The Persistent Homology of Dual Digital Image Constructions
Bea Bleile, Adélie Garin, Teresa Heiss, Kelly Maggs, Vanessa Robins
TThe Persistent Homology of Dual Digital Image Constructions
Bea Bleile , Ad´elie Garin , Teresa Heiss , Kelly Maggs , Vanessa Robins School of Science and Technology, University of New England, Armidale, Australia Laboratory for Topology and Neuroscience, ´Ecole polytechnique f´ed´erale deLausanne (EPFL), Lausanne, Switzerland. Institute of Science and Technology (IST) Austria, Klosterneuburg, Austria Research School of Physics, Australian National University, Canberra, Australia.
Abstract
To compute the persistent homology of a grayscale digital image one needs to build a simplicialor cubical complex from it. For cubical complexes, the two commonly used constructions(corresponding to direct and indirect digital adjacencies) can give different results for the sameimage. The two constructions are almost dual to each other, and we use this relationship toextend and modify the cubical complexes to become dual filtered cell complexes. We derive ageneral relationship between the persistent homology of two dual filtered cell complexes, andalso establish how various modifications to a filtered complex change the persistence diagram.Applying these results to images, we derive a method to transform the persistence diagramcomputed using one type of cubical complex into a persistence diagram for the other construction.This means software for computing persistent homology from images can now be easily adaptedto produce results for either of the two cubical complex constructions without additional low-levelcode implementation.
Persistent homology [10,24] allows us to compute topological features of a space via a nested sequenceof subspaces or filtration by returning a persistence diagram which reflects the connectedcomponents, tunnels, and voids appearing and disappearing as we sweep through the filtration. Ithas a wide range of applications in diverse contexts including digital images, used for example tostudy porous materials [20], hurricanes [21] or in medical applications [8]. While it is common toapply persistent homology to simplicial complexes arising from point clouds, digital images are madeup of pixels (in dimension d = 2) or voxels (for d ≥
2) rendering cubical complexes the natural choiceas they reflect the regular grid of numbers used to encode the image.There are two ways to construct a cubical complex from an image I : The V-construction V ( I )represents voxels by vertices and the T-construction T ( I ) represents voxels by top-dimensionalcubes. These constructions are closely related to two different voxel connectivities of classical digitaltopology. The V-construction corresponds to what is known in computer science as direct connectivity,where voxels are connected if and only if their grid locations differ by 1, so that each voxel has 2 d neighbours. For d = 2, pixels are 4-connected and the direct neighbours are to the left and right aswell as above and below. The T-construction corresponds to indirect connectivity, where voxels arealso connected diagonally, every voxel has 3 d − a r X i v : . [ m a t h . A T ] F e b t is well known that the choice of direct or indirect adjacency has an impact on the overalltopological structure of a binary image and the critical points of a grayscale image function. Theeffects are particularly significant when the image has structure at a similar length-scale to the digitalgrid. It will be no surprise then that the persistent homology can also be dramatically different whencomputed using the V- and T-constructions for the same image, see Figure 1 for an example. Afurther issue in classical digital topology is that a single choice of adjacency cannot be applied toboth the foreground and background of a binary image (or the sub-level and super-level sets of agrayscale image) in a topologically consistent way. If the sub-level set is given the direct adjacency,the super-level set must take the indirect adjacency (and vice versa) or the Jordan curve theoremwill fail to hold, for example. This suggests the existence of a duality-like relationship between the V-and T-constructions applied respectively to the sub-level and super-level sets of a grayscale image.The resulting cubical complexes are almost dual. Vertices in the V-construction correspond totop-dimensional cells in the T-construction and interior vertices of the T-construction correspondto top-cells in the V-construction. This paper establishes the precise nature of the duality-likerelationship between the cubical complexes of the T- and V-constructions, filtrations of these inducedby an image and its negative, and their persistence diagrams. We use these results to define simplealgorithms that return the persistence diagram for V ( I ) from software that computes diagramsbased on the T-construction and vice versa. Our results are based on a combinatorial notion of dualcell complexes and dual filtrations. We explain how the V- and T-constructions can be modifiedto obtain dual filtrations of the d -sphere. The relationship between the persistence diagrams of agrayscale digital image obtained via the V- and T-constructions then follows from an investigation ofthe effects of the required modifications on the persistence diagrams and the relationship betweenthe persistent homology of dual filtrations on dual cell complexes.Our duality results make it possible to use the advantages of different software packages evenwhen the cubical complex type is not the preferred one for the application at hand. For example,by using a streaming approach, the