Convexity Shape Constraints for Image Segmentation
Loic A. Royer, David L. Richmond, Carsten Rother, Bjoern Andres, Dagmar Kainmueller
CConvexity Shape Constraints for Image Segmentation
Loic A Royer ∗ , David L Richmond ∗ , Carsten Rother ,Bjoern Andres , and Dagmar Kainmueller MPI-CBG, TU Dresden, MPI for Informatics [email protected]
Abstract
Segmenting an image into multiple components is a central task in computervision. In many practical scenarios, prior knowledge about plausible componentsis available. Incorporating such prior knowledge into models and algorithms forimage segmentation is highly desirable, yet can be non-trivial. In this work, weintroduce a new approach that allows, for the first time, to constrain some or allcomponents of a segmentation to have convex shapes. Specifically, we extend theMinimum Cost Multicut Problem by a class of constraints that enforce convexity.To solve instances of this APX-hard integer linear program to optimality, weseparate the proposed constraints in the branch-and-cut loop of a state-of-the-artILP solver. Results on natural and biological images demonstrate the effectivenessof the approach as well as its advantage over the state-of-the-art heuristic.
Image segmentation is a challenging task for which often times the use of suitable prior knowledgeabout the shape of the sought objects plays an important role. One interesting shape prior isconvexity [9, 10, 7, 6]. In natural images, it often occurs that there are multiple convex structuresof the same or different classes present in one image, such as wheels or various fruit, as well ascomposite structures constructed from convex parts, such as bricks and floor tiles. Biology similarlygives numerous examples of multiple convex structures. For example, many cell types are convex,such as bacteria, yeast, and more complicated cells densely packed into tissue. Numerous subcellularstructures, including nuclei and various types of vesicles, are also convex. Despite the clear relevanceof this situation to the task of image segmentation, respective priors have, to the best of our knowledge,not yet been addressed in the literature.Existing methods for segmenting multiple convex structures are specifically designed for certainshapes like ellipsoids or rods, as e.g. [5]. Such methods do not enforce generic convexity, but insteademploy priors of specific shapes that happen to be convex. Furthermore, such methods commonlysegment multiple structures sequentially, and neither the reconstruction of individual structures northe resulting segmentation of multiple structures is globally optimal w.r.t. the underlying objective.Beyond specific shape priors, there has been recent interest in generic convexity priors for binaryimage segmentation.
Star convexity priors were introduced in [10], where convexity is defined withrespect to all rays emanating from a central, user-defined seed point. This approach was generalizedto the case of Geodesic Star Convexity by [7], which defines convexity with regards to Geodesicpaths. Truly convex objects were first handled by [9]. However, this approach is limited to a singleforeground class, which must be explicitly modeled. The task of segmenting convex foregroundobjects without the requirement of user input or explicit modeling is studied in [6]. They propose agraphical model with triple-cliques that encode convexity constraints as 1-0-1 label sequences alongstraight lines in the image. This formulation captures the global nature of convexity. However, it ∗ Shared first authors a r X i v : . [ c s . C V ] S e p igure 1: Method overview. (a) The Multicut Problem 1 expresses image decomposition as a binaryedge labeling problem in a pixel grid graph. Components are inferred from the edge labels viaconnectivity analysis. (b) Cycle constraints enforce that edges between pixels in the same componentcannot be cut. (c) We enrich the Multicut Problem by convexity constraints . Our constraintsimplement the set theoretic definition of convexity on a discrete grid.implies that only one connected component of the foreground class can be present. Furthermore, thecomplexity of the problem requires the use of approximate solvers which may lead to local minima.None of the above methods is able to segment many generic convex objects of multiple foregroundclasses, fully automatically, without the need for user-defined seed points. Contributions.
In this work we propose the first model and solver for pixel-level segmentation ofmany generic convex objects of multiple foreground classes. We introduce new models that includeconvexity constraints into multicut problems , which are ILP formulations of image decompositionproblems. We consider two multicut problems, one equivalent to the correlation clustering prob-lem [2], another one equivalent to the Potts model. For efficiency, following the idea in [8, 1], weiteratively incorporate only the constraints that are violated per instance, and when no more violationsoccur, we are guaranteed the globally optimal solution. To the best of our knowledge, our models arethe first to handle many convex objects and multiple foreground classes. Our models can be solved toglobal optimality in small yet practical cases. Figure 1 gives an overview of the multicut problems aswell as our proposed convexity constraints, as described in detail in Sections 2 and 3.
