Perceptual Closure

Abstract

Perceptual closure is a form of visual perception: the tendency to perceive a complete, bounded figure when the sensory evidence for it is fragmentary or interrupted. Gestalt psychologists named closure as a law of grouping, and modern work distinguishes modal completion, in which an illusory surface with vivid subjective contours is seen across empty regions, from amodal completion, in which an occluded object is perceived as continuing behind its occluder. A relatability theory specifies the geometric conditions under which separated contour fragments are joined, and single neurons in early visual cortex respond to illusory edges as though a real contour were present. This article covers the two forms of completion, the interpolation rules that govern them, their neural substrates and computational models, and interactive demonstrations of Kanizsa figures, amodal occlusion, and contour relatability.

Keywords: perceptual closure, amodal completion, illusory contours

Perceptual closure is the visual system's tendency to register a whole, bounded object even when its contours are incomplete, interrupted, or hidden. A circle drawn as a broken ring is seen as a circle; a cat behind a picket fence is seen as one cat, not a row of vertical slices; a bright triangle can appear where no triangle has been drawn. The Gestalt psychologists identified this as one of the laws of perceptual organization, arguing that the visual field is organized into the simplest, most regular, most enclosed figures the stimulus allows, so that a nearly closed form is completed into a closed one (Wertheimer, 1923). Closure matters to cognitive psychology because it is a clear case in which perception delivers more than the retinal image contains: the seen figure is a construction, built to the regularities of surfaces and objects, and that construction can be studied precisely, tied to specific interpolation rules and to identifiable neurons in the visual cortex.

Modal and Amodal Completion

Perceptual completion comes in two forms that look different phenomenally but are now thought to share a mechanism. In modal completion the missing contour is filled with a visible quality: an illusory surface appears, usually seeming to lie in front of its inducers and often brighter than the surrounding page, bounded by subjective contours that cross regions of uniform luminance. In amodal completion the missing part is represented without any visible filling-in; a partly occluded object is perceived as a single continuous thing that carries on behind the occluder, even though the hidden portion is not seen in any sensory modality. The word amodal marks exactly this: the completion has no modal, sensory content, yet it decisively shapes what object is perceived.

Sekuler and Palmer traced the time course of amodal completion by briefly showing a partly occluded shape and then testing which completion had been formed. They found that the visual system settles on the completed, whole-object interpretation within about 200 milliseconds, passing through an early stage in which the visible fragments are represented before the amodal completion consolidates, evidence that completion is a rapid constructive process rather than a static inference (Sekuler & Palmer, 1992). The two forms differ in depth as well as in visibility (Table 1): the modally completed surface is seen as nearer than its inducers, while the amodally completed object recedes behind the occluder. Bence Nanay has argued that amodal completion is not a rare laboratory curiosity but a pervasive feature of ordinary seeing, present whenever any opaque object is viewed, since its far side and its occluded parts are always perceptually represented (Nanay, 2018).

Illusory Contours and Subjective Figures

The most striking demonstration of modal completion is the Kanizsa triangle, in which three black disks each with a wedge cut out, the inducers, are arranged so that their straight cut edges are collinear across the gaps between them. Observers see a bright white triangle lying on top of three complete disks, its three sides marked by crisp contours that cross regions where the page is a single uniform white. Gaetano Kanizsa, who made these figures central to the study of perception, emphasized that the illusory surface is not merely inferred but appears with a definite brightness and a definite edge, a subjective contour that has all the phenomenal force of a real one while corresponding to no luminance change in the image (Kanizsa, 1976). The illusion depends on the precise alignment of the inducers: rotating them so that their edges no longer meet across the gaps destroys the triangle, which is why the effect is a completion of relatable fragments rather than a general tendency to see triangles.