persistence software cubicle [23] is able to handle particularlylarge images that do not even need to fit into memory. However, it has only been implemented forthe T-construction. Thanks to Section 6.2, cubicle can now be used to compute the V-constructionpersistence of an image that does not fit into memory and therefore cannot be processed by existingV-construction persistence software. At its most basic level, the algebraic relationship between the persistent homology of two dual filteredcell complexes is similar to that between persistent homology and persistent relative cohomology. Thelatter corresponds to taking the anti-transpose of the boundary matrix [5] or equivalently, leavingthe boundary matrix as is and applying the row reduction algorithm instead of the column-reductionalgorithm [5, 11]. Lemma 3.1 shows that the same relationship applies to the boundary matrices ofdual filtered cell complexes. Therefore, Theorem 3.3 can be viewed as a translation of the knownbijection between the persistence pairs of persistent homology and relative persistent cohomologyinto the setting of dual filtrations. This theorem is the first step towards establishing the mappingbetween persistence diagrams of the T- and the V-construction of images in Section 6. Furthermore,it applies more generally to the persistent homology of dual filtered cell complexes without using theconnection to persistent relative cohomology.The symmetry of extended persistence diagrams [3] is also closely related to our Theorem 3.3.The mathematical setting for extended persistence is a filtered simplicial complex whose underlyingspace, X , is a manifold without boundary. It extends the homology sequence derived from a filtrationof X by the sub-level sets of a function X s = ( f − ( ∞ , s ]), to continue with the relative homology of2he pair ( X, f − [ r, ∞ )), where s is an increasing threshold and r is a decreasing one. As observedin [12], the cubical complex constructions used in digital image analysis do not provide a suitabletopological structure for extended persistence because of the inconsistencies that arise if both thesub- and super-level sets are treated as closed. The authors of [12] overcome this by constructing asimplicial complex from the digital image that consistently reflects the connectivity of both sub- andsuper-level sets, and use this to obtain the expected symmetries in the extended persistence diagram.In contrast, we work with the existing widely-implemented cubical complex constructions of digitalimages and establish results that permit a simple high-level algorithm to transform between tworegular (not extended) persistence diagrams.This paper is a follow-up of an extended abstract [13] published in the Young Researcher Forumof SoCG 2020. A version with appendices included is available on arXiv. These results have alreadybeen cited as the basis for an algorithm implemented by the developers of the software cubicalripser [15]. Our results are aimed at both pure and applied mathematicians who want to understand and usethe relationship between the persistent homology of dual filtered cell complexes and particularly thetwo standard constructions of cubical complexes from digital images.In Section 2 we define dual cell complexes and dual filtered complexes as used throughout thispaper, and provide a brief outline of the definitions and results of persistent homology. The reader whois not familiar with the theory of persistent homology should refer to [9, 24] for a more comprehensiveintroduction. Section 3 establishes the relationship between persistence diagrams of two dual filteredcell complexes.In Section 4, we describe and formalise the two standard cubical complexes used in topologicalcomputations on digital images. We explain how these two complexes (the T- and V-constructionsdescribed earlier) must be extended and modified to form dual filtered cell complexes with underlyingspace homeomorphic to the d -sphere. The effects these modifications have on persistence diagramsare derived in Section 5. For the investigation of one of these effects we use the long exact sequenceof a filtered pair of cell complexes arising from the category theoretic view of persistence modules.The last Section 6 states the results for persistence diagrams of digital images and explains howto compute the persistence diagram of the T-construction by simple manipulation of a persistencediagram computed using the V-construction, and vice versa. This gives a practical method foradapting the output from existing software packages that use one or the other construction to obtainthe persistence diagram for the dual construction. CW-complexes [17] generalize simplicial complexes to allow cells that are not necessarily simplicesbut homeomorphic to open discs or balls, for example cubes instead of tetrahedra. A CW-complex is regular if the closure of each k -cell is homeomorphic to the closed k -dimensional ball D k . For theremainder of this paper a cell complex is a finite regular CW-complex. The dimension dim( X ) ofthe cell complex X is the maximum dimension of cells in X and, writing X for the topological spaceas well as for the set of its cells, we obtain dim( X ) = max { dim( σ ) | σ ∈ X } .Let X be a cell complex with cells τ and σ . If τ ⊆ σ then τ is a face of σ , and σ is a coface of τ , written as τ (cid:22) σ . The codimension of a pair of cells τ (cid:22) σ is the difference in dimension,3im( σ ) − dim( τ ). If σ has a face τ of codimension 1, we call τ a facet of σ , and write τ (cid:67) σ . Afunction f : X → R on the cells of X is monotonic if f ( σ ) ≤ f ( τ ) whenever σ (cid:22) τ . Definition 2.1.
