Invariants underlying shape inference are elusive: A variety of shapes can give rise to the same image, and a variety of images can be rendered from the same shape. The occluding contour is a rare exception: It has both image salience, in terms of isophotes, and surface meaning, in terms of surface normal. We relax the notion of occluding contour and, more accurately, the rim on the object that projects to it, to define closed extremal curves. This new shape descriptor is invariant over different renderings. It exists at the topological level, which guarantees an image-based counterpart. It surrounds bumps and dents, as well as common interior shape components, and formalizes the qualitative nature of bump perception. The invariants are biologically computable, unify shape inferences from shading and specular materials, and predict new phenomena in bump and dent perception. Most important, working at the topological level allows us to capture the elusive aspect of bump boundaries.

*the*given image directly to

*the*surface that gave rise to it. Thus far, this goal has been elusive. Instead, to find a well-defined image-to-surface map, computational researchers constrain the problem by, for example, limiting reflectance (say, to Lambertian materials), limiting surfaces (say, to elliptical patches), or training deep neural networks with limited data sets (references below). We propose an alternative approach. Instead of limiting the problem in this fashion, we seek a more general solution. To achieve this, we sacrifice the unique surface as our goal and instead postulate an intermediate stage between the image and the surface. This intermediate stage is topological in nature and different from previous approaches. With it we are able to identify those areas of the image that are most informative about shape and on which shape inferences could be anchored. This leads to a novel invariant between collections of images and collections of surfaces. The images could derive from different material, lighting, and rendering physics or from different observers. The invariant provides an explanation for certain aspects of shape psychophysics. Perhaps the most radical aspect of our invariant, for the perception community, is that it is defined over a neighborhood—a portion of the image or surface—that surrounds certain types of curves. These are the extremal contours mentioned in the title and which are developed in this article.

*critical contours*(Kunsberg & Zucker, 2018). Since the earlier work was more formal, the first part of this article is an informal review of the relevant material. The hope is this will make the material more accessible to the perception community. The new technical contribution in this article is the specialization of our invariant to defining bumps and dents (i.e., protrusions up to the convex/concave ambiguity).

^{1}This specialization is nontrivial, since it involves showing that the definition of a bump is generic. While bumps are classical components of shapes, and we all seem to “know one when we see one,” finding a definition for them has been problematic. In fact, there is even disagreement around where the boundary of a bump should be located (Watt & Morgan, 1983; Georgeson et al., 2007; Morgan, 2011; Subedar & Karam, 2016). We embrace this ambiguity; in fact, it is precisely what our topological invariant captures.

*Occluding contours are the projections of rim curves of maximal slant*. But therein also lies their limitation; occluding contours exist only at exterior boundaries and self-occlusions, that is, at very special locations within the image. Our goal in this article is to generalize such constructs to interior shape features, specifically bumps (and dents). In effect, we seek to “extend” the occluding contour into the interior of a shape. Just as the occluding contour bounds shapes, we study how to bound bumps (and dents and ridges, etc.). Importantly, the well-defined crispness of the occluding contour is relaxed into a neighborhood on the shape and thus into a neighborhood in the image. This is specifically where our sacrifice (from the opening paragraph) manifests: It is within these neighborhoods that the image and surface salient properties are defined.

*the*shape, the uniqueness requirement is relaxed (partly). Bumps, in our theory, have a boundary but it is only implicit; it may differ for individuals, for renderings, and for viewings, but it always “lives” within a band of uncertainty—a neighborhood. This neighborhood has a small diameter (an \(\epsilon\)-neighborhood of the bump boundary); for illustration, see Figure 1b–d. The red “sausage” is the heart of our approach. Regardless of how the ridge bends, there is a sausage that surrounds it. But there is more: The sausage has a special geometric structure, through which contours can be defined. When the sausage is over the image, we call the contour a

*critical contour*. Its counterpart on the surface is an extremal curve of slant. The sausage contains an image pattern relating to a surface property that is invariant of the rendering function. In the spirit of how an artist’s drawing can convey the impression of shape rather than its exact coordinates, the critical contour constrains the possible shapes. Taken in cross section, the surface is ridge-like but only partly constrained (see Figure 1d, top). As the rendering changes a little, or the viewpoint changes a little, the critical contour will always be in a similar red sausage. As the ridge becomes steeper, the sausage gets thinner; the limiting process is defined in Kunsberg and Zucker (2018). The image in the interior of the sausage may change a little with changes in material or lighting, but the existence of the boundary and the presence of the sausage are invariant. In effect, the meaning derives from the sausage and what it implies. The technical contribution in this article studies what happens when the critical contours are closed (i.e., when they surround bumps).