Given the pixel grid graph G = ( V, E ) of an image, image segmentation tasks are often modeledas the problem of assigning, to each node v ∈ V , one label from a label set L = { , . . . , k } so asto minimize some objective function. A widely used objective function is the energy of a pairwiseconditional random field. A particular instance of this type of objective is the well-known Potts model.Its pairwise terms are non-zero only for edges that connect nodes with different labels. A special caseof the Potts model is the Correlation Clustering or Partitioning model [2]. The respective energyneglects unary terms, and the number of labels equals the number of nodes.In the following we describe two
Multicut Problems . The first one is equivalent to CorrelationClustering. The second one is equivalent to the Potts model. We also discuss respective solversas proposed in [8, 1]. These Multicut Problems are ILPs that form the basis of our contribution:In Section 3 we take the set theoretic definition of convexity of the connected components of asegmentation, and directly translate it into inequality constraints that we add to the respective ILPs.
The Minimum Cost Multicut Problem.
The minimum cost multicut problem [4] is equivalentto the correlation clustering problem. Its feasible solutions are all decompositions of the graph G .A decomposition of G is a partition Π of V such that, for every U ∈ Π , the subgraph of G inducedby U is connected. A subgraph of G that is connected and induced by a node set U ⊆ V is called a component of G . Any decomposition Π of G is characterized by the subset of edges that straddledistinct components: E Π = { vw ∈ E | ∀ U ∈ Π : v / ∈ U ∨ w / ∈ U } . Such a subset of edges is calleda multicut of G . There is exactly one multicut related to each decomposition of G . The followingTheorem forms the basis of multicut problems: Theorem 1
The multicuts of G are precisely the subsets Y ⊆ E for which every cycle C ⊆ E in G satisfies | C ∩ Y | (cid:54) = 1 . y ∈ { , } E be a 01-encoding of a multicut Y . It makes explicit, for any pair uv ∈ E ofneighboring nodes, whether u and v are in distinct components, namely iff y uv = 1 . These edgelabels y e allow for an equivalent formulation of | C ∩ Y | (cid:54) = 1 as linear inequality constraints (2),which leads to the following Definition: Definition 1 [4] Given a finite, simple, non-empty graph G = ( V, E ) and a map c : E → R (thatis, for any pair vw ∈ E of neighboring nodes, a cost or reward c vw for v and w being in distinctcomponents), the instance of the Minimum Cost Multicut Problem (MC) with respect to G and c isthe ILP min y ∈{ , } E (cid:88) e ∈ E c e y e (1)subject to ∀ C ∈ cycles ( G ) ∀ e ∈ C : y e ≤ (cid:88) e (cid:48) ∈ C \{ e } y e (cid:48) (2)Constraints (2) are referred to as as cycle constraints . It is sufficient to consider only the chordlesscycles of G [4]. The Minimum Cost Multicut Problem with Node Labels.
We also consider a Multicut Problemthat is equivalent to the Potts model. This is a more constrained optimization problem in whichevery node assumes precisely one out of finitely many labels, and neighboring nodes are in the samecomponent iff they have the same label:
Definition 2
Given a finite, simple, non-empty graph G = ( V, E ) , a map c : E → R , a finite set L (cid:54) = ∅ and a map d : V × L → R (that is, for any node v and any label l , a cost or reward d vl for v being labeled l ), the instance of the Minimum Cost Multicut Problem with Node Labels (MCN) withrespect to G , L , c and d is the ILP min x ∈{ , } V × L y ∈{ , } E (cid:88) e ∈ E c e y e + (cid:88) v ∈ V (cid:88) l ∈ L d vl x vl (3)subject to ∀ v ∈ V : 1 ≤ (cid:80) l ∈ L x vl (4) ∀ v ∈ V ∀{ l, l (cid:48) } ∈ (cid:0) L (cid:1) : x vl + x vl (cid:48) ≤ (5) ∀ vw ∈ E ∀{ l, l (cid:48) } ∈ (cid:0) L (cid:1) : x vl + x wl (cid:48) − ≤ y vw (6) ∀ vw ∈ E ∀ l ∈ L : y vw ≤ − x vl − x wl (7)Here, any feasible solution ( x, y ) is constrained such that every node v is assigned at least and atmost one label, namely the unique l ∈ L such that x vl = 1 , by (4) and (5). It is also constrained suchthat, for any edge vw ∈ E , y vw = 1 if and only if v and w have distinct labels, by (6) and (7). Thus, y is the characteristic function of a multicut of G . It defines uniquely a decomposition of G . Solvers.