The d -dimensional cell complexes X and X ∗ are combinatorially dual if there isa bijection X → X ∗ , σ (cid:55)→ σ ∗ between the sets of cells such that1. (Dimension Reversal) dim( σ ∗ ) = d − dim σ for all σ ∈ X .2. (Face Reversal) σ (cid:22) τ ⇐⇒ τ ∗ (cid:22) σ ∗ for all σ, τ ∈ X . Definition 2.2. A filtered (cell) complex ( X, f ) is a cell complex X together with a monotonicfunction f : X → R . A linear ordering σ , σ , . . . , σ n of the cells in X , such that σ i (cid:22) σ j implies i ≤ j , is compatible with the function f when f ( σ ) ≤ f ( σ ) ≤ . . . ≤ f ( σ n ) . Note that the monotonicity condition implies that, for r ∈ R , the sub-level set X r := f − ( −∞ , r ]is a subcomplex of X . The value f ( σ ) determines when a cell enters the filtration given by thisnested sequence of subcomplexes. The definition of a compatible ordering also implies that each stepin the sequence ∅ ⊂ { σ } ⊂ { σ , σ } ⊂ · · · ⊂ { σ , σ , . . . , σ n } = X is a subcomplex, and every sub-level set f − ( −∞ , r ] appears somewhere in this sequence: f − ( −∞ , r ] = f − ( −∞ , f ( σ i )] = { σ , σ , . . . , σ i } for i = max { i = 0 , . . . , n | f ( σ i ) ≤ r } . Definition 2.3.
Two filtered complexes (
X, f ) and ( X ∗ , g ) are dual filtered complexes if X and X ∗ are combinatorially dual to one another and if there exists a linear ordering σ , σ , . . . , σ n of thecells in X that is compatible with f and its dual ordering σ ∗ n , σ ∗ n − , . . . , σ ∗ is compatible with g . Proposition 2.4.
Suppose two functions f : X → R and f ∗ : X ∗ → R satisfy f ∗ ( σ ∗ ) = − f ( σ ) .Then ( X, f ) and ( X ∗ , f ∗ ) are dual filtered complexes. When working with data, standard topological quantities can be highly sensitive to noise and smallgeometric fluctuations. Persistent homology addresses this problem by examining a collection ofspaces, indexed by a real variable often representing an increasing length scale. These spaces aremodelled by a cell complex X with a filter function f : X → R assigning to each cell the scale atwhich this cell appears. Given a filtered complex (
X, f ), we obtain inclusions f − ( −∞ , r ] → f − ( −∞ , s ] of sub-level sets for r ≤ s . Applying degree- k homology with coefficients in Z / Z to these inclusions yields linear mapsbetween vector spaces H k ( f − ( −∞ , r ]) → H k ( f − ( −∞ , s ]) . The resulting functor H k ( f ) : ( R , ≤ ) → Vec Z / Z from the poset category ( R , ≤ ) to the category ofvector spaces over the field Z / Z is called a persistence module , for details see [2].4s discussed in [2], Gabriel’s Theorem from representation theory implies that the persistencemodule H k ( f ) decomposes into a sum of persistence modules consisting of Z / Z for r ∈ [ b, d )connected by identity maps, and 0 elsewhere, called interval modules I [ b,d ) : H k ( f ) ∼ = (cid:77) l ∈ L I [ b l ,d l ) . Each interval summand I [ b l ,d l ) represents a degree- k homological feature that is born at r = b l and dies at r = d l . If the final space X has non-trivial homology there are features that never die. Thesehave d l = ∞ and the interval is called essential .The degree- k persistence diagram of f is the multiset Dgm k ( f ) = { [ b l , d l ) | l ∈ L } . We write [ b l , d l ) k ∈ Dgm k ( f ) to denote the homological degree of an interval and define the persis-tence diagram of f as the disjoint union over all degrees: Dgm ( f ) = dim( X ) (cid:71) k =0 Dgm k ( f ) . Writing
Dgm F ( f ) for the multiset of finite intervals with d l < ∞ , and Dgm ∞ ( f ) for the remainingessential ones, we obtain Dgm ( f ) = Dgm F ( f ) (cid:116) Dgm ∞ ( f ). To compute the persistence diagram