*the isophotes change drastically with changes in lighting or reflectance (e.g., Todd et al., 2014). While this observation is true in many—perhaps most—places in an image, it is not true in the red sausage neighborhood around critical contours.*This is the image side of the invariant we shall be developing.

*for the entire object.*Here topology again provides the payoff: The isophote

^{2}structure within the sausage follows a particular template for a wide range of renderings. While the intensity distribution changes

*almost*everywhere, it does not change everywhere. It follows a plan within the sausage, as is illustrated by the blue and red regions in Figure 1 and first characterized in Kunsberg and Zucker (2018), Kunsberg et al. (2018). Notice in particular how the (image) gradient points away from the sausage on both sides (Figure 1d). Consistency in this gradient orientation will be important in linking different sausages together and in finding the generic surface configuration corresponding to it.

*critical contour*—that cuts across the isophotes in a particular fashion (Figure 2e). This could be a boundary of the bump as seen in the image. In the scene domain, there is a corresponding contour—what we shall later describe as an

*extremal curve of slant*—that anchors the bump in surface terms (Figure 2f). Although in this artificial example, the critical contour in the image and the extremal curve of slant on the surface correspond closely, in general, they are only formally guaranteed to be contained in the sausage (Figure 2d).

^{3}Importantly, the presence of either one implies the existence of the sausage, which implies the existence of the other. Changing the rendering solely moves the critical contour around inside the sausage. All of this is proved in Kunsberg and Zucker (2018). We summarize this informally with the slogan:

*chairs*to

*cars*is limited (Sitzmann et al., 2019). While particular solutions engineered for particular applications can be useful (Parhi & Nowak, 2021), especially when the data set is acquired from natural scenes, it can be difficult to understand why the interpolation works (Hutson, 2018). One might speculate that “faces” work because there is an underlying invariance—a template (Blanz & Vetter, 1999)—that guides the interpolation. In any case,the robustness of our visual systems remains elusive.

*depends on which images are chosen*. Elliptical patches were a special case where a few curvature parameters specified the solution.

*critical contours*(Kunsberg & Zucker, 2018) are a concentration of shading. Imagine a drawing of a ridge: A thin line, perhaps drawn by an artist, would be the limiting case in which the ridge has infinitely steep intensity “walls” surrounding it (see Figure 7 in Kunsberg et al., 2018). Informally, critical contours can be viewed as a sketch of the shading inside a shape, just as an occluding contour is a sketch of the boundary of a shape. Qualitative judgments, such as relative depths, can still be made in some circumstances (Koenderink et al., 2015). Formally, the critical contours are those edges (technically, 1-cells) of the M-S complex that have large transverse second derivatives.

^{4}

^{5}It results, in effect, in an abstract, graphical version of what was just described. The mountain range becomes the value of a scalar function \(f(x,y)\). The nodes of the graph (

*0-cells*) are the extrema of this scalar function, or places where its derivative is 0; edges in the graph (

*1-cells*) connect maxima to saddles and saddles to minima. Notice, in particular, how cycles of four edges (

*2-cells*) are formed, connecting a maximum, a minimum, and two saddles in alternating order. These 2-cell quadrilaterals segment the mountain range into characteristic domains. We shall shortly be modifying these components to develop the abstract definition of a bump as a special type of domain.

*Remark 1.*The Morse-Smale complex is a topological description of a function; it makes certain of its shape features explicit but does not specify the precise function values everywhere. Values on the complex can be used to get a “weak” representation of the original function.

*critical contours*(Kunsberg & Zucker, 2018) are a concentration of shading. Imagine a drawing of a ridge: A thin line, perhaps drawn by an artist, would be the limiting case in which the critical contour has infinitely steep intensity “walls” surrounding it (see Figure 7 in Kunsberg et al., 2018). Informally, critical contours can be viewed as a sketch of the shading inside a shape, just as an occluding contour is a sketch of the boundary of a shape. Formally, the critical contours are those edges (technically, 1-cells) of the Morse-Smale complex that have large gradients surrounding them. Critical contours are computable from the image, and a main result of that theory is critical contours are abstractly invariant to changes in the rendering function. They are contained within the sausage and, in effect, they define a type of scaffold on which a shape can be built.

