Crowther criterion¶

The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion.

Version: v1 · 2026-09-28 · History
Domain-specific #: 8794
Domain group: Natural Sciences
Origin domain: Biology & Ecology
Subdomains: Structural Biology, Electron Tomography → Biology & Ecology

Core Idea¶

Crowther criterion is treated here as the recurring cross_domain_models_structures_representations identity summarized by this source-grounded definition: The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion.

The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. The minimum number of views, m , to reconstruct a particle of diameter D to a resolution of d is given by. where d = 1/R , with R the reciprocal space radius corresponding to the maximum meaningfully filled frequency in Fourier space.

For Crowther criterion, the abstraction is narrower than the article's general subject matter: a positive case must preserve The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in cross_domain_models_structures_representations, which is why this identity is domain-specific rather than prime.

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How Many Pictures You Need

Imagine trying to figure out the shape of a toy you can't touch, only by taking pictures of it from different sides. The bigger the toy and the more tiny details you want to see, the more pictures you need. The Crowther criterion tells scientists the fewest pictures needed for the detail they want.

Views Needed for a Sharp 3D Picture

In tomography, scientists build a 3D picture of an object by combining many 2D views taken from different directions. The Crowther criterion is the usual way to figure out how sharp that 3D picture can be. It says how many views you need to rebuild a particle of a certain size with a certain level of detail. The bigger the particle, or the finer the details you want to see, the more views you need. If you don't have enough views, the fine details can't be filled in properly.

Views Needed for Tomographic Resolution

In tomography, a 3D reconstruction is built from 2D projection images taken at different viewing angles. Each view fills in part of the object's information in 'Fourier space,' a way of describing the object by its spatial frequencies, where higher frequencies correspond to finer detail. The Crowther criterion is the conventional way to judge the resolution a reconstruction can reach: it gives the minimum number of views m needed to reconstruct a particle of diameter D to a resolution d. Here d = 1/R, where R is the largest radius in Fourier space that is meaningfully filled by the data. More views are needed as the particle gets larger or the target resolution gets finer.

The Crowther criterion is the conventional method for evaluating the achievable resolution of a tomographic reconstruction. By the projection-slice theorem, each projection view samples a central section of the object's Fourier transform, so a finite number of views leaves gaps between sections at high spatial frequency. The criterion gives the minimum number of views m needed to reconstruct a particle of diameter D to resolution d, with m scaling in proportion to D/d; in its usual form m = πD/d. Here d = 1/R, where R is the reciprocal-space radius corresponding to the maximum meaningfully filled frequency in Fourier space. Conversely, given the number of views and the particle size, it bounds the resolution that the reconstruction can meaningfully claim.

Structural Signature¶

Sig role-phrases:

Defining carrier — The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion.
Constitutive relation — The minimum number of views, m , to reconstruct a particle of diameter D to a resolution of d is given by.
Operating condition — where d = 1/R , with R the reciprocal space radius corresponding to the maximum meaningfully filled frequency in Fourier space.
Recognition evidence — The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion.
Admissible variation — The minimum number of views, m , to reconstruct a particle of diameter D to a resolution of d is given by.
Characteristic consequence — where d = 1/R , with R the reciprocal space radius corresponding to the maximum meaningfully filled frequency in Fourier space.
Failure boundary — The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion.

What It Is Not¶

Not the whole field of cross_domain_models_structures_representations. The node requires the specific identity stated by The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion.
Not an over-broad reading. The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion.
Not an over-broad reading. The minimum number of views, m , to reconstruct a particle of diameter D to a resolution of d is given by.
Not an over-broad reading. where d = 1/R , with R the reciprocal space radius corresponding to the maximum meaningfully filled frequency in Fourier space.
Not automatically Repetition method. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Crowther criterion applies literally inside cross_domain_models_structures_representations wherever the source-defined carrier and relation can be established. Its documented habitats include:

Documented setting. The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion.
Documented setting. The minimum number of views, m , to reconstruct a particle of diameter D to a resolution of d is given by.
Documented setting. where d = 1/R , with R the reciprocal space radius corresponding to the maximum meaningfully filled frequency in Fourier space.
Documented setting. The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion.
Documented setting. The minimum number of views, m , to reconstruct a particle of diameter D to a resolution of d is given by.
Documented setting. where d = 1/R , with R the reciprocal space radius corresponding to the maximum meaningfully filled frequency in Fourier space.