Dgm ( f ) we choose an ordering σ , σ , . . . , σ n of the cells in X that is compatible with f . Cells σ i and σ j appear at the same step in the nested sequence of sub-levelsets (cid:0) f − ( ∞ , r ] (cid:1) r ∈ R if f ( σ i ) = f ( σ j ). For the following computations however, we must add exactlyone cell at every step: ∅ ⊂ { σ } ⊂ { σ , σ } ⊂ · · · ⊂ { σ , σ , . . . , σ n − } ⊂ { σ , σ , . . . , σ n } = X. When adding the cells one step at a time, a cell of dimension k causes either the birth of a k -dimensional feature or the death of a ( k − birth ora death cell . A pair ( σ i , σ j ) of cells where σ j kills the homological feature created by σ i is called a persistence pair . A persistence pair ( σ i , σ j ) corresponds to the interval [ f ( σ i ) , f ( σ j )) ∈ Dgm F ( f ).Note that this interval can be empty, namely if f ( σ i ) = f ( σ j ). Empty intervals are usually neglectedin the persistence diagram. A birth cell σ i with no corresponding death cell is called essential , andcorresponds to the interval [ f ( σ i ) , ∞ ) ∈ Dgm ∞ ( f ).Recall that presentations for the standard homology groups are found by studying the imageand kernel of integer-entry matrices that represent the boundary maps taking oriented chains ofdimension k to those of dimension ( k −
1) [19]. In persistent homology, we work with the Z / Z totalboundary matrix D , which is defined by D i,j = 1 if σ i (cid:67) σ j and 0 otherwise. Define r D ( i, j ) = rank D ji − rank D j − i − rank D ji +1 + rank D j − i +1 where D ji = D [ i : n, j ] is the lower-left sub-matrix of D attained by deleting the first rows up to i − j + 1. Theorem 2.5 (Pairing Uniqueness Lemma [4]) . Given a linear ordering of the cells in a filtered cellcomplex X , ( σ i , σ j ) is a persistence pair if and only if r D ( i, j ) = 1 . R and using the property that rank D ji = rank R ji under the operations of thealgorithm. The persistence pairs can then be read off easily since r R ( i, j ) = 1 if and only if the i thentry of the j th column of the reduced matrix is the lowest 1 of this column. However, in this paper,we can work directly with r D . Corollary 2.6. If r D ( i, j ) (cid:54) = 1 and r D ( j, i ) (cid:54) = 1 for all j then the cell σ i is essential.Proof. The fact that every cell is either a birth or a death cell implies that σ i must be an unpairedbirth or death cell. However, as every filtration begins as the empty set, there are no unpaired deathcells. Recall again that in standard homology and cohomology the coboundary map is the adjoint ofthe boundary map. Hence, given a consistent choice of bases for the chain and cochain groups,their matrix representations are related simply by taking the transpose. In [5], another algebraicrelationship is established between persistent homology and persistent relative cohomology, basedon the observation that the filtration for relative cohomology reverses the ordering of cells in thetotal (co)boundary matrix. The same reversal of ordering holds for the dual filtered cell complexesdefined here, so we obtain a similar relationship between the persistence diagrams. Our proof ofthe correspondence between persistence pairs in dual filtrations uses the matrix rank function andpairing uniqueness lemma in a similar way to the combinatorial Helmoltz-Hodge decompositionof [11]. Nonetheless, Theorem 3.3 interprets the underlying linear algebra in the setting of dualfiltered complexes and only uses the concept of persistent homology without using the connection topersistent relative cohomology, which makes it more accessible.For this section suppose (
X, f ) and ( X ∗ , g ) are dual filtered cell complexes with n + 1 cells.Suppose that a linear ordering σ , σ , . . . , σ n of the cells in X is compatible with the filtration ( X, f ),and that σ ∗ n , σ ∗ n − , . . . , σ ∗ is the dual linear ordering compatible with g . Let D be the total boundarymatrix of X and D ∗ be the total boundary matrix of X ∗ with their respective orderings. Remark.
