A map can have five-metre pixels while its underlying response spreads a narrow feature over much more than five metres. Pixel spacing tells you where values are placed. A point-spread or line-spread function describes how a small target contributes to surrounding measurements. Both are needed before a fine grid can be called detailed evidence.
We will work through an original one-dimensional blur model and a separate Gaussian response model. The numbers are hypothetical and do not describe a named satellite. Real image products can include optics, detector integration, motion, reconstruction and later processing, each with its own spatial effects.
Find the question you need
Choose a route below or read in order. All teaching chapters remain open; the index groups let you jump directly to a worked example or check.
Route 1 · Separate sampling from response
Chapters 1–2: Separate sampling from response.
Route 2 · Calculate a small blur and its ambiguities
Chapters 3–4: Calculate a small blur and its ambiguities.
Route 3 · Measure width in physical units
Chapters 5–6: Measure width in physical units.
Route 4 · Interpret processing and calibration evidence
Chapters 7–8: Interpret processing and calibration evidence.
Route 5 · Practise and choose the next route
Chapters 9–12: Practise and choose the next route.
Full chapter index · Try the transfer checks · Remote Sensing subject root
Chapter contents
Separate sampling from response · Chapters 1–2
Calculate a small blur and its ambiguities · Chapters 3–4
Measure width in physical units · Chapters 5–6
Interpret processing and calibration evidence · Chapters 7–8
Practise and choose the next route · Chapters 9–12
CHAPTER 1 OF 12 · Separate sampling from response
1. Name the grid and the effective response
Ground sample distance, GSD, is commonly used for the spacing of image samples projected onto the ground. A delivered grid also has a pixel spacing. Reprojection or resampling can change the latter without repeating the original observation.
A point-spread function describes the response to an ideal spatial point under a specified imaging model. A line-spread function describes the response associated with a narrow line or an integrated direction. Widths and frequency-response measures provide different ways to characterise that response.
There is no single number that establishes every aspect of spatial performance. Along-track and across-track responses can differ; geometry can vary across a scene; contrast and noise affect what can be detected. A product’s nominal pixel label leaves those questions open.
The root’s remote-sensing guide introduces resolution, scale and mixed pixels. Here we make the next distinction calculable: which features can become similar or indistinguishable after the same blur, even if the output grid is very finely drawn?
CHAPTER 2 OF 12 · Separate sampling from response
2. Define a three-tap teaching filter
Imagine equally spaced one-dimensional scene values xj, with centres ten metres apart. Define measured values by yj = 0.25x(j−1) + 0.50xj + 0.25x(j+1). Each output is a weighted average of three neighbouring input values.
The weights sum to one, so a constant input remains constant. The filter is symmetric, so it does not deliberately move the centre of an isolated symmetric feature. It still changes local contrast and the relation between a target value and the surrounding outputs.
This is a complete discrete response model, not a claim that every sensor uses this kernel. For the examples, take an infinite line or a region far enough from image boundaries that edge-handling choices do not enter.
The order of the operation matters. We first define the scene samples and apply the filter; sampling and interpolation in a real optical system may occur through a more complex chain. The small model is useful because every output and every ambiguity can be checked directly.
CHAPTER 3 OF 12 · Calculate a small blur and its ambiguities
3. Follow one narrow feature into three outputs
Set x0 = 1 and every other xj = 0. The nonzero measured values are y−1 = 0.25, y0 = 0.50 and y1 = 0.25. The isolated unit feature has become a wider, lower response.
The sum of those outputs remains one. Its weighted position remains centred at zero. Peak preservation is a different property: the peak fell from one to half. Conserving a total and a centroid does not mean that the local feature stayed sharp.
A hypothetical detection rule requiring a measured value greater than 0.6 would find no output above its threshold. That result reflects this response and chosen decision rule. It does not prove that the feature is absent from the scene.
| Output centre | Position | Measured response |
|---|---|---|
| j=−1 | −10 m | 0.25 |
| j=0 | 0 m | 0.50 |
| j=1 | 10 m | 0.25 |
| All other j | Elsewhere | 0 |
For real narrow targets, signal strength, shape, background and the full response matter. A feature smaller than a nominal pixel can influence a measured value, while distinguishing two close features or measuring their widths can remain difficult. Detection, separation and accurate measurement are different tasks.
CHAPTER 4 OF 12 · Calculate a small blur and its ambiguities
4. Find a scene pattern that the filter erases
Now let scene samples alternate indefinitely between one and zero. At a location with xj = 1, both neighbours are zero, so yj = 0.50. At a location with xj = 0, both neighbours are one, again giving yj = 0.50.
The entire output is a constant half. A genuinely constant input xj = 0.50 produces exactly that same output. Under this specified filter, the two scenes are indistinguishable from the output values alone.
