Chapter map

1. 1. Point-cloud formulation
2. 2. Feature harmonisation
3. 3. Initial similarity pairs
4. 4. Overlap filtering
5. 5. Centroid subtraction
6. 6. SVD rigid transform
7. 7. Coarse translation
8. 8. Dual kNN graphs
9. 9. Edge feature updates
10. 10. Dynamic neighbourhoods
11. 11. Multiscale aggregation
12. 12. Cross-slice soft map
13. 13. Temperature softmin
14. 14. Joint loss
15. 15. Final rigid refinement
16. 16. Governance and learning

Visit the How Mathematics Works hub for the wider series and its connections to education, reasoning, science, computing, design and careers.

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1. Point-cloud formulation

The central mathematical idea is concrete. Coordinates and molecular features represent each slice as paired matrices.

A concrete situation

A Xenium source has 50,000 cells and a Visium target has 4,000 spots.

What the mathematics does not license

Different resolutions alter density and neighbourhoods. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Write dimensions, units and shared features. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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2. Feature harmonisation

Here the geometry becomes operational. Comparable omics profiles are constructed before matching.

A concrete situation

Shared genes are normalized across platforms.

What the mathematics does not license

Batch effects can mimic spatial difference. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Repeat with stable feature panels. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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3. Initial similarity pairs

This step deserves a careful pause. High feature similarity proposes source-target correspondences.

A concrete situation

A neuronal source cell matches a molecularly similar target region.

What the mathematics does not license

Similar cell types may occur in several locations. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Measure ambiguity and reciprocity. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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4. Overlap filtering

The useful question is not only how to calculate, but what the calculation assumes. Stitching discards candidate pairs inconsistent with the overlapping region.

A concrete situation

Two partially imaged fields share a central strip.

What the mathematics does not license

Bad filtering removes real overlap. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Plot retained and rejected pairs. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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5. Centroid subtraction

A small example makes the mechanism visible. Matched coordinate clouds are centred before rotation.

A concrete situation

Both pair sets are translated to zero mean.

What the mathematics does not license

Outliers move centroids. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use robust and trimmed variants. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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6. SVD rigid transform

This is where modelling choices begin to affect evidence. A cross-covariance matrix yields the least-squares rotation.

A concrete situation

The source rotates 90 degrees before translation.

What the mathematics does not license

Reflection handling must be explicit. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Check determinant and residuals. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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7. Coarse translation

The central mathematical idea is concrete. Centroids and rotation determine the shift vector.

A concrete situation

The rotated source moves 240 micrometres right.

What the mathematics does not license

Coordinate units may differ. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Standardize physical scale. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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8. Dual kNN graphs

Here the geometry becomes operational. Separate graphs describe spatial and omics neighbourhoods.

A concrete situation

Each spot has ten coordinate neighbours and ten molecular neighbours.

What the mathematics does not license

One k cannot fit every density. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Sweep k by region. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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9. Edge feature updates

This step deserves a careful pause. Dynamic graph convolution compares each node with neighbours.

A concrete situation

An edge uses the node vector and neighbour difference.

What the mathematics does not license

Noisy neighbours propagate errors. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Ablate graph updates. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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10. Dynamic neighbourhoods

The useful question is not only how to calculate, but what the calculation assumes. kNN sets are recomputed after learned layers.

A concrete situation

Two homologous domains become neighbours in embedding space.

What the mathematics does not license

Early mistakes can reinforce themselves. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Track neighbour turnover. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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11. Multiscale aggregation

A small example makes the mechanism visible. Layer outputs are concatenated to retain local and global cues.

A concrete situation

Short- and longer-range tissue structure share one vector.

What the mathematics does not license

High-dimensional concatenation may overfit. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Compare layer subsets. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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12. Cross-slice soft map

This is where modelling choices begin to affect evidence. Embeddings generate weights from every source to possible targets.

A concrete situation

One source spot distributes weight across three nearby targets.

What the mathematics does not license

Weights are model scores, not biological probabilities. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Report entropy and concentration. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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13. Temperature softmin

The central mathematical idea is concrete. Exponentiated negative distances create attention weights.

A concrete situation

Lower temperature sharpens the nearest region.

What the mathematics does not license

Too low becomes brittle. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Publish a temperature curve. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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14. Joint loss

Here the geometry becomes operational. Pearson-correlation dissimilarity and coordinate distance combine through alpha.

A concrete situation

Alpha 0.6 emphasizes molecular agreement under the documented convention.

