A Computational Approach to Edge Detection
Readings in Computer Vision, 1987 · paper · AI and machine learning
Defines criteria for edge detection (detection, localization, single response) and derives optimal detectors from them. For step edges, the result is marking maxima in gradient magnitude of a Gaussian-smoothed image, extended with multiple scales and elongated operators.
Checked: PDF from andrew.cmu.edu, fetched 5 Oct 2026. Page numbers are those of that PDF.
6 findings
Substance Eq. (37) p.686: a2 and a4 both end in "+ α(α − β))/4(ω² sinh² α − α² sin² ω)", contrary to the claim that a4 follows from a2 by α → −α, and the printed pair does not satisfy the boundary condition a2 + a4 + c = 0 of Eq. (36).
- p. 8 Eq. (37), a2 and a4 last terms; following text
similarly for a4 from a2
Only the authors can say what is right here.
Judge's note: My working: under α→−α every term of a2 maps onto the corresponding printed a4 term (cosh is even; sinh and e^{-α}sinh α change as required) except the last, which becomes α(α+β). Summing the numerators with the last a4 term taken as α(α+β): −2α²cos2ω − 4ω²sinh²α + 2α² = 4α²sin²ω − 4ω²sinh²α = −4D. That gives a2+a4 = −c, as (36) requires. With the printed terms there is an extra −2αβc/(4D), so a2+a4+c ≠ 0 in general. One of a2 or a4 is misprinted in its last term. The text layer agrees with the image that both last terms are α(α − β).
Second check: another model agreed
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Reference or label Introduction p.1 cites "shape from contour [27] and shape from texture [31]", but [31] is A. P. Witkin, "Shape from contour" and [27] is K. A. Stevens, "Surface perception from local analysis of texture and contour"; the two citation numbers appear swapped.
- p. 1 Introduction, right column
shape from contour [27] and shape from texture [31]
- p. 20 Reference [31]
A. P. Witkin, "Shape from contour,"
- p. 20 Reference [31]
A. P. Witkin, "Shape from contour," M.I.T. Artificial Intell. Lab.
- p. 19 Reference [27]
"Surface perception from local analysis of texture and contour,"
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: what the two technical reports actually cover. On p.1 the image reads 'shape from contour [27] and shape from texture [31]'. The reference list (p.19 [27] and p.20 [31]) has the titles quoted. Going by the titles in the paper, [31] fits contour; [27] mentions both topics. Only the author can say whether the pairing was intended.
Second check: another model agreed
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Reference or label p.3: "the use of the product simplifies the analysis for step edges, as should become clear in Section III"; the step-edge analysis showing the product is scale-invariant is in Section IV (Eqs. 21-22).
- p. 3 Sec. II-A, after Eq. (9)
become clear in Section III
- p. 3 Sec. II.A, paragraph after Eq. (9)
the use of the product simplifies the analysis for step edges, as should become clear in Section III
- p. 6 Section IV heading
IV. A DETECTOR FOR STEP EDGES
- p. 3 Sec. II-A, after Eq. (9)
simplifies the analysis for step edges, as should become clear in Section III
Judge's note: Section III (pp.682-683) covers numerical optimisation for ridge and roof profiles. It mentions step edges only in passing and itself says the step operator is 'derived in Section IV'. Section IV on p.684 gives the argument that the product of (19) and (20) is invariant under changes in scale. The forward pointer should be to Section IV.
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Reference or label Fig. 7 caption: "(d) Image thresholded with hysteresis using both the thresholds in (a) and (b)."; (a) is the unthresholded parts image, and the two thresholds T1 and 2T1 are those of panels (b) and (c).
- p. 12 Fig. 7 caption
both the thresholds in (a) and (b).
- p. 12 Fig. 7 caption
(d) Image thresholded with hysteresis using both the thresholds in (a) and (b).
- p. 12 Fig. 7 caption
Image thresholded with hysteresis using both the thresholds in (a) and (b).
Judge's note: The caption crop confirms: '(a) Parts image, 576 by 454 pixels. (b) Image thesholded at T1. (c) Image thresholded at 2 T1.' The pointer should be to (b) and (c).