Branch-and-cut algorithms for Problem 1 are proposed in [8, 1]. They find globallyoptimal solutions in reasonable run-time in many practical cases by including constraints (2) perinstance of the problem only in case they are violated. Problem 2 is solved by [8] by transformingit into an equivalent Minimum Cost Multicut Problem on a modified graph. This transformationinvolves flipping the meaning of node label variables resulting in the so-called multiway cut problem . This Section provides the methodological contribution of our work. In Section 3.1, we propose amodel for image decomposition under the constraint that each component of the resulting partitionhas to be convex. We achieve this via additional inequality constraints that we include into theMinimum Cost Multicut Problem 1. Furthermore, we propose a respective model with node labels. Note that in [4] the interpretation of the 01-encoding is flipped, i.e. y e = 0 means that an edge is an elementof the multicut. { k, l } ∈ (cid:0) L (cid:1) , that a component of label k does not contain any node labeled l in its convex hull. This is more general than “simply” enforcingconvexity of components. We achieve this, again, via inequality constraints that we include into theMinimum Cost Multicut Problem with Node Labels 2. In Section 3.2 we propose a solver for theabove optimization problems. Let P ⊂ E denote an arbitrary open path in G = ( V, E ) . All components of an image decomposition Π are convex iff for any path P that does not contain any edges in E Π , the straight line between theend points of P also does not contain any edges in E Π . We discretize this set theoretic definition ofconvexity as follows: In a pixel grid graph with the usual embedding into the 2d plane, where pixelsare Voronoi regions of graph nodes (cf. Fig. 1a), a component is discrete convex iff for every path P that does not contain any edges in E Π , the interior of the loop formed by P and the straight linebetween its end points does not enclose any nodes of a distinct component. See Figure 2 for a sketch.We call the set of nodes enclosed by this loop the hull of P . Let S ( P ) denote the path in G that runsalong the boundary of the discrete hull of P and connects the first and the last node covered by P (cf.Fig. 2). All components of an image partition are discrete convex iff ∀ P ∈ paths ( G ) : (cid:88) e ∈ P y e = 0 ⇒ (cid:88) e ∈ S ( P ) y e = 0 . (8)For a sketch, see Figure 1c. Note that in general, the Bresenham line [3] is different from S ( P ) , assketched in Figure 2. We propose to formulate (8) as linear inequality constraints, which enables usto formulate the task of finding the optimal decomposition of G into convex components as an ILP: Definition 3
Given a finite, simple, non-empty graph G = ( V, E ) and a map c : E → R , the instanceof the Minimum Cost Convex Component Multicut Problem (Convex-MC) with respect to G and c isthe ILP min y ∈{ , } E (cid:88) e ∈ E c e y e subject to (2) (9)and ∀ P ∈ paths ( G ) : | S ( P ) | · (cid:88) e ∈ P y e ≥ (cid:88) e ∈ S ( P ) y e (10)We also refer to this model as correlation clustering with convexity constraints , to put it into well-known terminology. Lemma 1
Constraints (10) are equivalent to (8) . A proof of Lemma 1 is given in the Appendix.Now we consider image decomposition with node labels, which is equivalent to the Potts model(cf. Section 2). Let V P denote the nodes covered by a path P , and V S ( P ) the nodes covered by therespective straight line S ( P ) but not by P (i.e., all but the “end points” of S ( P ) ). Given an imagedecomposition with node labels x , all components assigned to label k are convex iff ∀ P : (cid:88) v ∈ V P x vk = | V P | ⇒ (cid:88) v ∈ V S ( P ) x vk = | V S ( P ) | . (11)More generally, components assigned to label k have in their convex hulls only nodes with labelsfrom a subset of L k ⊂ L (where k ∈ L k ) iff ∀ P : (cid:88) v ∈ V P x vk = | V P | ⇒ (cid:88) v ∈ V S ( P ) ,l ∈ L k x vl = | V S ( P ) | . (12)4his holds because (5) entails x vk = 1 ⇒ (cid:80) l ∈ L k x vl = 1 . Constraints (11) are a special case of(12), namely with L k = { k } . We propose to formulate (12) as linear inequality constraints and henceyield an ILP that models image decomposition, with node labels, into convex components: Definition 4