Outside cross_domain_models_structures_representations, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Pattern or should be marked as analogy.

Clarity¶

A clear use of Crowther criterion names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. The strongest recognition evidence in the frozen account is: The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Crowther criterion compresses multiple cross_domain_models_structures_representations details into a stable diagnostic relation. The source shows both the central mechanism—the minimum number of views, m , to reconstruct a particle of diameter D to a resolution of d is given by.—and the practical consequence—where d = 1/R , with R the reciprocal space radius corresponding to the maximum meaningfully filled frequency in Fourier space. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the cross_domain_models_structures_representations entities to which the claim applies.
State the relation. Use the source-grounded identity: The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion.
Check operation and conditions. where d = 1/R , with R the reciprocal space radius corresponding to the maximum meaningfully filled frequency in Fourier space.
Demand recognition evidence. The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion.
Test variation. Change an implementation or setting while preserving the minimum number of views, m , to reconstruct a particle of diameter D to a resolution of d is given by.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Pattern.

Knowledge Transfer¶

Within the home domain. Knowledge about Crowther criterion transfers literally when a new case preserves the same carrier type, relation, and recognition test. The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. The minimum number of views, m , to reconstruct a particle of diameter D to a resolution of d is given by.

Beyond the home domain. No canonical parent is asserted for Crowther criterion. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion; recognition evidence → The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion

Applied / In Practice¶

The minimum number of views, m , to reconstruct a particle of diameter D to a resolution of d is given by. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → the applied context; invariant → The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion; boundary → the case exits the class when the conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion

Structural Tensions¶

T1 — Stable identity versus admissible variation. The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. The minimum number of views, m , to reconstruct a particle of diameter D to a resolution of d is given by. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. where d = 1/R , with R the reciprocal space radius corresponding to the maximum meaningfully filled frequency in Fourier space. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Crowther criterion literally, co-instantiate Pattern, or only resemble it?

T6 — Autonomy versus reduction. The minimum number of views, m , to reconstruct a particle of diameter D to a resolution of d is given by. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Crowther criterion distinguish that the broader parent Pattern leaves together?

Terminal boundary synthesis. For Crowther criterion, the terminal identity test begins with the definition The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion.. A reviewer must then establish the carrier and operation described by The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. and The minimum number of views, m , to reconstruct a particle of diameter D to a resolution of d is given by.. Recognition is constrained by where d = 1/R , with R the reciprocal space radius corresponding to the maximum meaningfully filled frequency in Fourier space., while admissible variation is limited by The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. and the collapse boundary The minimum number of views, m , to reconstruct a particle of diameter D to a resolution of d is given by.. The source-domain setting in cross domain models structures representations matters because The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. and The minimum number of views, m , to reconstruct a particle of diameter D to a resolution of d is given by. specify where those roles have literal occupants. The strongest negative controls are The node requires the specific identity stated by The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. and The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion.; a case satisfying either exclusion should not be rescued merely because its label or examples look familiar.

Terminal adjudication sequence. First, bind the claimed instance to a concrete carrier and state the criterion by which The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. is recognized. Second, vary implementation, scale, notation, and example while holding The minimum number of views, m , to reconstruct a particle of diameter D to a resolution of d is given by. fixed; persistence supports one identity rather than several topic fragments. Third, remove where d = 1/R , with R the reciprocal space radius corresponding to the maximum meaningfully filled frequency in Fourier space. or trigger The minimum number of views, m , to reconstruct a particle of diameter D to a resolution of d is given by. and verify that the classification fails. Fourth, compare the result with the two negative controls instead of relying on name similarity. Fifth, check scope against The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. and record any qualification supplied by cross domain models structures representations. Finally, audit the graph claim. The approved unparented placement prevents a weak lexical resemblance from becoming a false ontological claim; a later edge must preserve every constitutive role stated here. This sequence makes the entry rejectable, keeps analogy separate from literal transfer, and exposes which fact would require revision.