Second check: another model agreed
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Wording p.689: "g2(x) = k exp(− x/2σ²)"; the step response of operator (42) is the Gaussian k exp(−x²/2σ²), consistent with the printed R22(x) = k2 exp(−x²/4σ²).
- p. 11 Sec. VI, Wiener filter derivation
g2(x) = k exp(− x/2σ²)
Judge's note: The p.11 image shows 'x/2σ²' with no square on x, and the text layer agrees. My working: integrating (42) gives exp(−x²/2σ²), and its autocorrelation is ∝ exp(−τ²/4σ²), which matches R22 as printed. The missing exponent is a misprint; the following result (R22) stands.
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Wording Sec. IX: the edge is I(x, y) = Au₋₁(y), so y is normal to the edge, yet the text says it scales the projection function (along the edge) by "replacing f(x, y) by f_l(x, y) = f(x, (y/l))", which stretches the filter normal to the edge.
- p. 15 Sec. IX
I(x, y) = Au−1(y)
- p. 17 Sec. IX, before Eq. (49)
replacing f(x, y) by fl(x, y) = f(x, (y/l))
Only the authors can say what is right here.
Judge's note: Page 15: 'step edge in the x direction ... I(x,y) = Au₋₁(y)'; the signal integral runs over y from −∞ to 0 and x over all values, so y is the cross-edge coordinate. Page 17: 'We have already seen what happens if we scale the function normal to the edge (21). We now do the same to the projection function by replacing f(x,y) by f(x,(y/l))'. My reworking: substituting y=l·y1 gives a signal of l·∫∫f dx dy1 and a noise of n00·sqrt(l·∫∫f² dx dy1), so the ratio is sqrt(l)Σ. With f(x/l,y) the same sqrt(l) ratio results, because the half-plane integral is invariant to scaling either axis. So the SNR result stands, but the substitution as printed scales the normal direction, not the projection (along-edge) direction. That matters for the accompanying claim that localization also improves as sqrt(l). The step is mislabelled, but the stated SNR result stands, so S3.
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20 minor (typos and the like; most checked by the judge only)
Reference or label Reference [21] (T. Poggio, H. Voorhees, and A. Yuille, "A regularized solution to edge detection") appears in the reference list but is not cited anywhere in the text.
- p. 19 Reference [21]
A regularized solution to edge detection
Only the authors can say what is right here.
Judge's note: A grep of the full text for '[21]', 'Voorhees', 'Yuille' and 'regulariz' finds only the reference-list entry. I checked the page 1 introduction image, where the stereo citations are [19], [20] and there is no [21]. Other references in that range ([19], [20], [22], [23]) are cited. Only the authors can say where it was meant to be cited.
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Wording p.684: "the changes are inversely related, that is, both criteria either increase or decrease by √w"; by Eq. (21) one criterion is multiplied by √w and the other divided by it, so 'both ... increase or decrease' misdescribes the relation.
- p. 6 Sec. IV, after Eq. (21)
both criteria either increase or decrease by √w
Judge's note: The image of p.6 confirms the wording. Eq. (21) gives Σ(f_w)=√w Σ(f) and Λ(f'_w)=Λ(f')/√w. The sentence can be read loosely as 'each changes by a factor √w', but as written it suggests both move in the same direction, which contradicts 'inversely related'.
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Wording Eq. (33) p.686 is labelled "J1[f + g] = ∫ 2g² + 2λ1g'² + 2λ2g''² dx ≥ 0" but it is the second variation, defined just above as δ²J = (ε²/2)J2[f, g].
- p. 8 Eq. (33)
J1[f + g] =
Judge's note: I checked the image of p.8. The text says J1 is the integral of g times the Euler equation, and defines δ²J through J2. The integrand of (33) is built from the second partial derivatives of Ψ, so it is J2, not J1. The result that follows (4λ2 > λ1²) is unaffected.
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Wording p.686: the solution (35) is stated for "the range [−W, 0]" with conditions "f(−W) = 0 ... f'(−W) = 0", but the second and fourth lines of (36) are f and f' evaluated at x = +1 (e.g. "a1e^α sin ω + a2e^α cos ω + a3e^−α sin ω + a4e^−α cos ω + c = 0").
- p. 8 Eq. (35)-(36)
f(−W) = 0
- p. 8 Eq. (36), second and fourth lines
a1e^α sin ω + a2e^α cos ω + a3e^−α sin ω
Only the authors can say what is right here.