What the mathematics does not license

Scaling changes the effective trade-off. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Normalize and report both terms. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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15. Final rigid refinement

This step deserves a careful pause. Soft-map-derived paired coordinates produce another SVD transform.

A concrete situation

A two-degree correction removes remaining rotation.

What the mathematics does not license

Rigid refinement misses stretching. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Map local residual vectors. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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16. Governance and learning

The useful question is not only how to calculate, but what the calculation assumes. Version data, graphs, k, alpha, temperature, filters and transforms.

A concrete situation

A student solves a three-point SVD alignment.

What the mathematics does not license

A low loss is conditional evidence. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Write an alignment audit card. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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A seven-day practice lab

1. Day 1: define the decision, observation unit, X variables and Y or reference condition.
2. Day 2: reproduce the smallest worked example and check every dimension, unit and sign.
3. Day 3: build a transparent baseline and write what it assumes.
4. Day 4: fit the main method inside a training-only pipeline; record every parameter.
5. Day 5: challenge one assumption with a deliberately difficult dataset or operating regime.
6. Day 6: compare validation metrics with the practical workload created by errors.
7. Day 7: present one supported conclusion, one unresolved uncertainty and one safe next experiment.

Questions for a student, parent or mentor

• Which part of the model is a mathematical definition, and which part is a choice?
• What small calculation would catch a sign, scale or dimension error?
• What evidence comes from training, validation and the untouched test?
• What benign condition could look like a fault or prediction failure?
• How would the answer change for another response, asset, school project or time period?
• What should a human review before acting on the score?

A method-specific audit before deployment

SANTO separates alignment into a fast coarse stage and a learned fine stage. The coarse stage uses omics similarity to propose one-to-one spot pairs, can filter pairs to estimate an overlap region for stitching, and uses singular-value decomposition to estimate a rotation and translation. This supplies a useful starting pose before the harder local refinement.

The fine stage builds separate k-nearest-neighbour graphs for coordinates and molecular features. Dynamic graph convolution recomputes neighbourhoods through the network, aggregates multiscale local and global signals, and produces a one-to-many soft map. A temperature-controlled softmin gives greater weight to nearer candidate partners.

Its loss combines molecular similarity, expressed through Pearson correlation, with Euclidean coordinate distance. Alpha balances the two components. The resulting alignment remains conditional on preprocessing, k, temperature, alpha, overlap filtering and rigid-transform assumptions, so synthetic rotations, independent landmarks and sensitivity curves are essential.

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Frequently asked questions

What is SANTO?

SANTO is a coarse-to-fine method for aligning or stitching spatial-omics point clouds in two or three dimensions.

What happens in coarse alignment?

Feature similarity proposes matching spots, optional filtering estimates an overlap region, and an SVD-based calculation estimates rotation and translation.

Why use dynamic graphs?

Neighbourhoods are recomputed in learned feature space so local relations can change as the representation improves.

What is the soft mapping?

It is a one-to-many weighted relation from each source spot to a region of target spots rather than a forced single partner.

What do alpha and temperature control?

Alpha balances molecular correlation and spatial distance in the loss; temperature controls how concentrated softmin weights are.

Does SANTO recover non-rigid deformation?

The described coarse and fine transformations use rotations and translations; local correspondence learning can improve pose estimation but does not by itself make every tissue deformation physically correct.

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Primary source and fact-check note

Haoyang Li, Yingxin Lin et al., “SANTO: a coarse-to-fine alignment and stitching method for spatial omics,” Nature Communications 15, 6048 (2024), DOI 10.1038/s41467-024-50308-x. Definitions and core claims in this article were checked against this primary publication. Calculations shown here were independently recomputed. Examples are educational illustrations; they are not reported industrial performance results.

Useful next reading

• SLAT heterogeneous slices
• STAligner multi-slice maps
• PASTE probabilistic alignment
• How Mathematics Works

Keep the options open. Mathematics education supports clearer questions, transferable problem-solving skills and informed choices across science, engineering, computing, AI, statistics, design and many other careers. It does not guarantee intelligence, income, admission or employment; those outcomes depend on far more than one subject or model.

Extended laboratory: topic-specific validation and responsible use

Input audit

Record coordinate units, shared features and platform resolution.

Coarse audit

Plot initial pairs, overlap filtering and SVD residuals.

Graph audit

Sweep spatial and omics k separately.

Dynamic audit

Track neighbourhood changes across layers.

Loss audit

Report alpha, temperature and component scales.

Mapping audit

Measure soft-map entropy and reciprocity.

Landmark audit

Use independent anatomy and synthetic transformations.