Counterfactual boundary matrix. Evaluate Crowther criterion under four controlled substitutions. In the carrier substitution, replace the concrete entities while retaining The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion.; the identity should persist only if the new carrier has the same operative type. In the operation substitution, replace The minimum number of views, m , to reconstruct a particle of diameter D to a resolution of d is given by. while preserving surface vocabulary; the identity should fail unless the replacement entails the same relation. In the evidence substitution, change the instrument, representation, or witness used for where d = 1/R , with R the reciprocal space radius corresponding to the maximum meaningfully filled frequency in Fourier space.; classification may persist when the new evidence warrants the same fact. In the scope substitution, move the case outside The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. and ask whether The minimum number of views, m , to reconstruct a particle of diameter D to a resolution of d is given by. still gives the roles literal occupants. These four tests separate constitutive structure from implementation, evidence, and familiar examples. They also identify the exact revision needed when a source expands or narrows the recognized class.

Neighbor and residual test. The negative controls The node requires the specific identity stated by The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. and The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. define two directions of possible overreach. A reviewer should construct one case that satisfies the first control but not Crowther criterion, one that satisfies Crowther criterion but not the control, and the corresponding pair for the second control. If no such asymmetric pair can be stated, the candidate may duplicate a neighbor or the distinction may depend only on wording. When the specialist identity fails but a thinner relation remains, record that residual separately instead of stretching Crowther criterion. The approved unparented placement prevents a weak lexical resemblance from becoming a false ontological claim; a later edge must preserve every constitutive role stated here. The resulting decision trail makes later DAG densification possible without treating today's uncertainty as a hierarchy fact.

Structural–Framed Character¶

Crowther criterion is mixed or framed-leaning. Its structural side is the repeatable organization summarized by The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. Its framed side is the cross_domain_models_structures_representations vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: where d = 1/R , with R the reciprocal space radius corresponding to the maximum meaningfully filled frequency in Fourier space. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Pattern. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. The minimum number of views, m , to reconstruct a particle of diameter D to a resolution of d is given by. It further constrains recognition and variation through: where d = 1/R , with R the reciprocal space radius corresponding to the maximum meaningfully filled frequency in Fourier space. The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion.

What is domain-bound. cross domain models structures representations supplies the operative entities, technical vocabulary, warrants, and exceptions that make Crowther criterion literal. Its documented scope includes the condition that The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. Another bounded application condition is that The minimum number of views, m , to reconstruct a particle of diameter D to a resolution of d is given by. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—The minimum number of views, m , to reconstruct a particle of diameter D to a resolution of d is given by.—and future graph densification may discover a defensible relation only if it preserves that boundary.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Crowther criterion. The reviewed identity is: The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Neighborhood in Abstraction Space¶

Crowther criterion sits in a sparse region of the domain-specific corpus (85^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2551 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Pattern. The parent omits the specialist differentia. Tell: Can the case establish The conventional method to evaluate the resolution of a tomography reconstruction is determined by the Crowther criterion?
Repetition method. A surveying procedure that measures the same horizontal angle successively around an instrument circle so accumulated readings can be divided and systematic effects reduced. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Resolution Matching. Match the resolving granularity of an instrument, effector, representation, or control surface to the smallest distinction the task must preserve; a coarser resolver collapses consequential differences, while unnecessary fineness spends capacity without improving the decision. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Widely applicable information criterion. A Bayesian predictive-fit criterion combining log pointwise posterior predictive density with a variance-based effective-complexity penalty. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Crowther criterion remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside cross_domain_models_structures_representations lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Pattern?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Crowther_criterion (revision 1332538830).

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.