Drawing extra interpolated pixels between these outputs cannot choose which original scene occurred. The missing distinction was removed by the response. An additional assumption or an independently informative observation would be required to identify it.
This is a precise information-loss example, not a declaration that every blur makes all reconstruction impossible. Some components may be recoverable with suitable data and a known response, while noise and weak response can make inversion unstable. Our alternating component is exactly erased by this particular kernel.
CHAPTER 5 OF 12 · Measure width in physical units
5. Work a blurred edge and a threshold boundary
Let the scene be zero for j < 0 and one for j ≥ 0. The nominal step lies halfway between centres j = −1 and j = 0, at −5 metres. The filter gives y−2 = 0, y−1 = 0.25, y0 = 0.75 and y1 = 1.
If we linearly interpolate these output samples, the half-height crossing is halfway between −10 and zero metres: −5 metres. In this symmetric constructed case, the half-height boundary stays at the nominal step position even though the edge becomes less sharp.
Change the threshold to 0.8. Between y0 = 0.75 at zero and y1 = 1 at ten metres, the crossing is 0 + 10×(0.8 − 0.75)/(1 − 0.75) = 2 metres. It lies seven metres to the high-valued side of the nominal boundary.
The example separates blur, boundary convention and displacement. A changed threshold can change an inferred feature boundary without any movement in the scene. For temporal comparisons, keep both the response and the extraction rule consistent before interpreting a changing outline.
CHAPTER 6 OF 12 · Measure width in physical units
6. Calculate a Gaussian response width
For a separate continuous model, take a Gaussian line response proportional to exp[−x²/(2σ²)], with σ = 10 m. Its full width at half maximum, FWHM, is the distance between the two positions where the response is half its peak.
Solve exp[−x²/(2σ²)] = 1/2. Taking logarithms gives |x| = σ√(2 ln 2). Therefore FWHM = 2√(2 ln 2)σ ≈ 2.35482σ, giving about 23.55 m for this example.
With ten-metre grid spacing, that width covers about 2.355 sample intervals. Re-expressing the same response on a five-metre grid gives about 4.710 intervals across the same physical 23.55 m width. The number of samples across the response changed; its physical width did not.
FWHM is one response metric. It is not automatically the minimum detectable object width or the separation needed for every pair of targets. The criterion, target geometry, contrast, noise and processing must accompany an application claim.
CHAPTER 7 OF 12 · Interpret processing and calibration evidence
7. Keep equal grids separate from equal sharpness
Consider two hypothetical products both delivered on a ten-metre grid. Product A has Gaussian σ = 5 m, giving FWHM ≈ 11.77 m. Product B has σ = 10 m, giving FWHM ≈ 23.55 m. Their grid spacing matches while their modeled response widths differ by a factor of two.
Now resample B onto a five-metre grid without additional observations. Its response remains the broader one. Smaller cells can support alignment, display and numerical integration, but the label five metres does not turn B into A.
| Product/model | Grid spacing | Physical σ | FWHM | FWHM/grid spacing |
|---|---|---|---|---|
| A | 10 m | 5 m | 11.77 m | 1.177 |
| B | 10 m | 10 m | 23.55 m | 2.355 |
| B resampled | 5 m | 10 m | 23.55 m | 4.710 |
Finer sampling of a genuinely sharper observation can be valuable. The distinction is whether additional detail was measured and retained, rather than whether an output file contains more columns and rows. Read the acquisition and product history before comparing grid specifications.
A sharpening operation may make edges look crisper. Assess its response, noise and assumptions separately. Appearance and estimated spatial performance can change in different ways, especially when a method uses external imagery or learned scene expectations.
CHAPTER 8 OF 12 · Interpret processing and calibration evidence
8. Calculate what an additional smoothing step changes
Apply the three-tap filter twice. Combining the two weighted averages gives a five-tap response with weights (1, 4, 6, 4, 1)/16. The weights still sum to one, but the isolated-feature peak is now 6/16 = 0.375 instead of 0.50.
For continuous Gaussian responses, convolution adds variances when the stated Gaussian model applies. If an existing σ = 10 m response is additionally smoothed with Gaussian σ = 5 m, the combined σ is √(10² + 5²) ≈ 11.18 m.
Its FWHM is approximately 26.33 m. Adding the two FWHM values directly would be wrong for this model; variances are the quantities that add. The result concerns an effective chain of responses, not necessarily a single optical component.
Record smoothing and resampling steps when comparing products. A classifier or mean estimate may benefit from some noise reduction while losing narrow-feature contrast. An improvement in one analysis outcome does not establish that every spatial question now has a better observation.