Final claim

A fitted pose is conditional on the selected evidence.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Point-cloud formulation

Begin with this mechanism: Coordinates and molecular features represent each slice as paired matrices. Work through the concrete case: A Xenium source has 50,000 cells and a Visium target has 4,000 spots. Make the limitation visible: Different resolutions alter density and neighbourhoods. Complete the evidence step: Write dimensions, units and shared features. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 2: Feature harmonisation

Begin with this mechanism: Comparable omics profiles are constructed before matching. Work through the concrete case: Shared genes are normalized across platforms. Make the limitation visible: Batch effects can mimic spatial difference. Complete the evidence step: Repeat with stable feature panels. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 3: Initial similarity pairs

Begin with this mechanism: High feature similarity proposes source-target correspondences. Work through the concrete case: A neuronal source cell matches a molecularly similar target region. Make the limitation visible: Similar cell types may occur in several locations. Complete the evidence step: Measure ambiguity and reciprocity. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 4: Overlap filtering

Begin with this mechanism: Stitching discards candidate pairs inconsistent with the overlapping region. Work through the concrete case: Two partially imaged fields share a central strip. Make the limitation visible: Bad filtering removes real overlap. Complete the evidence step: Plot retained and rejected pairs. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 5: Centroid subtraction

Begin with this mechanism: Matched coordinate clouds are centred before rotation. Work through the concrete case: Both pair sets are translated to zero mean. Make the limitation visible: Outliers move centroids. Complete the evidence step: Use robust and trimmed variants. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 6: SVD rigid transform

Begin with this mechanism: A cross-covariance matrix yields the least-squares rotation. Work through the concrete case: The source rotates 90 degrees before translation. Make the limitation visible: Reflection handling must be explicit. Complete the evidence step: Check determinant and residuals. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 7: Coarse translation

Begin with this mechanism: Centroids and rotation determine the shift vector. Work through the concrete case: The rotated source moves 240 micrometres right. Make the limitation visible: Coordinate units may differ. Complete the evidence step: Standardize physical scale. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 8: Dual kNN graphs

Begin with this mechanism: Separate graphs describe spatial and omics neighbourhoods. Work through the concrete case: Each spot has ten coordinate neighbours and ten molecular neighbours. Make the limitation visible: One k cannot fit every density. Complete the evidence step: Sweep k by region. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 9: Edge feature updates

Begin with this mechanism: Dynamic graph convolution compares each node with neighbours. Work through the concrete case: An edge uses the node vector and neighbour difference. Make the limitation visible: Noisy neighbours propagate errors. Complete the evidence step: Ablate graph updates. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 10: Dynamic neighbourhoods

Begin with this mechanism: kNN sets are recomputed after learned layers. Work through the concrete case: Two homologous domains become neighbours in embedding space. Make the limitation visible: Early mistakes can reinforce themselves. Complete the evidence step: Track neighbour turnover. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 11: Multiscale aggregation

Begin with this mechanism: Layer outputs are concatenated to retain local and global cues. Work through the concrete case: Short- and longer-range tissue structure share one vector. Make the limitation visible: High-dimensional concatenation may overfit. Complete the evidence step: Compare layer subsets. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 12: Cross-slice soft map

Begin with this mechanism: Embeddings generate weights from every source to possible targets. Work through the concrete case: One source spot distributes weight across three nearby targets. Make the limitation visible: Weights are model scores, not biological probabilities. Complete the evidence step: Report entropy and concentration. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 13: Temperature softmin

Begin with this mechanism: Exponentiated negative distances create attention weights. Work through the concrete case: Lower temperature sharpens the nearest region. Make the limitation visible: Too low becomes brittle. Complete the evidence step: Publish a temperature curve. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 14: Joint loss

Begin with this mechanism: Pearson-correlation dissimilarity and coordinate distance combine through alpha. Work through the concrete case: Alpha 0.6 emphasizes molecular agreement under the documented convention. Make the limitation visible: Scaling changes the effective trade-off. Complete the evidence step: Normalize and report both terms. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 15: Final rigid refinement

Begin with this mechanism: Soft-map-derived paired coordinates produce another SVD transform. Work through the concrete case: A two-degree correction removes remaining rotation. Make the limitation visible: Rigid refinement misses stretching. Complete the evidence step: Map local residual vectors. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 16: Governance and learning

Begin with this mechanism: Version data, graphs, k, alpha, temperature, filters and transforms. Work through the concrete case: A student solves a three-point SVD alignment. Make the limitation visible: A low loss is conditional evidence. Complete the evidence step: Write an alignment audit card. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.