Illusory contours became a proving ground for theories of edge perception because they dissociate the seen contour from any local luminance cue, forcing an explanation in terms of longer-range interpolation. Lesher, reviewing several decades of psychophysics and modelling, catalogued how illusory-contour strength varies with inducer support, alignment, and spatial scale, and argued that any adequate account must combine local orientation signals with a cooperative process that links them across the gap (Lesher, 1995). The clarity of a subjective figure is graded, not all-or-none, and that graded strength is what the interpolation rules and neural models set out to predict.

The Relatability Account of Interpolation

The decisive theoretical step was to specify exactly when two visible edge fragments will be joined by a completion. Kellman and Shipley proposed that the same process underlies illusory contours and occluded objects, and defined the geometric condition, relatability, under which two edges are interpolated into one perceived contour. Two edge segments are relatable when a smooth, monotonic curve, bending through no more than ninety degrees, can connect them without reversing direction; edges that would require a sharper turn or a reversal are not relatable and no unified contour is seen (Kellman & Shipley, 1991). Their unifying claim was that modal and amodal completion are two expressions of a single interpolation mechanism: whether the result is a visible illusory edge or an unseen amodal boundary depends only on the depth relations and figural conditions in the display, not on two separate processes.

Relatability makes closure a matter of measurable geometry rather than of vague good form. It explains why the collinear inducers of a Kanizsa triangle produce such a strong contour, the interpolated edges turn through zero degrees, and why misaligning them abolishes the figure, the required connection is no longer smooth and monotonic. It also predicts the completion of curved and occluded contours, and it places the study of closure on a footing where quantitative predictions about which fragments will unite can be tested against what observers report.

Neural Substrates of Completion

Perceptual closure has an unusually direct neural correlate. Recording from the visual cortex of the alert monkey, von der Heydt, Peterhans, and Baumgartner found neurons in area V2 that respond to an illusory contour placed in their receptive field, firing to the subjective edge of a Kanizsa-type figure at the same orientation to which they respond for a real luminance edge, even though no local contrast falls on the receptive field (Figure 1). Neurons in the neighbouring primary area V1 largely lacked this property, locating the first stage of contour completion at the level of V2 (von der Heydt et al., 1984). This was among the earliest demonstrations that a constructed percept, rather than the physical stimulus, drives single cortical neurons. The same area also solves the problem that makes the completed surface look like a figure lying in front: many V2 neurons signal border ownership, coding not merely that an edge is present but which of the two abutting regions owns it and is therefore the figure, a one-sided assignment that fixes the depth ordering the modal surface depends on. Later work established that this figure-ground signal is not confined to tidy laboratory shapes but operates on the cluttered borders of natural scenes, so the same mechanism that assigns the front surface of a Kanizsa triangle also parses everyday occlusion (Williford & von der Heydt, 2016).

Human neuroimaging has since traced completion across the early visual areas. Murray and colleagues found that when a set of fragments is perceived as a single completed shape, activity in the lateral occipital object area rises while activity in primary visual cortex falls, consistent with a model in which a higher-level interpretation of the whole reduces the representational work left to V1 (Murray et al., 2002). Komatsu, reviewing the physiology of perceptual filling-in, distinguished the several mechanisms by which the brain assigns a surface quality, brightness, colour, or texture, to regions where the retinal image provides none, and located them across retinotopic visual cortex (Komatsu, 2006). Higher-resolution work has confirmed early involvement: de Haas and Schwarzkopf found spatially selective responses in human visual cortex to both Kanizsa illusory figures and occluded regions, showing that the cortical representation of completed and occluded space is laid out retinotopically as early as V1 and V2 (de Haas & Schwarzkopf, 2018).

Figure 1

A V2 Neuron Responds to an Illusory Contour as to a Real Edge

Computational Models of Boundary Completion

The psychophysics and physiology together pose a computational question: how does a network of orientation-selective units link fragments across a gap to yield a single, continuous boundary? Grossberg and Mingolla answered it with the Boundary Contour System, a model in which oriented cells at each location cooperate with aligned cells at nearby locations and compete with differently oriented ones, so that a completed boundary emerges as the stable state of a cooperative-competitive network even where the image supplies no local edge. In their account illusory contours, the grouping of collinear elements, and the sharpening of noisy edges are all products of the same boundary-completion circuit, later elaborated as the FACADE theory of form perception (Grossberg & Mingolla, 1985).