*Remark 2.*Critical contours provide a topological signature of key interior shape components, stable under generic lighting and rendering variations.

^{6}Now, viewed from above, notice how the slant is minimal (Figure 5f), then increases to its maximum on the steep sides, and then decreases again. While this maximum is not \(\pi /2\), it is large. The M-S complex for this is shown in Figure 5j. A blue extremal curve passes through a maximum (of the slant function) (yellow dot) and a (green) saddle point. The circular curve is the extremal line of slant, because this is the only part of the M-S complex that has “steep sides” of the slant function. The minimum is somewhere inside it. The template for this pattern (Figure 5b) schematizes this; in effect, this template is the definition of a bump in the slant domain. By passing from the max to saddles, it illustrates how the extremal curve is a relaxation of the occluding contour. The M-S complex on the slant function shows how the extremal contour encircles the bump, with a slant minimum (and no other maximum) inside it.

**Definition 3.1.**Slant extremal contours are the saddle-maxima 1-cells of the M-S complex of the slant function.

**Definition 3.2.**Extremal rings are closed slant extremal contours.

**Definition 3.3.**A bump/valley is an interior region within an extremal ring.

*surface meaning*and

*image salience*, provided the surface and rendering are generic.

^{7}The solution with higher Gaussian curvature will be the solution that is less smooth, more dependent on lighting direction (Freeman, 1994), and with a higher chance of occlusion. We compare the relative likelihood of the two solutions \(\frac{L_1}{L_2}\) by considering the inverse of the ratio \(\frac{T_2}{T_1}\). A simple calculation shows

*Remark 3.*The surface normal along an extremal contour points consistently to the interior or exterior, almost always. When closed, the normal points consistently to a bump or a dent.

**Definition 4.1.**Critical contours are gradient flows in the image with large transversal second derivatives.

*Remark 4.*Critical contours are computable from the image gradients, whereas extremal contours are computable from the slant gradients.

**Theorem 1.**

*Given a surface normal field*\(N(x, y)\)

*and any two choices of generic rendering functions*\(F_1, F_2 \in \mathcal {C}\),

*construct*\(I_1 = F_1(N(x, y)), I_2 = F_2(N(x, y))\).

*If a critical contour is present in*\(I_1\),

*then there is a arbitrarily close critical contour in*\(I_2\).

*Persistence simplification*(Edelsbrunner & Harer, 2010; Carlsson, 2009) is one of the important developments in computational topology. Basically, take a discrete structure, cover it with a smooth object, and then use it to calculate topological features such as critical points. By the above arguments, sampling and noise can introduce singularities that are irrelevant to the actually continuous structure. Just as blurring can smooth over tiny holes due to noise in an image, persistence simplification is a globally consistent way to reveal overall structure while removing those tiny holes and “irrelevant” noisy details that derive from quantization and discretization. Critical points that are extremely fine scale and local are eliminated in a kind of structure-preserving smoothing. Algorithms to compute the Morse-Smale complex from discretized images (i.e., a mesh) have been developed by Reininghaus and Hotz (2011), Sahner et al. (2008), and Weinkauf et al. (2010), among others. We use the algorithm of Reininghaus and Hotz (2011) in all of our experiments. However, because persistence only involves how much smoothing a critical point can survive, the explicit “steep sides” in the critical contours definition are not always incorporated. This explains the variation in the cobblestone image in Figure 9. Some artifacts (noise/discretization loops) were eliminated, and by and large, almost all bumps are localized in every example. But the result is not perfect and more work would be required to fully implement the critical contours “steep sides” criterion.

^{8}through it. Notice how this curve surrounds the bumps just as the extremal ring did. The extent of the parallel flow correlates with the sausage around the contour, which could provide a means for estimating it. Since these flows are a type of oriented texture, we speculate that working with nested flows also supports a generalization from shape-from-shading to shape-from-texture. Formally, this follows because texture compression induces similar flows (Cholewiak et al., 2014). Finally, we show a number of specular examples, in which the oriented flows arise from compression of the visual scene around the object (Fleming et al., 2004; Mooney & Anderson, 2014); see Figure 12.