A useful indexing observation is that σ ∗ i is the ( n − i ) -th cell of the dual filtration. We denote by D ⊥ the anti-transpose of the matrix D , that is the reflection across the minordiagonal: D ⊥ i,j = D n − j,n − i . Anti-transposition is also the composition of standard matrix transpositionwith a reversal of the order of the columns and of the rows.
Lemma 3.1.
The matrix D ∗ is the anti-transpose D ⊥ of D , that is, D ∗ i,j = D n − j,n − i = D ⊥ i,j . Proof.
The equivalences below follow from the definition of D , of dual cell complexes, and the aboveremark. D n − j,n − i = 1 ⇔ σ n − j (cid:67) σ n − i ⇔ σ ∗ n − i (cid:67) σ ∗ n − j ⇔ D ∗ i,j = 1 . Lemma 3.2.
The sub-matrices defined in Section 2.2.2 satisfy ( D ji ) ⊥ = ( D ⊥ ) n − in − j nd thus rank D ji = rank ( D ⊥ ) n − in − j and r D ( i, j ) = r D ⊥ ( n − j, n − i ) . Proof.
The first statement follows from( D ji ) ⊥ = ( D [ i : n, j ]) ⊥ = D ⊥ [( n − j ) : n, n − i )] = ( D ⊥ ) n − in − j . The second statement follows because anti-transposition is attained by composing the rank preservingoperations of transposition and row and column permutations. The third statement follows from thesecond through: r D ( i, j ) = rank D ji − rank D j − i − rank D ji +1 + rank D j − i +1 = rank( D ⊥ ) n − in − j − rank( D ⊥ ) n − in − j +1 − rank( D ⊥ ) n − i − n − j + rank( D ⊥ ) n − i − n − j +1 = r D ⊥ ( n − j, n − i ) . Theorem 3.3 (Persistence of Dual Filtrations) . Let ( X, f ) and ( X ∗ , g ) be dual filtered complexeswith compatible ordering σ , σ , . . . , σ n . Then1. ( σ i , σ j ) is a persistence pair in the filtered complex ( X, f ) if and only if ( σ ∗ j , σ ∗ i ) is a persistencepair in ( X ∗ , g ) .2. σ i is essential in ( X, f ) if and only if σ ∗ i is essential in ( X ∗ , g ) .Proof. Lemma 3.2 implies that r D ( i, j ) = r D ∗ ( n − j, n − i ). Therefore, r D ( i, j ) = 1 ⇔ r D ∗ ( n − j, n − i ) = 1 . By the Pairing Uniqueness Lemma 2.5, the above implies that ( σ i , σ j ) is a persistence pair wheneverthe ( n − j )-th cell of the dual filtration ( X ∗ , g ) is paired with the ( n − i )-th, thus proving Part (1).For Part (2), Lemma 3.2 also tells us that the following two statements are equivalent: • Both r D ( i, j ) (cid:54) = 1 and r D ( j, i ) (cid:54) = 1 for all j . • Both r D ∗ ( n − j, n − i ) (cid:54) = 1 and r D ∗ ( n − i, n − j ) (cid:54) = 1 for all n − j .By Corollary 2.6, this means that σ i is an essential cell in ( X, f ) if and only if the ( n − i )-th cell σ ∗ i is essential in the dual filtration ( X ∗ , g ). Corollary 3.4.