Given a finite, simple, non-empty graph G = ( V, E ) , a map c : E → R , a finite set L (cid:54) = ∅ , a map d : V × L → R , and a set L k ⊂ L for each k ∈ L with k ∈ L k , the instance of the Minimum Cost Convex Component Multicut Problem with Node Labels (Convex-MCN) with respectto G , L , c , d , and { L k : k ∈ L } is the ILP min x ∈{ , } V × L y ∈{ , } E (cid:88) e ∈ E c e y e + (cid:88) v ∈ V (cid:88) l ∈ L d vl x vl subject to (4) , (5) , (6) , (7) (13)and ∀ P : | V S ( P ) | · (cid:32) | V P | − (cid:88) v ∈ V P x vk (cid:33) ≥ | V S ( P ) | − (cid:88) v ∈ V S ( P ) ,l ∈ L k x vl (14)We also refer to this model as Potts model with convexity constraints , to put it into well-knownterminology. The proof of equivalence between the convexity constraints (12) and (14) worksanalogously to the proof of Lemma 1 that we give in the Appendix.
Similar to [8, 1] we pursue a cutting plane approach in which violated cycle- and convexity constraintsare separated and added incrementally. Our algorithm starts by solving MC (see Def. 1) or MCN(see Def. 2) with the method described in [8]. Given the resulting image decomposition, we solve theSeparation Problem w.r.t. convexity constraints as described below. However, solving an ILP withadded convexity constraints can entail new violations of cycle constraints. Hence we propose thealgorithm defined in Figure 3. Upon termination, the algorithm gives the globally optimal feasiblesolution. In practice we can, optionally, allow for ILPs to be solved up to some relative gap of (cid:15) % . Inthis case our algorithm gives a feasible solution whose energy is at most (cid:15) last % away from the globaloptimum, where (cid:15) last % is the gap obtained in the last iteration, i.e. directly before termination.Figure 3: Our proposed solver for our Convex Component Multicut Problems (Definitions 3 and 4). Separation of Convexity Constraints.
Given an image decomposition, assuming a 2D pixel grid,we detect violated convexity constraints by scanning along sequences of nodes, { v , . . . , v n } , eachdefined by a start node v = ( x, y ) and an offset ( a, b ) , with a, b ∈ N co-prime. In case of theConvex-MC problem 3, convexity is violated iff two nodes v i , v j along a sequence lie in the samecomponent, and there exists a node v z in the sequence with i < z < j such that v z lies in a differentcomponent. In case of the Convex-MCN problem 4, there is a violation of (generalized) convexity iff v i , v j lie in the same component, labeled k , and there exists a node v z with i < z < j such that v z islabeled l (cid:54)∈ L k .The cardinality of the set of orientations of sequences, A := { ( a, b ) co-prime, − N x ≤ a ≤ N x , N y ≤ b ≤ N y } , is bounded by O ( N ) , where N = N x · N y denotes the number of pixelsin a 2d image. Scanning for violations along a particular sequence can be done greedily. Foreach orientation, the union of the sets of sequences to consider covers the whole image. Hencethe computational complexity for processing one particular orientation is in O ( N ) . Overall, thecomputational complexity of scanning for violations is bounded by O ( N ) . Optionally, one maywant to consider only a subset of orientations, A ⊂ A . Then the effort reduces to O ( | A | · N ) .5iven a pair of nodes ( v i , v j ) that constitutes a violation of convexity, we need to find a respectivepath P and straight line S ( P ) . As for a path P from v i to v j through their component: In generalthere are many such paths. To form a constraint, we pick just one such path. We explore