*individually*.

*Remark 5.*This is not the concave/convex (“hollow face”) illusion in disguise because the surface portion outside the extremal ring is stably perceived as convex in depth. This is due to the portions of the occluding contour that are visible at the edges of the image. This example could be generalized to include \(N\) extremal “rings” leading to possible \(2^N\) ambiguous perceptions. Rather than a single concave/convex ambiguity on the global object, we could have concave/convex ambiguity on individual parts governed by the extremal curves.

- 1.
*Bumps exist in the interior*of a shape. The occluding boundary delimits the full extent of an object. We showed that bumps have a description that is a relaxation of the occluding contour in a precise mathematical sense. - 2.
*Bumps are qualitative*. While bumps have a boundary, precisely where it lies is less clear. Evidence suggests that shape perception is qualitative: That is, while different subjects agree on certain basic properties of shape, they disagree on quantitative details. We encompass this qualitative aspect of shape inferences in a topological representation. The uncertainty in precision of the bump boundary was represented as a topological sausage around the critical contour. - 3.
*Bumps are global objects*; they are defined not at a point, such as the curvature at the peak, but over a neighborhood. Like a mountain, a bump is a collection of material that builds to a peak; they can be climbed from many sides. We characterized the global nature of bumps with the Morse-Smale complex, a global topological descriptor for surfaces. The M-S complex was attractive because the level sets in climbing a bump are nested and increasing. - 4.
*Bumps have both image and scene signatures.*To be perceivable, there must be some image signature to the bump. To define this, we built on previous theoretical ideas of critical contours. Bumps were defined using extremal curves of slant. We prove invariance to many aspects of lighting and material changes. - 5.
*Bumps are distinct parts*of a shape. As such, they are bounded from one another and should be manipulable separately. We introduced several visual illusions that illustrate this multistability. - 6.
*Bumps have consistent normals*The M-S complex also indicates how surface normals fit together, providing a constraint system that operates on the surface normal and can be extended to the occluding contour.

*Journal of Vision,*3(10), 4, https://doi.org/10.1167/3.10.4. [CrossRef]

*Beyond the third dimension*. New York, NY: Scientific American Library.

*IEEE Conference on Computer Vision and Pattern Recognition*. IEEE, pp. 334–341.

*Neural Computation,*16(3), 445–476. [CrossRef]

*IEEE Computer Graphics and Applications,*36(4), 56–66. [CrossRef]

*ACM Computing Surveys,*40(4), 12:1–12:87. [CrossRef]

*IEEE Transactions on Pattern Analysis & Machine Intelligence*, (8), 775–790.

*SIGGRAPH '99: Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques*, pp. 187–194, https://doi.org/10.1145/311535.311556.

*Bulletin of the American Mathematical Society,*3(3), 907–950. [CrossRef]

*Publications Mathématiques de l'IHÉS,*68, 99–114. [CrossRef]

*Perception,*26(11), 1353–1366. [CrossRef]

*IEEE Conference on Computer Vision and Pattern Recognition*, 782–789.

*Perspectives in shape analysis*. New York: Springer.

*Bulletin of the American Mathematical Society,*46(2), 255–308. [CrossRef]

*Journal of Vision,*14(10), 1113. [CrossRef]

*Vision Research,*37(11), 1441–1449. [CrossRef]

*Vision Research,*36(10), 1399–1410. [CrossRef]

*Vision Research,*42(16), 2013–2020. [CrossRef]

*ACM Transactions on Graphics (Proc. SIGGRAPH),*22(3), 848–855. [CrossRef]

*Nature,*329(6138), 438–441. [CrossRef]

*Vision Research,*47(12), 1608–1613. [CrossRef]

*Computational topology: An introduction*. American Mathematical Soc: New York, Springer, 2016.

*Journal of Vision,*15(2), 24. [CrossRef]

*Vision Research,*33(7), 981–991. [CrossRef]

*Journal of Vision,*13(5), 10, https://doi.org/10.1167/13.5.10. [CrossRef]

*Proceedings of the National Academy of Sciences,*108(51), 20438–20443. [CrossRef]

*Journal of Vision,*4(9), 10. [CrossRef]

*Calculus of Variations: The Rice Undergraduate Colloquium*. American Mathematical Society.