Let ( X, f ) and ( X ∗ , g ) be dual filtered complexes. Then1. [ f ( σ i ) , f ( σ j )) ∈ Dgm k F ( f ) ⇔ [ g ( σ ∗ j ) , g ( σ ∗ i )) ∈ Dgm d − k − F ( g ) . [ f ( σ i ) , ∞ ) ∈ Dgm k ∞ ( f ) ⇔ [ g ( σ ∗ i ) , ∞ ) ∈ Dgm d − k ∞ ( g ) . Proof.
Note that for a persistence pair ( σ i , σ j ), found for an ordering compatible with the function f ,the birth value is f ( σ i ) and the death value is f ( σ j ). The result then follows directly from Theorem3.3. 7 emark. It is worth noting that there is a dimension shift between essential and non-essential pairscoming from the fact that the birth cell defines the dimension of a homological feature. For finitepersistence pairs, the birth cell changes from σ i (of dimension k ) to σ ∗ j (of dimension d − ( k + 1) )in the dual, while for an essential cycle, the birth cell in the dual is σ ∗ i . This dimension shift alsoappears in our results on images later on. As described in the introduction, the motivating application for the duality results of this paper isgrayscale digital image analysis. This section begins with the definition of grayscale digital imagesand describes the two standard ways to model such images by cubical complexes as well as themodifications required to make these dual filtered complexes.
Definition 4.1. A d -dimensional grayscale digital image of size ( n , n , . . . , n d ) is an R -valuedarray I ∈ M n × n × ... × n d ( R ). Equivalently, it is a real-valued function on a d -dimensional rectangulargrid I : I = (cid:74) , n (cid:75) × (cid:74) , n (cid:75) × . . . × (cid:74) , n d (cid:75) → R where (cid:74) , n i (cid:75) is the set { k ∈ N | ≤ k ≤ n i } . The index set, I , of I is also called the imagedomain .Recall that elements p ∈ I are called pixels when d =2, voxels if d ≥
3, and the value I ( p ) ∈ R is the grayscale value of p .One would like to use persistent homology to analyse such images via their sub-level sets. However,the canonical topology on I ⊆ Z d ⊂ R d makes it a totally disconnected discrete space. To induce ameaningful topology on the image that better represents the perceived connectivity of the voxels,grayscale digital images are modelled by regular cubical complexes [16]. Definition 4.2. An elementary k -cube σ ⊂ R d is the product of d elementary intervals, σ = e × e × . . . × e d such that k of the intervals have the form e i = [ l i , l i + 1] and d − k are degenerate, e i = [ l i , l i ].A cubical complex X ⊂ R d is a cell complex consisting of a set of elementary k -cubes, suchthat all faces of σ ∈ X are also in X , and such that all vertices of X are related by integer offsets. There are two common ways to build a filtered cubical complex from an image I : I −→ R . Onemethod is to represent the voxels as vertices of the cubical complex as in [20]. We call this cubicalcomplex the vertex construction, or V-construction for short. The second method takes voxels astop-dimensional cells: we call it the top-cell construction, or T-construction. It is shown in [16], thatthe vertex construction corresponds to the graph-theoretical direct adjacency used in traditionaldigital image processing and the top-cell construction to the indirect adjacency model. Theseadjacency models are also respectively referred to as the open and closed digital topologies.An example of how each construction is built from an image is given in Figure 1. The explicitdefinitions of such constructions are given below. Definition 4.3.
Given a d -dimensional grayscale digital image I : I → R , of size ( n , n , . . . , n d ),the V-construction is a filtered cell complex ( V ( I ) , V ( I )) defined as follows.8 construction -constructionImage Figure 1: Top: The V- and T-constructions generated by an image I : I → R with the values of I indicated on the vertices and the top-dimensional cells, respectively.Middle: the filtration V ( I ) : V ( I ) → R and the corresponding persistence pairs.Bottom: the filtration T ( I ) : T ( I ) → R and the corresponding persistence pairs.9. V ( I ) is a cubical complex built from an array of ( n − × . . . × ( n d −
1) elementary d -cubesand all their faces.2. The vertices υ (0) ∈ V ( I ) are indexed exactly by the elements p ∈ I , and we define the function V ( I ) firstly on these vertices as, V ( I )( υ (0) ) = I ( p ) . Then for an elementary k -cube σ , the function takes the maximal value of its vertices V ( I )( σ ) = max υ (0) (cid:22) σ V ( I )( υ (0) ) . This ensures that V ( I ) is monotonic with respect to the face relation on V ( I ). Definition 4.4.