twovariants: In case of Convex-MC (see Def. 3), a shortest path in terms of number of edges; In case ofConvex-MCN (see Def. 4), a cheapest path in terms of unaries of the label assigned to the component.Dijkstra’s algorithm yields an optimal path for either variant.We compute the straight line S ( P ) as follows: (1) Determine the orientation of the loop formedby path P and the (continuous) straight line v i , v j . (2) Infer the direction of the normal on the(continuous) straight line v i , v j that points to the interior of this loop. Without loss of generality, weassume that P and the continuous straight line do not intersect. (3) Greedily find the discrete shortestpath with minimal distance of nodes to the continuous straight line, subject to the constraint that nonode lies outside the loop. We present proof-of-concept results of our method on various photographs and biological images.For each exemplary image, in addition to the result of our method, we also show the respective resultobtained without convexity constraints. Finally, we provide a comparison to state of the art [6] on twoexemplary images. We employ a four-connected grid graph in all experiments. If not noted otherwise,we check violations of convexity in 8 discrete directions, and set the stopping criterion for the ILPsolver to 2%. This is not a hard stopping criterion, and hence smaller gaps are achieved per instance.Table 1 lists the gaps, number of iterations, run-times, and energies obtained for each of the examples.
Potts Model.
Figure 4 shows examples to which we apply two-label Potts models enriched byconvexity constraints on the foreground label by means of the Convex-MCN problem 4. Figure 4(a)shows a biological image of ellipsoidal cell nuclei in the tissue of a nematode worm. The imageshows four whole nuclei, and a fraction of a fifth one at the bottom. Convexity constraints allow fora perfect segmentation of the nuclei. In contrast, a Potts model without convexity constraints failsto split apart the nuclei, and yields holes within components. Figure 4(d) shows a biological imageof polygonal cells in a fly wing. Again, convexity constraints allow for a perfect segmentation ofthe nuclei, while a respective Potts model without convexity constraints is not able to correctly splitapart the cells. Figure 5(a) shows a photograph to which we apply a three-label Potts model withconvexity constraints on one foreground label. Our method is able to accurately segment two convexforeground objects on top of a second, non-convex foreground label.Results in Figures 4 and 5(c) were achieved with “simple” convexity constraints as captured by (11).An example for the more general constraints (12) is given in Figure 5(d-f). Here, we enforcecomponents labeled “apple” to be convex as such, while we enforce components labeled “orange”to be convex only w.r.t. the background. In other words, we allow concavities in orange-labeledcomponents as long as they are filled exclusively by apple-labeled nodes. Consequently suchconcavities appear as desired. However, incorrect spurious apple components also appear withinorange components. This result is interesting despite the fact that our model does not achieve aperfect segmentation, because it shows potential implications of the generalized constraints.
Correlation Clustering.
Figure 6 shows two exemplary results for correlation clustering enrichedby convexity constraints by means of the Convex-MC problem 3. In both examples, densely packedobjects, namely bricks in a stone wall and cells in a fly wing, are nicely separated due to convexityconstraints. At the same time, the space between these objects is tesselated into convex components.Although these results might not be of direct use as segmentations of the respective objects, they maywell serve as convex supervoxelizations to be used as input for further processing.
Comparison to State-Of-The-Art on Two Exemplary Images.