CHAPTER 9 OF 12 · Practise and choose the next route
9. Read edge and line targets as measurement evidence
A calibration target is not an ideal mathematical impulse merely because it looks narrow. Its physical width, orientation, contrast and surroundings can contribute to the recorded profile. A finite bridge across dark water can broaden an observed line response.
An edge-spread profile records the transition across a sufficiently suitable edge. Under the relevant linear, shift-invariant model, differentiating that edge response yields a line-spread response. Estimating the derivative from noisy samples requires its own method and uncertainty account.
NASA’s evaluations explicitly distinguish target width from the underlying response and use both line-like features and edge targets depending on image detail. Do not take an arbitrary road profile and label its apparent width the instrument’s true spatial resolution.
Use multiple suitable targets and directions where appropriate. State the scene, band, processing level and response metric. A response estimated for one acquisition or product chain should not silently be assigned to every band, resampled output or viewing geometry.
CHAPTER 10 OF 12 · Practise and choose the next route
10. Connect response differences with the final claim
A small mapped object can change size after smoothing or thresholding even when the physical scene is constant. A time comparison that changes effective response can therefore create a processing-related boundary difference, distinct from location misalignment.
The co-registration and false-boundary-change guide isolates displacement. The categorical raster resampling guide isolates how inappropriate interpolation can invent labels. This branch isolates loss and redistribution of signal before those interpretations.
Spatial response also affects mixed pixels and reference matching. A sharp reference boundary may not represent the same footprint as a blurred measurement. More precise reference coordinates do not remove the response mismatch.
Before applying an accuracy result, ask whether its targets and scales match the intended use. The spatial validation guide concerns generalisation across places. A clean split and a sharp sensor address different parts of the evidence chain; neither substitutes for the other.
CHAPTER 11 OF 12 · Practise and choose the next route
11. Practise changed kernels and changed units
Check one: replace the weights with (0.10, 0.80, 0.10). An isolated unit input now gives a central peak of 0.80. For an alternating one-zero input, outputs alternate between 0.80 and 0.20, so this kernel does not erase that pattern.
Check two: keep a Gaussian physical σ of ten metres, but halve the output spacing from ten to five metres. FWHM stays 23.55 m while the width in grid intervals doubles. Explain the changed denominator before claiming an increase in observational detail.
Check three: apply the original filter to a constant value seven. Every output is 0.25×7 + 0.50×7 + 0.25×7 = seven. A filter can preserve constant regions while strongly changing narrow objects and high-frequency patterns.
For transfer, examine two maps with identical pixel dimensions but different processing histories. Ask for response evidence, target scale, resampling and the exact detection or measurement criterion. The conclusion should follow what the chain preserves, not the fineness of the display.
CHAPTER 12 OF 12 · Practise and choose the next route
12. Questions, reading routes and sources
Is pixel spacing useless?
No. It defines sampling geometry and affects storage, alignment and analysis. It is one part of spatial performance rather than a complete response description.
Can a subpixel object be detected?
Its signal can affect measurements and may be detectable under suitable contrast and noise conditions. Detecting a contribution does not automatically resolve its shape, distinguish close objects or measure its dimensions accurately.
Does interpolation recover erased detail?
Interpolation alone cannot distinguish the two exactly identical outputs in our alternating-pattern example. Reconstructions using additional observations or assumptions need separate evidence for their claims.
Choose the next question
Follow spatial meaning and processing
- Why Averaging NDVI Differs from Averaging Reflectance
- Co-Registration Offsets and False Change at a Boundary
- Why Bilinear Resampling Can Invent Land-Cover Classes
- Spatial Validation and the Nearby Pixels That Make a Test Too Easy
- Stratified Validation and Area-Adjusted Class Estimates
- Rare Change, False Alarms and the Reliability of a Mapped Alert
Check the observation and comparison population
- Emissivity and Brightness-Temperature Bias
- A Shared Baseline and the Uncertainty in Comparing Two Changes
- Paired Validation and Uncertainty in a Net Change-Area Estimate
- Coverage Thresholds and the Footprint Behind a Regional Mean
- Cloud Masks and the Seasonal Population Behind a Clear-Sky Mean
- Revisit Gaps, Cloud Cover and Missed Short Events
Sources and scope
NASA’s sensor-resolution and grid-sizing poster distinguishes response width from grid spacing. NASA’s evaluation of optical sensor spatial resolution explains line-spread, edge-spread and target-width evidence. NASA’s astronaut-imagery resolution FAQ adds the role of contrast and scene detail. The kernels, null-pattern example, Gaussian calculations and threshold exercise here are original teaching cases.
Sources checked 22 October 2026. Return through The Gold Standard of Remote Sensing, The Gold Standard of Education and How X Works to practise separating the measured response from a visual impression.