version of f ( r ) is given by f ( r − Δ r ). Because the FT of the

), translation is a band-preserving operator.
Without loss of generality, one can assume that all components of the vector Δ r
are in the open interval ]0 , 1[ because otherwise the integer translation is performed
first. Integer translations are conveniently implemented as a permutation. Using Eq.
(8.15), the remaining noninteger translation is implemented by sampling the recon-
structed and shifted signal at the grid points as

translated image is exp( − iΔ r T ω ) F ( ω

Δ r )=
l

f ( r j − f ( r l ) μ ( r j − Δ r − r l ) (8.20)