We compare our method to thestate-of-the-art for segmentation with convexity constraints [6]. The method of Gorelick et al. [6] isable to handle only one convex structure of one foreground label, and we chose exemplary imagesaccordingly (Figure 7). We use the code provided by the authors. First we study a synthetic image(Figure 7 top). The method of [6] initializes via the Graph Cuts solution, yielding a hole in theforeground object. In the process of resolving this high energy configuration, the method breaks theouter boundary of the object and settles to a sub-optimal solution. In contrast, our method is able toobtain the globally optimal solution. On a second, biological image (Figure 7 bottom), we ran [6]6a) (b) (c) (d) (e) (f)Figure 4: Examples for convexity constraints in a two-label Potts model: Excerpts of microscopicimages of (a) cell nuclei of a nematode worm, and (d) densely packed cells in a fly wing. (b/e) Two-label Potts model. (c/f) Convexity constraints on foreground label of the respective Potts model. Notethat for the nuclei we check violations of convexity in all discrete directions.(a) (b) (c) (d) (e) (f)Figure 5: Examples for convexity constraints in three-label Potts models. (a) Photograph of two friedeggs in a pan (by Matthew Hurst on flickr, CC BY-SA 2.0). (b) Three-label Potts model. (c) Convexityconstraints on yolk label of the same three-label Potts model. (d) Image of an apple occluding anorange. (e) Three-label Potts model. (f) Generalized convexity constraints, with L apple = { apple } and L orange = { apple, orange } . Thus the apple label cannot have anything but apple in it’s convexhull, while the orange label is allowed to have apple in it’s convex hull, but not background.(a) (b) (c) (d) (e) (f)Figure 6: Examples for convexity constraints in correlation clustering models: (a) Natural image ofa stone wall. (d) Densely packed cells in excerpt of microscopic image of fly wing. (Same imageas in Figure 4(d).) (b/e) Correlation clustering model. (c/f) Convexity constraints in the respectivecorrelation clustering model. Experiment Res E ConvexE
Table 1: For each experiment, we list the image resolution in pixels (Res), the energy of the solutionobtained without convexity constraints (E), the energy of the solution obtained with convexityconstraints (ConvexE), the number of times that convexity constraints are iteratively added to the ILP(
We proposed a new approach that introduces convexity constraints into two multicut problems thatare equivalent to Correlation Clustering and the Potts Model, respectively. Our approach handlesconvexity constraints for many connected components of multiple different classes, and additionallyfor pre-specified convexity relationships between objects of different classes. All concepts describedin this paper extend in a straightforward way to 3D. In future work we will explore strategies forimproving the run-time, e.g. via warm-start of subsequent ILPs, for the application to larger data.
Appendix
Proof of Theorem 1.
First we show that every multicut satisfies | C ∩ E Π | (cid:54) = 1 for all cycles.Given a decomposition and related multicut E Π , assume there exists a cycle C with C ∩ E Π = { uv } .Then uv straddles distinct components because uv ∈ E Π . On the other hand, the path C \ { uv } connects u and v with edges that are not in E Π , and hence u and v belong to the same component. Bycontradiction, this proves that | C ∩ E Π | (cid:54) = 1 for all cycles. Second we show that every Y that satisfies | C ∩ Y | (cid:54) = 1 for all cycles is a multicut of G . Given such a Y , we construct a decomposition Π andrelated multicut E Π of G as follows: uv (cid:54)∈ E Π : ⇔ there exists a path P between u and v such that P ∩ Y = ∅ . Now we show that Y = E Π . If uv (cid:54)∈ Y , then uv (cid:54)∈ E Π , because there is a path between u and v such that P ∩ Y = ∅ , namely P = { uv } . If uv (cid:54)∈ E Π , assume uv ∈ Y . Then, because uv (cid:54)∈ E Π , there exists a path between u and v such that P ∩ Y = ∅ . Hence the cycle C := P ∪ { uv } satisfies | C ∩ Y | = 1 . By contradicting uv ∈ Y , this proves that if uv (cid:54)∈ E Π , then uv (cid:54)∈ Y . Proof of Lemma 1.
Given a path P . In case (cid:80) e ∈ P y e = 0 it follows from y e ≥ that (10) ⇔ (cid:80) e ∈ S ( P ) y e ≤ ⇔ (cid:80) e ∈ S ( P ) y e = 0 ⇔ (8). Case (cid:80) e ∈ P y e (cid:54) = 0 entails (cid:80) e ∈ P y e ≥ because y e ∈ { , } . Hence | S ( P ) | · (cid:80) e ∈ P y e ≥ | S ( P ) | . And, | S ( P ) | ≥ (cid:80) e ∈ S ( P ) y e because y e ≤ . So, true ⇔ (10) ⇔ (8). 8 eferences [1] B. Andres, J. Kappes, T. Beier, U. K¨othe, and F. Hamprecht. Probabilistic image segmentation withclosedness constraints. In Computer Vision (ICCV), 2011 IEEE International Conference on , pages2611–2618, Nov 2011.[2] N. Bansal, A. Blum, and S. Chawla. Correlation clustering.
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