E09 · Electron acquisition / diffraction

Raster Scan & Drift Recovery

Can aligning repeated frames undo the distortion that drift causes in a scanned image? Only partly. Aligning frames on features that vary in both directions corrects the shift from one frame to the next. A featureless surface gives nothing to align on, and parallel stripes leave the shift along them undetermined. One shift per frame also cannot fix distortion that happens within a frame.

Acquire a stack of frames, estimate how far each frame has shifted from its counts and compare the plain average with the aligned result. Then try to break the estimate with stripes, a featureless surface, slow scanning and a narrow search window.

Assigned within-frame motion $0\,\mathrm{nm}$

Accepted-frame common coverage $0\,\mathrm{\%}$

Frames used · includes first frame $0$

Registration status $\text{—}$

More measurements and target

Stored frame time $0\,\mathrm{s}$

Stored acquisition duration $0\,\mathrm{s}$

Assigned first-to-last motion $0\,\mathrm{nm}$

Incident-count relative noise · all frames $0\,\mathrm{\%}$

Stored 6 nm grating sampling $\text{—}$

Estimated first-to-last horizontal shift $0\,\mathrm{nm}$

Estimated first-to-last vertical shift $0\,\mathrm{nm}$

Lowest accepted image correlation $0$

Effective committed search radius $0\,\mathrm{\mathrm{pixel}}$

Experiment task $\text{—}$

Simulated examples; each plot identifies its data source.

More examples

Show image-estimated alignmentSwitch between the raw average and the counts aligned in the last alignment step.Pixel spacing

p

4\,\mathrm{nm}

Pixel dwell

t_p

20\,\mathrm{\mu s}

Number of frames

N_f

4

Horizontal stage drift

v_x

12\,\mathrm{nm\,s^{-1}}

Advanced settings

Even rows, then odd rowsPlan the next acquisition; change the time assigned to each row.Counting noisePlan seeded count noise for the next stack; stored counts stay fixed.Line flyback

t_b

250\,\mathrm{\mu s}

Vertical stage drift

v_y

4\,\mathrm{nm\,s^{-1}}

Line deflection error amplitude

A_l

0\,\mathrm{nm}

Specimen: landmarks / stripes / uniform

S

0

Registration search radius

R

12\,\mathrm{\mathrm{pixel}}

Learning goal and model limits

Scan an image and correct its drift

Raster Scan & Drift Recovery

Resolve the 6 nm grating in a stored stack of at least 8 frames. Every frame must be accepted and the alignment shown, the assigned within-frame motion must stay below 1 nm, and common coverage must stay above 90 percent. A high correlation alone does not prove the result is accurate.

Model assumptions

The specimen is an assigned analytic pattern, scanned by a 1 nm Gaussian probe with a fixed 2 pA beam. Shifts are found by normalized correlation of low-pass filtered images against the first frame; the estimator never sees the true drift or the specimen reference. The acceptance thresholds are rules of thumb for teaching. Accepted frames are resampled bilinearly from their original counts, and areas not covered by every frame are masked. First-frame distortion, periodic ambiguities, undersampling and within-frame shear remain in the result. Count noise is Poisson below 40 counts and Gaussian above that. Detector read noise, dose damage and scan-coil dynamics are not modeled.

Start with the essentials

Focus question: When can a shift be worked out from the images alone?
One-sentence intuition: Features that vary in two directions fix the shift in both directions. A uniform surface carries no shift information, and stripes leave the shift along them undetermined. One translation per frame cannot remove the shear that builds up within a frame.

Core mathematical model

Every pixel has its own time

t_{f,r,c}=fT_f+\ell(r)T_l+(c+\tfrac12)t_p,\quad T_l=Nt_p+t_b,\quad T_f=NT_l

Times are in seconds. A positive stage drift lowers the specimen coordinate that the probe samples. Changing the order of the rows changes the distortion.

Estimate a shift from image correlation

\widehat{\mathbf s}_f=\underset{|s_x|,|s_y|\le R}{\arg\max}\;\frac{\sum_{\Omega}(A-\overline A)(B_f(\mathbf r+\mathbf s)-\overline B_{f,\mathbf s})}{\sqrt{\sum_{\Omega}(A-\overline A)^2\sum_{\Omega}(B_f(\mathbf r+\mathbf s)-\overline B_{f,\mathbf s})^2}}

A is the filtered first frame and B is a filtered stored frame. The window stays fixed inside the image, so nothing wraps around the edges. A coarse search followed by local subpixel refinement can miss a competing peak. Thresholds set for teaching reject estimates that are weak, ambiguous or at the edge of the search range.

Average the accepted frames

J(\mathbf r)=\frac{1}{|\mathcal A|}\sum_{f\in\mathcal A}I_f(\mathbf r+\widehat{\mathbf s}_f),\quad\widehat{\mathbf s}_0=\mathbf0,\quad p<3\,\mathrm{nm}

Only accepted frames are averaged, with their original counts resampled by bilinear interpolation. Pixels outside the area that every accepted frame covers are masked. The first frame is always kept as the coordinate reference; it was never checked as a match itself. To sample the 6 nm grating, the pixel spacing must be strictly below 3 nm.

Common difficulties

A high correlation does not guarantee accuracy

Typical misconceptionIf a match has high correlation and passes the checks, it must be accurate.

Better mental modelThe acceptance thresholds are rules of thumb, not probabilities or calibrated error bars. That is why the task also checks whether every frame was accepted, the sampling, the motion within each frame and the common coverage.

The origin is the first frame

Typical misconceptionAlignment gives back the true, undistorted specimen coordinates.

Better mental modelThe first recorded frame sets the origin and keeps its own distortion. The estimator is never given the absolute drift or the ideal specimen.

Run the experiment

01
Record the frames first
Load Recovery target, change the dwell time and confirm that the stored images do not change. Click Acquire next frames, then Estimate and align.
What to observe: The acquisition controls set up the next stack. The search radius takes effect the next time the same counts are processed. The alignment toggle only changes what is displayed.
02
Compare the set drift with the estimated shifts
Compare the assigned first-to-last motion with the estimated horizontal and vertical shifts. Turn the alignment display off and on without acquiring again.
What to observe: The estimated shifts come from image correlation, so they carry errors from count noise and from the model. They are measured relative to the first frame, and the motion within each frame is still there.
03
Make the estimate fail
Load Periodic ambiguity, No landmarks and Search too narrow. Read the number of frames used together with the common coverage. Then try Slow drifting scan and Interlaced scan.
What to observe: A coverage of 100 percent can mean that only the reference frame was kept. Rejected frames are left out of the aligned average. A high image correlation does not prove that the shift is spatially accurate.
04
Tell sampling apart from drift recovery
Load Aliasing, reduce the pixel spacing to 2 nm, then acquire and process again.
What to observe: Neither more frames nor interpolation can restore a grating that was undersampled. A shift estimate can pass even when the fine detail is aliased.

Core mathematical model

Every pixel has its own time

Estimate a shift from image correlation

Average the accepted frames

Common difficulties

A high correlation does not guarantee accuracy

The origin is the first frame

Run the experiment

Record the frames first

Compare the set drift with the estimated shifts

Make the estimate fail

Tell sampling apart from drift recovery