Judge's note: My working: f(1) from (35) is a1e^α sinω + a2e^α cosω + a3e^{−α} sinω + a4e^{−α} cosω + c, which is the printed line. f(−1) = −a1e^{−α} sinω + a2e^{−α} cosω − a3e^{α} sinω + a4e^{α} cosω + c, which is not. The fourth line likewise matches f'(1). W is also set to 1 without comment. Because f is extended antisymmetrically, the equations would be consistent if they were derived on [0, W], so the stated range and conditions disagree with the equations rather than the result being affected.
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Wording Eq. (43) p.687 appears to print "|f'(0)| = 1/σs", whereas for f in (42) f'(0) = −1/σ², the value that the stated ΣΛ = √(8/3π) and r = √(4/15) require.
- p. 9 Eq. (43)
|f'(0)| = 1/σs
Only the authors can say what is right here.
Judge's note: My working: with f = −(x/σ²)exp(−x²/2σ²), f'(0) = −1/σ². The other three integrals in (43) check out: 1, √π/2σ, 3√π/4σ³. ΣΛ = √(2σ/√π)·(1/σ²)·√(4σ³/3√π) = √(8/3π) only if |f'(0)| = 1/σ². At the available resolution the glyph on p.9 looks like a subscript s, and the text layer reads it the same way as σ_s in (40)-(41). Because the glyph is small I am not fully certain; a higher-resolution scan would settle it.
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Wording Sec. VII says operator (47) locates "zero-crossings in the second derivative in the edge direction", but the paper defines "edge direction" as the tangent to the contour, and (47) is ∂²/∂n² with n normal to the edge.
- p. 13 Sec. VII, right column
second derivative in the edge direction
- p. 12 Sec. VII, first paragraph
the direction of the tangent to the contour
- p. 13 Sec. VII, right column
by locating the zero-crossings in the second derivative in the edge direction
- p. 12 Sec. VII, first paragraph
we will use the term "edge direction" to mean the direction of the tangent to the contour
Judge's note: Page 12 defines 'edge direction' as 'the direction of the tangent to the contour'. Page 13 Eq. (45) and the text say n 'should be oriented normal to the direction of an edge', and Eq. (47) is ∂²/∂n² G*I = 0. Elsewhere the paper writes 'differentiating normal to the edge direction'. The sentence on p.13 therefore contradicts the paper's own definition and should say 'normal to the edge direction' (or 'in the direction n').
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Wording "(The Gaussian is the unique function with minimal product of bandwidth and frequency.)" should say bandwidth and spatial width.
- p. 13 Sec. VII, right column, parenthetical
minimal product of bandwidth and frequency
- p. 13 Sec. VII, right column, parenthetical
Gaussian is the unique function with minimal product of bandwidth and frequency.
- p. 13
minimal product of bandwidth and frequency.)
Judge's note: Page 13 image confirms the text. The preceding sentence speaks of 'very low bandwidth for a given spatial width', so the intended product is spatial width × bandwidth; as printed, bandwidth and frequency are the same domain.
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Typo p.684 Sec. IV: "the case where the input G(x) is step edge" is missing the article 'a'.
- p. 6 Sec. IV, first sentence
the input G(x) is step edge
Judge's note: The page image of p.6 shows the wording exactly as quoted.
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Typo pp.686-687: "no single best function for arbitrary ω, but a class of functions which were obtained by scaling a prototype function by ω" uses ω for the spatial scale factor, which is w elsewhere (Eqs. 21, 24-25) and ω is the frequency parameter of (35)-(37) and Fig. 4.
- p. 8 Sec. IV, last paragraph
no single best function for arbitrary ω
- p. 9 Sec. IV, top of left column
totype function by ω
Judge's note: The images of p.8 (last paragraph) and p.9 (first line) show the Greek ω. In Section IV the scale factor is the Latin w, f_w(x)=f(x/w).
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Typo p.686: "Since f(x) is asymmetric, we can extend the above definition to the range [−W, W] using f(−x) = −f(x)"; the property meant is antisymmetric.