Such models make the relatability geometry mechanistic: the ninety-degree, monotonic limit on which edges combine falls out of the range and orientation tuning of the long-range horizontal connections that let one oriented cell excite another. Lesher's review compared this class of cooperative model against the alternatives and against the psychophysical data, concluding that the combination of local orientation measurement with a longer-range grouping stage captures the principal regularities of illusory-contour strength, even as the exact circuitry remained to be pinned down (Lesher, 1995). The models thus close the loop from the geometry of relatable edges, through a plausible cortical circuit, to the tuned V2 responses recorded physiologically.

The Development and Dynamics of Closure

Closure is not instantaneous, and its time course is informative. Ringach and Shapley measured how long the visual system takes to complete an illusory or occluded contour by masking a Kanizsa figure at varying delays, and found that boundary completion builds up over roughly 100 to 200 milliseconds, with the same temporal signature for the visible illusory contour and for the amodal completion of an occluded shape, further evidence that a common interpolation process serves both (Ringach & Shapley, 1996). The graded, time-extended character of the effect matches the cooperative-network models, in which a boundary settles as activity spreads and stabilizes across the gap.

The ability to close incomplete figures also has a developmental course, and it long served as a test of perceptual and cognitive integrity. Gollin measured recognition of pictures of common objects degraded to different degrees of incompleteness and found that younger children need more of the contour present before they recognize the object than older children or adults do, so that the amount of physical support a viewer requires for closure decreases with development (Gollin, 1960). Incomplete-figure tests of this kind became standard instruments in perceptual and neuropsychological assessment, precisely because the threshold at which a fragmented figure snaps into a recognized whole is a stable, measurable index of the closure process.

Worked Example

The strength of a completion can be quantified by how much of the final contour is physically specified. For a Kanizsa triangle, each side of the illusory figure runs between two inducers, and the only parts of that side actually present in the image are the straight cut edges of the inducers at its two ends; the stretch across the central gap is interpolated. A convenient measure is the support ratio: the physically specified length of a contour divided by its total length once the interpolated portion is included.

Take a Kanizsa triangle whose sides are each L = 6.0 cm long, formed by inducers whose straight cut edge contributes r = 1.2 cm of real contour at each end of a side. The physically specified length per side is then s = 2r = 2 × 1.2 = 2.4 cm, and the interpolated gap is L − s = 6.0 − 2.4 = 3.6 cm. The support ratio is s ÷ L = 2.4 ÷ 6.0 = 0.40, meaning that 40 percent of each perceived side is real contour and 60 percent is completion. Now enlarge the inducers so that each cut edge contributes r = 1.8 cm while the triangle keeps the same 6.0 cm sides: the specified length becomes s = 2 × 1.8 = 3.6 cm and the support ratio rises to 3.6 ÷ 6.0 = 0.60. The relatability geometry is unchanged, the interpolated edges are still collinear and turn through zero degrees, so both figures are completed; but the second, with more of its contour physically given, yields the clearer, stronger illusory edge. The point of the example is that closure is graded and that its strength tracks a simple ratio the display makes explicit, which is what lets the phenomenon be studied quantitatively rather than only described. Shipley and Kellman confirmed this empirically, showing that the perceived strength of an interpolated edge rises with its support ratio (Shipley & Kellman, 1992).

