P2-T5 — Edges, Contours and Simple Geometric Features

Learning objectives

After this lesson, you should be able to:

• Compute the image gradient with the Sobel operator and state its magnitude and direction.
• Explain why gradients amplify noise and describe how smoothing and thresholds with hysteresis address it.
• Describe the stages of the Canny detector and what a contour is.
• Compute area, centroid, bounding-box aspect ratio and circularity of a region and say what each can and cannot distinguish.

The idea: look at the change, not the level

Walk along a row of pixels across the edge of a bright block. The values are low, then jump to high. A uniform surface, whatever its brightness, gives nearly constant values. What identifies the boundary is the change. Taking differences of neighbouring pixels converts “where is the boundary?” into “where is the difference large?”.

A derivative responds to change, and noise is change. That is why every edge detector smooths first.

Edges are useful for several reasons. They are less sensitive to a uniform change of lighting than raw brightness (an edge between two surfaces stays an edge when the light dims, though its strength falls). They mark the outlines of objects. And they are the starting point of many geometric methods: lines, corners (Module 4), outlines for shape measurement.

From edges to contours to numbers

1. Smooth to suppress noise (Lesson P2-T3).
2. Differentiate to get the gradient: its magnitude says how strong the change is; its direction says which way the intensity increases.
3. Threshold the magnitude to keep strong edges, ideally with thin, connected results (Canny).
4. Trace the boundaries to get contours: ordered lists of boundary pixels around a region.
5. Measure the region: area, centroid, bounding box, perimeter, and ratios of these.

The mathematics: gradient, Sobel, Canny and shape measures

The gradient

The gradient of the image intensity I is the vector of its partial derivatives. Its magnitude and direction are

Gx, Gy

the rate of change of intensity along the columns and along the rows, in grey levels per pixel.

|G|

the edge strength at the pixel: large where brightness changes quickly.

θ

the gradient direction: perpendicular to the edge, pointing from dark to bright.

The Sobel operator

The derivative is approximated by a difference of neighbouring values, and the Sobel kernels combine that with a small amount of smoothing in the other direction:

Applying Kx gives Gx (the difference between the right and left columns, with the middle row counted twice), and Ky gives Gy. For a vertical step edge from level a to level b, Gx at the edge is (1 + 2 + 1)(b − a) = 4(b − a) and Gy = 0. The factor of 4 is the sum of the weights on one side; some libraries divide it out.

Noise amplification

If each pixel has independent noise of standard deviation σ_n, the difference of two neighbours has standard deviation √2 σ_n. The Sobel response sums eight weighted pixels, so its noise standard deviation is σ_n√(1 + 4 + 1 + 1 + 4 + 1) = σ_n√12 ≈ 3.5 σ_n, compared with a signal of 4(b − a) at a step. The signal-to-noise ratio of an edge is therefore about 4(b − a)/(3.5 σ_n) ≈ 1.15 (b − a)/σ_n: weak edges, with a step not much larger than the noise, cannot be separated from it by a threshold. Impulse noise is worse: one pixel with an extra 185 grey levels contributes up to 2 × 185 = 370 to the response of its neighbours, more than a genuine step of 90.

The Canny detector

The Canny detector (1986) is the standard pipeline built on the gradient. In outline:

1. Smooth with a Gaussian to reduce noise.
2. Compute the gradient magnitude and direction.
3. Non-maximum suppression: keep only pixels that are a local maximum of |G| along the gradient direction, which thins edges to one pixel.
4. Hysteresis thresholding: use two thresholds. Pixels above the high threshold are certain edges; pixels between the two are kept only if they connect to a certain edge. This keeps weak but connected edge segments and discards isolated noise.

Contours and shape measures

A contour is the boundary of a region, as an ordered list of pixels. For a binary region (a mask component) the quantities commonly used are:

A

the area in pixels (multiply by the pixel footprint squared for a physical area).

centroid

the mean position of the region’s pixels, a robust position for the object.

aspect ratio

a measure of elongation: near 1 for squares and discs, large for bars.

P

the perimeter. In a pixel image it depends strongly on how it is measured: counting pixel edges makes every diagonal boundary a staircase and overestimates its length, so treat circularity as approximate.

C

circularity: 1 for a perfect disc and smaller for other shapes. A square gives π/4 ≈ 0.785.

These are cheap, interpretable and invariant to translation; area and perimeter are not invariant to scale (the same object nearer is bigger), but aspect ratio and circularity are. Hough transforms extend the idea to find lines and circles from edge points by voting, and are a standard tool where the shape is known.

A worked example: Sobel on a step and shapes from a mask

Take a 3 × 3 patch across a vertical step from 60 on the left to 180 on the right:

Gx = (180 + 2·180 + 180) − (60 + 2·60 + 60) = 720 − 240 = 480. Gy = (60 + 2·60 + 60) − (60 + 2·60 + 60) = 0. So |G| = 480 and θ = 0° (the intensity increases to the right). With an edge threshold of 220 as in Lab P2-L3, this pixel is an edge; a hot pixel of 255 on a floor of 70 in a smooth region gives Gx up to 2 × 185 = 370 at its horizontal neighbours, which is also above 220 and produces a false edge. A median filter removes the hot pixel and the false edge before the gradient is taken.