- p. 8 Sec. IV, after Eq. (35)
Since f(x) is asymmetric
Judge's note: The image of p.8 confirms the wording. The relation f(−x) = −f(x) in the same sentence is antisymmetry, which the paper elsewhere calls 'antisymmetric'.
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Typo Eq. (42) p.687: "f(x) = − x/χ² exp(−x²/2σ²)" prints the prefactor denominator with a χ-like glyph instead of σ².
- p. 9 Eq. (42)
f(x) = − x/χ² exp
Judge's note: On the p.9 image the denominator glyph looks visibly different from the σ in the exponent and in (43). The intended symbol is evidently σ². This is a typesetting slip.
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Typo p.688: "r = √(4/15) ≈ 0.51"; √(4/15) = 0.516, which rounds to 0.52.
- p. 10 Eq. (44) and text below Fig. 6
r = \sqrt{4/15} ≈ 0.51
Judge's note: I checked the p.10 image. 0.515² is below 4/15, so √(4/15) > 0.515 and it rounds to 0.52. The printed value is a truncation, whereas the neighbouring √(8/3π) ≈ 0.92 is rounded normally. I also rederived r = √(4/15) from (41) and (43). The comparison 'worse by about 10 percent' against 0.576 is unaffected.
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Typo p.689: "will be determined mainly the noise energy" is missing 'by'.
- p. 11 Sec. VI, right column
will be determined mainly the noise
- p. 11 Sec. VI, right column
will be determined mainly the noise energy
- p. 11 Sec. VI
determined mainly the noise
Judge's note: The p.11 image (right column) shows the wording exactly as quoted.
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Typo Fig. 7 caption: "(b) Image thesholded at T1." should read "thresholded".
- p. 12 Fig. 7 caption
Image thesholded at T
- p. 12 Fig. 7 caption
Image thesholded at T1
- p. 12 Fig. 7 caption
Image thesholded at
- p. 12 Fig. 7 caption
Image thesholded at T,.
Judge's note: Page 12 image shows 'thesholded' in the Fig. 7 caption, while the next sentence of the caption spells it 'thresholded'.
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Typo Sec. VIII: "the edges from the second smallest scale that were not marked by at the first" has a stray word; it should read 'marked by the first'.
- p. 14 Sec. VIII, last paragraph
marked by at the first
- p. 14 Sec. VIII
not marked by at the first
Judge's note: Page 14 image shows 'marked by at the first'.
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Typo Sec. VIII: "we synthesize the large operator outputs than would have been produced" should read 'outputs that would have been produced'.
- p. 14 Sec. VIII, feature synthesis paragraph
puts than would have been produced
- p. 14 Sec. VIII, right column
we synthesize the large operator outputs than would have been produced
- p. 14 Sec. VIII
large operator out-puts than would have been produced
Judge's note: Page 14 image confirms 'than'.
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Typo Sec. VIII: "preventing the large channels for responding to 'fuzzy' edges" should read 'from responding'.
- p. 14 Sec. VIII, right column
preventing the large channels for responding
- p. 14 Sec. VIII, right column
preventing the large channels for responding to "fuzzy"
Judge's note: Page 14 image confirms the wrong preposition.
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Typo Sec. IX: "a filter whose impluse response is f(x, y)" should read 'impulse'.
- p. 15 Sec. IX
impluse re-
- p. 15 Sec. IX, right column
a filter whose impluse re-
- p. 15 Section IX, first column
whose impluse re-
Judge's note: Page 15 image shows 'impluse'.
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Typo Reference [5] title: "based on noncasual Markovian random field models" should read 'noncausal'.
- p. 19 Reference [5]
noncasual Markovian random field models
Judge's note: Page 19 image shows 'noncasual'. The intended term in the Markov random field context is 'noncausal'.
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Typo In the otherwise alphabetical reference list, "[21] T. Poggio, H. Voorhees, and A. Yuille" precedes "[22] A. P. Pentland", which is out of alphabetical order (the entry is also uncited, see C025).
- p. 19 References [21]–[22]
[21] T. Poggio, H. Voorhees, and A. Yuille
Judge's note: Page 19 image: [20] Mayhew, [21] Poggio, [22] Pentland, [23] Prewitt. 'Pentland' sorts before 'Poggio', so [21] is out of order. The fact that it is uncited is the same defect as C025; the ordering point is separate and cosmetic.
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