Discussion

Perceptual closure occupies a central place in perceptual science because it shows, cleanly and repeatably, that seeing is constructive. The Gestalt observation that the visual field organizes itself into complete, enclosed figures (Wertheimer, 1923) was sharpened by Kanizsa's subjective figures into a phenomenon that dissociates the seen contour from any luminance cue (Kanizsa, 1976), and then given a precise theory by the relatability account, which specified the geometry under which fragments unite and unified modal and amodal completion as one interpolation process (Kellman & Shipley, 1991). That the two forms share a time course (Ringach & Shapley, 1996; Sekuler & Palmer, 1992) and that computational models built on cooperative orientation circuits reproduce the effect (Grossberg & Mingolla, 1985; Lesher, 1995) make closure one of the better-understood constructive achievements of vision.

The neural evidence anchors the construction in cortex. Illusory contours drive V2 neurons as real edges do (von der Heydt et al., 1984), completion changes activity as early as V1 in human imaging (Murray et al., 2002; de Haas & Schwarzkopf, 2018), and the several routes to filling in a surface quality have been catalogued physiologically (Komatsu, 2006). The open questions are less about whether closure happens than about how routinely: a recent review of amodal-completion neuroimaging stresses how much of the work remains to be localized and how varied the reported substrates are (Thielen et al., 2019), and the argument that amodal completion pervades ordinary perception (Nanay, 2018) implies that the mechanisms studied with Kanizsa figures are engaged whenever any solid object is seen. The distinction worth keeping sharp is that the stimulus specifies only fragments, whereas the bounded, whole figure, modal or amodal, is an achievement of a visual system built to recover objects from partial evidence.

Common Misconceptions

Illusory contours are a simple optical illusion with no real perceptual basis.

The subjective contour of a Kanizsa figure drives orientation-selective neurons in area V2 much as a real edge does, so it is a genuine product of cortical contour processing, not a mistake of the eye (von der Heydt et al., 1984).

Modal and amodal completion are two unrelated phenomena.

The relatability account treats the visible illusory edge and the unseen occluded boundary as two outcomes of a single interpolation process, differing only in the depth and figural conditions of the display (Kellman & Shipley, 1991).

Closure happens automatically for any incomplete figure.

Interpolation occurs only when the fragments are relatable, joinable by a smooth, monotonic curve turning through no more than ninety degrees; misaligned edges that fail this condition are not completed (Kellman & Shipley, 1991).

Glossary

Amodal completion.

The perception of a partly occluded object as a single continuous thing whose hidden part is represented without being seen in any sensory modality.

Border ownership.

The coding of which of the two regions flanking an edge owns it and is therefore the figure, a one-sided assignment carried by many V2 neurons that fixes figure-ground and depth ordering.

Boundary Contour System.

Grossberg and Mingolla's model in which a cooperative-competitive network of oriented cells completes boundaries across gaps, generating illusory contours and grouping.

Closure.

The Gestalt grouping tendency to perceive an incomplete, interrupted, or occluded form as a complete, bounded figure.

Filling-in.

The assignment of a surface quality such as brightness, colour, or texture to a region where the retinal image supplies none.

Gestalt psychology.

The school that formulated the laws of perceptual organization, holding that the visual field is organized into wholes rather than assembled from independent parts.

Gollin figures.

A graded set of incomplete object drawings used to measure the amount of contour a viewer needs before the whole object is recognized.

Illusory contour.

A perceived edge that crosses a region of uniform luminance, seen where the image contains no local contrast; the boundary of a modally completed surface.

Inducer.

One of the elements, such as a notched disk, whose aligned edges evoke an illusory figure across the space between them.

Interpolation.

The visual process that joins two separated edge fragments into a single perceived contour spanning the gap between them.

Kanizsa triangle.

The classic modal-completion figure in which three notched disks with collinear cut edges induce a bright illusory triangle bounded by subjective contours.

Modal completion.

Completion that produces a visible illusory surface with subjective contours, typically seen in front of and brighter than its inducers.

Occlusion.

The blocking of one object by another nearer to the observer, the condition that triggers amodal completion of the hidden object.

Relatability.

Kellman and Shipley's geometric condition for interpolation: two edges are joined when a smooth, monotonic curve turning through no more than ninety degrees connects them.