Telling shapes apart with three numbers

Consider three regions found in a mask: a 6 × 6 square, a 3 × 12 bar, and a disc of diameter 7 pixels (the pixels whose centres lie within 3.5 pixels of the centre). The perimeter is counted as the number of pixel edges between the region and the background, and the bounding box gives the aspect ratio:

The areas of the square and the bar are equal (36), so area alone cannot tell them apart; aspect ratio does (1.0 against 4.0). The surprising entry is the disc: its circularity (0.59) is lower than the square’s (0.785), although a true disc has circularity 1 and a true square π/4 ≈ 0.785. The reason is the staircase: a curved boundary on a pixel grid is made of many small steps. Counting pixel edges overestimates the length of a boundary by 41% along a 45° run and by about 27% on average over all directions (4/π, the mean of |cos θ| + |sin θ|), while a boundary aligned with the grid is measured exactly. A disc is penalised on every part of its outline and a grid-aligned square on none, so the disc’s ideal circularity of 1 becomes about 1/1.27² ≈ 0.62 (measured: 0.59), while the square keeps its ideal 0.785. At this size, circularity from pixel counts is not a reliable shape test; it needs regions that are tens of pixels across and a perimeter measured along a smoothed contour.

The lesson for design: choose a measure that separates the classes you actually have. For a robot sorting bars from cubes, aspect ratio is enough. For round against square parts it takes more pixels, or the corners found in Module 4.

Engineering implications and limitations

Smooth, then differentiate. The order in the pipeline is not optional. Choose the smoothing for the noise (median for impulses, Gaussian for fine grain), and scale it to the size of the edges you want.

Use hysteresis rather than a single threshold. A high threshold alone breaks real edges into pieces; a low threshold alone keeps noise. Two thresholds with connectivity give long edges with few false ones. The thresholds are in units of gradient magnitude and depend on the contrast and noise of your images.

Remember edges are not objects. Shadows, textures, reflections, printed patterns and the seams of a floor all produce edges. Edges tell you where brightness changes, not where objects end. They are a cue to combine with others.

Measure shapes in physical units when you can. Convert areas and lengths with the known pixel footprint (or the camera model) and compare against the real dimensions of the objects. A rule such as “a part has an area of 20 to 30 cm²” survives a change of zoom or distance; “area between 300 and 450 pixels” does not.

Know the regime of validity. Shape measures such as circularity break for regions of a few pixels, for regions cut by the image border, and for partially occluded objects. A shape-based classifier needs an estimate of how many pixels the object has and must refuse to decide when there are too few.

Edge strength falls with the light. A threshold on |G| tuned on a bright image misses edges in a dim one for the same reason an intensity threshold does (Lab P2-L2). Normalise by local contrast, or tune on the darkest condition you need.

Common misconceptions

Engineering takeaways

1. Edges are where brightness changes quickly; the gradient measures it: |G| = √(Gx² + Gy²), θ = atan2(Gy, Gx).
2. Sobel gives Gx = 4(b − a) at an ideal step; it amplifies noise about 3.5×, and impulse noise far more, so smooth first.
3. Canny = Gaussian smoothing, gradient, non-maximum suppression and hysteresis thresholds; it yields thin, connected edges.
4. Contours and shape measures (area, centroid, aspect ratio, circularity) turn regions into numbers; pick measures that separate your classes and use enough pixels.
5. Edges mark brightness changes, not objects; combine them with other cues and tune thresholds for the darkest condition.

References and further reading

Classical sources; the tables were computed for this lesson from the stated regions.

• Canny, J. (1986). A computational approach to edge detection. IEEE Transactions on Pattern Analysis and Machine Intelligence 8(6): the Canny detector.
• Sobel, I. and Feldman, G. (1968). A 3×3 isotropic gradient operator for image processing. Presented at the Stanford Artificial Intelligence Project: the operator, as commonly cited.
• Gonzalez, R. C. and Woods, R. E. (2018). Digital Image Processing, 4th ed. Pearson: edge detection, boundary description and region descriptors.
• Duda, R. O. and Hart, P. E. (1972). Use of the Hough transformation to detect lines and curves in pictures. Communications of the ACM 15(1).

Knowledge check

Three conceptual questions. Write an answer, then reveal the explanation. Your answers stay in your browser.

What’s next

Next: when it all breaks: the lighting, exposure, noise, shadow and glare failures that defeat image preprocessing, and how to diagnose them.

Related content

Nothing below is required to finish this lesson.

• Before
  P2-T4: Thresholding and Segmentation

  Segment an image with fixed and adaptive thresholds and connected components.

• Lab
  P2-L3: Diagnose a Failing Image-Processing Pipeline

  Find which stage of a filter-threshold-contour pipeline fails and repair it.