Given a d -dimensional grayscale digital image I : I → R , of size ( n , n , . . . , n d ),the T-construction is a filtered cell complex ( T ( I ) , T ( I )) defined as follows.1. T ( I ) is a cubical complex built from the array of n × . . . × n d elementary d -cubes and all theirfaces.2. The d -cells τ ( d ) ∈ T ( I ) are indexed exactly by the elements p ∈ I , and we define the function T ( I ) firstly on these top-dimensional cells as, T ( I )( τ ( d ) ) = I ( p ) . Then for an elementary k -cube σ , the function takes the smallest value of any adjacent d -cubes, T ( I )( σ ) = min σ (cid:22) τ ( d ) T ( I )( τ ( d ) ) . This ensures that T ( I ) is monotonic with respect to the face relation on T ( I ).The next section describes how to modify the original image and take quotients to obtain dualcomplexes and filtrations. The cubical complexes defined using the top-cell and vertex constructions are not strictly dual toeach other in the standard context of a rectangular digital image domain due to the presence ofa boundary. There are two methods to resolve this issue. One is to treat the image domain asperiodic and identify opposite faces of the rectangular domain. This makes V ( I ) and T ( I ) intodual cubical complexes with d -cubes and vertices in both cases indexed by I ; their underlying spaceis the d -torus. Taking V ( I ) as the function on V ( I ) and T ( −I ) as the function on T ( I ), we alsoobtain dual filtrations and Theorem 3.3 can be applied to deduce the persistence pairs of one filteredcomplex from the other.A more commonly used approach to handling the boundary of a convex domain in R d is to takethe quotient identifying the boundary to a point and thus treat the convex domain as a subsetof the d -sphere. To obtain dual cell complexes on the d -sphere, we increase the size of the imagedomain before taking the quotient modulo the boundary. The image function is assigned a largearbitrary value on these extra voxels and dual filtered complexes are obtained by considering I inone construction and −I in the other as detailed in the following definitions and results.Let I : I → R be a grayscale digital image with index set I = (cid:74) , n (cid:75) × . . . × (cid:74) , n d (cid:75) , and set N > max p ∈ I I ( p ) . efinition 4.5. The padded image I P : I P → R has image domain I P = (cid:74) , n +1 (cid:75) × . . . × (cid:74) , n d +1 (cid:75) and I P ( p ) = (cid:40) I ( p ) , for p ∈ IN, for p ∈ I P \ I As shown in Figure 2, the padded image is simply obtained by adding a shell of N -valued voxelsto I . We denote the V- and T-constructions over the padded image by V ( I P ) : V ( I P ) → R and T ( I P ) : T ( I P ) → R , respectively. Padded Image
Figure 2: The transformation of the V- and T-construction into dual cell complexes (right) using thepadded image (left) and a mapping from V ( I P ) to T ( I ) (center).Let T ( I ) (cid:116) ∂T ( I ) κ ( d ) denote the cell complex obtained from T ( I ) by attaching a d -cell κ ( d ) alongthe boundary ∂T ( I ) (see [14, p.5] for details). Furthermore, let V ( I P ) /∂V ( I P ) be the quotient cellcomplex obtained by identifying all points of the boundary (see [14, p.8] for details). Note that boththese modifications create cells that are not elementary cubes. Lemma 4.6.