Subjective contour.

A synonym for an illusory contour, emphasizing that the edge is contributed by the observer's visual system rather than by the stimulus.

Support ratio.

The proportion of a completed contour that is physically specified, equal to the real edge length divided by the total length including the interpolated portion.

Visual cortex area V2.

The second visual cortical area, whose neurons were the first found to respond to illusory contours as they do to real edges.

Key Researchers

Stephen Grossberg. Cognitive scientist at Boston University; with Ennio Mingolla he built the Boundary Contour System and FACADE theory, modelling illusory-contour formation and boundary completion as the emergent output of cooperative-competitive grouping circuits. Faculty Page - Google Scholar - Wikipedia - Wikidata

Rüdiger von der Heydt. Neuroscientist at the Johns Hopkins Mind/Brain Institute; with Peterhans and Baumgartner he discovered neurons in area V2 that respond to illusory contours as if a real edge were present, giving perceptual completion an early cortical substrate. Faculty Page - Google Scholar - ORCID

Gaetano Kanizsa (1913-1993). Italian psychologist at the University of Trieste; he made subjective contours central to perception with the Kanizsa triangle, in which aligned inducers evoke a vivid illusory surface crossing regions of uniform luminance. Wikipedia - Wikidata

Philip J. Kellman. Perceptual psychologist at UCLA; with Thomas Shipley he formulated the relatability theory of visual interpolation, specifying the geometry under which contour fragments unite and unifying modal and amodal completion under one process. Faculty Page - Google Scholar - Wikipedia - Wikidata

Bence Nanay. Philosopher of perception at the University of Antwerp; he argues that amodal completion is pervasive in ordinary seeing rather than exceptional, reframing perceptual closure as a constant contribution the visual system makes to represent occluded parts of objects. ORCID - Faculty Page - Google Scholar - Wikipedia - Wikidata

Max Wertheimer (1880-1943). Founder of Gestalt psychology; his laws of perceptual organization named closure as a grouping principle, formalizing the tendency to complete interrupted contours into whole, bounded figures. Wikipedia - Wikidata

Frequently Asked Questions

What is perceptual closure?
It is the visual system's tendency to perceive a complete, bounded figure when the sensory evidence is fragmentary, interrupted, or occluded, first identified by the Gestalt psychologists as a law of perceptual grouping (Wertheimer, 1923).

What is the difference between modal and amodal completion?
Modal completion produces a visible illusory surface with subjective contours seen in front of its inducers, whereas amodal completion represents an occluded object as continuing unseen behind the occluder; the relatability account treats both as one interpolation process (Kellman & Shipley, 1991).

What is a Kanizsa triangle?
It is a figure in which three notched disks with collinear cut edges induce a bright illusory triangle bounded by subjective contours that cross regions of uniform luminance, the classic demonstration of modal completion (Kanizsa, 1976).

When are two edges joined into one contour?
When they are relatable, meaning a smooth, monotonic curve turning through no more than ninety degrees can connect them; edges requiring a sharper turn or a reversal are not interpolated (Kellman & Shipley, 1991).

Where in the brain does contour completion occur?
Neurons in area V2 respond to illusory contours as they do to real edges, and human imaging shows completion changing activity as early as V1 (von der Heydt et al., 1984; de Haas & Schwarzkopf, 2018).

Is perceptual closure fast or slow?
It builds up over roughly 100 to 200 milliseconds, with the same time course for visible illusory contours and for the amodal completion of occluded shapes (Ringach & Shapley, 1996; Sekuler & Palmer, 1992).

How is the strength of a completion measured?
One measure is the support ratio, the proportion of the perceived contour that is physically specified; completions with a higher support ratio are seen as clearer and stronger (Shipley & Kellman, 1992).

Does closure change with development?
Yes; younger children require more of a figure's contour to be present before they recognize the whole object, so the physical support needed for closure decreases as perception matures (Gollin, 1960).

References