projections, include a tilt series alignment by cross correlation, as just described in
this chapter and not in its modified version proposed by [4]. When this approach is
applied to an experimental TET tilt series, the cross-correlation between two consecutive images, I i and I i þ 1 is computed and the corresponding displacements
(D i;i þ 1 ) identified. This estimated displacement is a good estimation of the real
shifts since in a tilt series, when the increment on the tilt angle is low, two consecutive images are almost the same (except for a shift, as illustrated in Fig. 7.2).
Once the consecutive shifts are identified, we may find the relative shift between
any two images i and j. For that, all images are translated to be centered with respect
to the 0-tilt image. Let us illustrate this by representing a tilt series from Àh max to
h max degrees with increments of Dh degrees (e.g., from −60° to 60° in steps of 1°),
with indexes from Ài max to i max ði max ¼ h max =DhÞ. The shift needed to align an
image with negative tilt to the 0-tilt image is just the accumulation from Ài to 0 of
all the consecutive shifts:
D Ài;0 ¼ D Ài;Ài þ 1 þ D Ài þ 1;Ài þ 2 þ Á Á Á þ D À1;0 ¼
X À1
j¼À1
D Àj;Àj þ 1
On the other side, for positive tilts, we need to accumulate the shifts in reverse
order
D i;0 ¼ ÀD iÀ1;i À D iÀ2;iÀ1 À Á Á Á À D 0;1 ¼ À
X iÀ1
j¼0
D j;j þ 1
With these translations we may produce a new set of centered images ~ I i .
Unfortunately, reality is not that easy for several reasons:
• Experimental images are extremely noisy, resulting in a noisy correlation
function whose maximum may be spuriously misplaced (see Fig. 7.4).
• If one of the images is in-plane rotated with respect to the other, the correlation
function is distorted with respect to the unrotated correlation function. This
distortion may produce a totally incorrect estimation of the displacement vector
(see Fig. 7.5).
• Local differences in the illumination conditions or the presence of a persistent
illumination pattern totally distorts the correlation pattern (see Fig. 7.6).
Acknowledging these difficulties, we may try to robustly estimate the shifts
between any two images. First, we can bandpass filter the images to remove any
persistent illumination pattern, smooth local illumination variations (low frequency)
as well as noise and small image details (high frequency) unnecessary to globally
align two images. Once the images are bandpass filtered, they can be safely
down-sampled, to reduce their size and speed-up calculations. Then, we may
construct a polar 2D correlation function (the correlation function when the images
are expressed in polar form). The location of the maximum in this polar correlation
map indicates the optimal rotation [7]. In this way, we can identify both rotations
and translations by alternating between looking for the best shift, then for the best
rotation, and iterating several times this sequence till convergence.
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A. Verguet et al.
this chapter and not in its modified version proposed by [4]. When this approach is
applied to an experimental TET tilt series, the cross-correlation between two consecutive images, I i and I i þ 1 is computed and the corresponding displacements
(D i;i þ 1 ) identified. This estimated displacement is a good estimation of the real
shifts since in a tilt series, when the increment on the tilt angle is low, two consecutive images are almost the same (except for a shift, as illustrated in Fig. 7.2).
Once the consecutive shifts are identified, we may find the relative shift between
any two images i and j. For that, all images are translated to be centered with respect
to the 0-tilt image. Let us illustrate this by representing a tilt series from Àh max to
h max degrees with increments of Dh degrees (e.g., from −60° to 60° in steps of 1°),
with indexes from Ài max to i max ði max ¼ h max =DhÞ. The shift needed to align an
image with negative tilt to the 0-tilt image is just the accumulation from Ài to 0 of
all the consecutive shifts:
D Ài;0 ¼ D Ài;Ài þ 1 þ D Ài þ 1;Ài þ 2 þ Á Á Á þ D À1;0 ¼
X À1
j¼À1
D Àj;Àj þ 1
On the other side, for positive tilts, we need to accumulate the shifts in reverse
order
D i;0 ¼ ÀD iÀ1;i À D iÀ2;iÀ1 À Á Á Á À D 0;1 ¼ À
X iÀ1
j¼0
D j;j þ 1
With these translations we may produce a new set of centered images ~ I i .
Unfortunately, reality is not that easy for several reasons:
• Experimental images are extremely noisy, resulting in a noisy correlation
function whose maximum may be spuriously misplaced (see Fig. 7.4).
• If one of the images is in-plane rotated with respect to the other, the correlation
function is distorted with respect to the unrotated correlation function. This
distortion may produce a totally incorrect estimation of the displacement vector
(see Fig. 7.5).
• Local differences in the illumination conditions or the presence of a persistent
illumination pattern totally distorts the correlation pattern (see Fig. 7.6).
Acknowledging these difficulties, we may try to robustly estimate the shifts
between any two images. First, we can bandpass filter the images to remove any
persistent illumination pattern, smooth local illumination variations (low frequency)
as well as noise and small image details (high frequency) unnecessary to globally
align two images. Once the images are bandpass filtered, they can be safely
down-sampled, to reduce their size and speed-up calculations. Then, we may
construct a polar 2D correlation function (the correlation function when the images
are expressed in polar form). The location of the maximum in this polar correlation
map indicates the optimal rotation [7]. In this way, we can identify both rotations
and translations by alternating between looking for the best shift, then for the best
rotation, and iterating several times this sequence till convergence.
188
A. Verguet et al.