*Nature,*368(6471), 542–545.

*Neuron,*27(2), 227–235.

*Journal of Vision,*7(13), 7.

*Attention, Perception, & Psychophysics,*83(6), 2709–2727, doi:10.3758/s13414-021-02282-5.

*Image and Vision Computing,*13(7), 543–557.

*Topological methods in data analysis and visualization III*. Springer, pp. 135–150.

*Journal of Mathematical Imaging and Vision,*60(6), 968–992.

*Shape from shading*. Cambridge, MA: The MIT Press.

*Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001*, pp. II-718–II-725.

*Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001*(Vol. 2, pp. 153–158). IEEE.

*Science,*360(6388), 861.

*ACM Transactions on Graphics,*26(3), 19.

*Perception,*36(8), 1191–1213.

*i-Perception*, 6(6), 2041669515615713. [PubMed]

*i-Perception,*6(6).

*Perception,*13(3), 321–330.

*Solid shape*. Cambridge, MA: The MIT Press.

*Journal of Modern Optics,*27(7), 981–996.

*Perception,*25(9), 1009–1026.

*Perception,*33(12), 1405–1420.

*Proceedings of the National Academy of Sciences,*90(16), 7495–7497.

*arXiv preprint arXiv:1503.03167*.

*Interface Focus,*8(4), 20180019.

*Encyclopaedia of computational neuroscience*. Springer, https://doi.org/10.1007/978-1-4614-7320-6_100661-110.1007/978-1-4614-7320-6_100661-1.

*SIAM Journal on Imaging Sciences,*11(3), 1849–1877.

*Journal of Physiology–Paris,*103(12), 18–36.

*Nature,*333(6172), 452–454.

*Vision Research,*40(2), 217–242.

*Journal of Cognitive Neuroscience,*33(10), 2017–2031.

*Mach bands: Quantitative studies on neural networks. Retina.*San Francisco, CA: Holden-Day.

*Vision Research,*36(15), 2351–2367.

*Current Biology,*29(2), 306–311.

*Current Biology,*25(6), R221–R222.

*An introduction to Morse theory*(Vol. 208). American Mathematical Society.

*The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science,*40(269), 421–427.

*Morse theory, annals of mathematics studies*. Princeton, NJ: Princeton University Press.

*Topology from the differentiable viewpoint*, Univ. Press of Virginia, Charlottesville, 1990.

*Biological Cybernetics,*53(3), 137–151.

*Current Biology,*24(22), 2737–2742.

*Elife,*8, e48214.

*The Poincaré Conjecture, Clay Mathematics Monographs*, vol. 3. American Math. Society, Providence, 521.

*Vision Research,*51(7), 738–753.

*Journal of Vision,*14(9), 15.

*Proceedings of the National Academy of Sciences,*108(30), 12551–12553.

*The American Mathematical Monthly,*114(9), 819–834.

*IEEE Transactions on Pattern Analysis and Machine Intelligence,*PAMI-6(4), 442–450.

*Journal of Vision,*17(10), 324–324.

*Acta Psychologica,*121(3), 297–316.

*JOSA A,*15(12), 2951–2965.

*i-Perception,*11(6), 2041669520982317.

*Elementary differential geometry*(Rev. 2nd ed.). Burlington, MA: Elsevier.

*Psychological Science,*19(1), 77–83.

*arXiv preprint arXiv:2105.03361*.

*Filling-in: From perceptual completion to cortical reorganization*. Oxford University Press.

*Scientific American,*259(2), 76–83.

*Nature,*331(6152), 163–166.

*CVGIP: Graphical Models and Image Processing,*53(2), 157–185.

*Topological Methods in Data Analysis and Visualization*(pp. 103–114). Springer, Berlin, Heidelberg.

*Computer Graphics Forum*(Vol. 27, pp. 735–742). Oxford, UK: Blackwell Publishing Ltd.

*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*(pp. 6296–6305).

*Vision Research,*38(23), 3805–3815.

*arXiv preprint arXiv:1906.01618*.

*Annals of Mathematics,*74(1), 199–206.

*Topological library: Part 1: Cobordisms and their applications*(pp. 251–268).

*Calculus on manifolds: A modern approach to classical theorems of advanced calculus*. Reading MA: Addison Wesley, Publiishing Company.

*Current Directions in Psychological Science,*30(2), 120–128.

*ACM Transactions on Applied Perception (TAP),*13(3), 1–13.

*Journal of Vision,*12(1), 12.

*arXiv preprint arXiv:1206.6445*.

*i-Perception,*5(6), 497–514.

*Current Biology,*19(8), R323–R324.

*ACM Transactions on Graphics,*26(3):77, 9, http://doi.acm.org/10.1145/1239451.1239528.

*Benjamin/Cummings series in the life sciences mathematics monograph series*(p. 130). W. A. Benjamin.

*Proceedings of the European Conference on Computer Vision (ECCV)*(pp. 52–67).

*Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition*(pp. 689–698).

*Vision Research,*23(12), 1465–1477.

*Computer Graphics Forum*(Vol. 29, pp. 1221–1230). Oxford, UK: Blackwell Publishing Ltd.

*Computer Graphics Forum,*(Vol. 28, pp. 1519–1528). Oxford, UK: Blackwell Publishing Ltd.

*Vision Research,*38(19), 2933–2947.

*Advances in Neural Information Processing Systems,*28, 127–135.

*IEEE Transactions on Pattern Analysis and Machine Intelligence,*21(8), 690–706.

*Journal of Neuroscience,*20(17), 6594–6611.

*PERCEPTION*(Vol. 48, pp. 69–69). London, England: Sage Publications Ltd.

*rim*of an object in \(\mathbb {R}^3\) is composed of all noninterior points where the view vector “glances” the object, that is, where the view vector lies in the tangent plane to the surface. The

*occluding contour*is defined as the projection onto the image of the rim of the object. A powerful (but often elusive) cue, it has been studied in Koenderink (1984, 1990) and Lawlor et al. (2009), among many others.

*surface meaning*. Second, the occluding contour has a consistent flow “signature,” so it also has

*image salience*. We develop these in turn, starting with a standard representation for surfaces.

*slant*\(\sigma (x, y)\) is the polar angle between the surface normal and the view direction. The

*tilt*\(\tau (x, y)\) is the azimuthal angle between the surface normal and the view direction; see Figure 18. Both can be considered scalar functions on the image domain. Of course, these functions are unknown when the surface is unknown.

*Remark. The slant achieves a global maximum on the occluding contour, since the slant of the visible surface must always be bounded by \(\pi /2\).*

^{9}It goes without saying that Morse’s actual contributions (Bott, 1980) and their implications (Bott, 1988) are far deeper and wide-reaching than the brief introduction here.

**Theorem.**

*Let M be a closed, compact, smooth submanifold of Euclidean space (of any dimension). Let*\(E: M \rightarrow \mathbb {R}\)

*be a smooth, real-valued function on M. Suppose that every critical point of E is nondegenerate. Then M can be built from a finite collection of cells, with exactly one cell of dimension i for each critical point of index i.*

*Morse function*: All its critical points are nondegenerate (meaning the Hessian at those points is nonsingular), and no two critical points have the same function value. Should the surface not be Morse, we can always perturb it slightly to obtain one.

*gradient*\(\nabla \sigma = \left( \partial f / \partial x, \partial f / \partial y \right)\) exists at every point. A point \(p \in \mathbb {R}^2\) is called a

*critical point*when \(\nabla \sigma (p) = 0\). This gradient field gives a direction at every point in the image, except for the critical points, which are rare (a set of measure zero). Following the vector field will trace out an

*integral line*. These integral lines must end at critical points, where the gradient direction is undefined. Thus, one can define an

*origin*and

*destination*critical point for each integral line.

*index*: The number of negative eigenvalues of the Hessian at that point. For scalar functions on \(\mathbb {R}^2\), there are only three types: a maximum (with index 2), a minimum (with index 0), and a saddle point (with index 1).

*1-cell*. It naturally must connect a saddle with either a maximum or a minimum. For example, a

*saddle-maximum 1-cell*connects a saddle and a maximum. The set of 1-cells will naturally segment the scalar field into different regions, called

*2-cells*. In addition, the scalar values on the 1-cells govern the values on the 2-cells. See Figure 3 for illustration.

*ascending manifold*is defined as the union of integral lines having that critical point as a common origin. Similarly, its

*descending manifold*is the union of integral lines with that critical point as a common destination.