Given a rectangular digital image domain, I , the quotient, V ( I P ) /∂V ( I P ) , of thepadded V-construction modulo its boundary is the combinatorial dual of T ( I ) (cid:116) ∂T ( I ) κ ( d ) .Proof. Each elementary k -cube, σ ∈ V ( I P ) takes the form σ = e × . . . × e d , e i = [ l i , l i + 1] or e i = [ p i , p i ]where k of the elementary intervals are non-degenerate with l i ∈ { , . . . , n i } and ( d − k ) are degeneratewith p i ∈ { , . . . , n i + 1 } . Note that σ ∈ ∂V ( I P ) if at least one degenerate interval has p i = 0 or( n i + 1). Now consider the following cell constructed from σ : σ ∗ = e ∗ × . . . × e ∗ d , e ∗ i = [ l i + , l i + ] or [ p i − , p i + ]with l i and p i as defined above. This cell has k degenerate intervals and ( d − k ) non-degenerate ones so σ ∗ is an elementary ( d − k )-cube. If we insist that σ (cid:54)∈ ∂V ( I P ), then we see that p i ∈ { , . . . , n i } , andthe degenerate coordinate values ( l i + ) ∈ { , . . . , ( n i + ) } . Thus we obtain a bijection between k -cells in V ( I P ) \ ∂V ( I P ) and ( d − k )-cells in T ( I ). Mapping the 0-cell [ ∂V ( I P )] ∈ V ( I P ) /∂V ( I P ) to11he d -cell attached to ∂T ( I ) yields a dimension reversing bijection between all cells of V ( I P ) /∂V ( I P )and those of T ( I ) (cid:116) ∂T ( I ) κ ( d ) .The next step is to confirm that the face relations between cells in V ( I P ) /∂V ( I P ) are mapped tocoface relations in T ( I ) (cid:116) ∂T ( I ) κ ( d ) . By the construction above, all interior face relations for V ( I P )map to coface relations for T ( I ). Given that only cells in the boundary belong to [ ∂V ( I P )], thiscorrespondence is inherited by the quotient.Hence, the last detail we need to check is that the vertex [ ∂V ( I P )] in V ( I P ) /∂V ( I P ) has dual facerelations to the d -cell κ ( d ) attached to the boundary of T ( I ) in T ( I ) (cid:116) ∂T ( I ) κ ( d ) . This is equivalentto the statement that[ ∂V ( I P )] (cid:22) σ in V ( I P ) /∂V ( I P ) ⇔ σ ∗ (cid:22) κ ( d ) in T (cid:116) ∂T κ ( d ) . Now, σ ∗ is a face of κ ( d ) if and only if σ ∗ ∈ ∂T ( I ), which means at least one of the degenerateelementary intervals of σ ∗ has l i + = or ( n i + ). This makes l i = 0 or n i , so the correspondingelementary interval in the dual cell σ is e i = [ l i , l i +1] = [0 ,
1] or [ n i , n i +1]. This forces σ ∩ ∂V ( I P ) (cid:54) = ∅ ,so that [ ∂V ( I P )] (cid:22) σ . The converse implication follows in the same manner, and we are done. Lemma 4.7.
Given a rectangular digital image domain, I , the quotient, T ( I P ) /∂T ( I P ) , of thepadded T-construction modulo its boundary is the combinatorial dual of V ( I P ) (cid:116) ∂V ( I P ) κ ( d ) .Proof. This follows from the same arguments as the previous lemma with the roles of T and Vreversed. Note that we pad the V-construction before attaching the cell κ ( d ) to account for the factthat T ( I ) naturally has more cells than V ( I ).We have described how the two cubical complex models can be augmented to form dual cellcomplexes of the d -sphere. We now show how to obtain dual filtered cell complexes by comparingthe image function on one construction with its negative on the other. The details are made precisein the lemmata below.First note that the function V ( −I P ) is constant on ∂V ( I P ) so it induces a function on thequotient space, (cid:101) V ( −I P ) : V ( I P ) /∂V ( I P ) → R with (cid:101) V ( −I P )([ ∂V ( I P )]) = − N and agreeing with V ( −I P ) on all other cells. Similarly, the function T ( I ) extends to a function (cid:98) T ( I ) on T ( I ) (cid:116) ∂T ( I ) κ ( d ) with (cid:98) T ( I )( κ ( d ) ) = N . Lemma 4.8.
For each σ ∈ T ( I ) (cid:116) ∂T ( I ) κ ( d ) and dual cell σ ∗ ∈ V ( I P ) /∂V ( I P ) we have − (cid:98) T ( I )( σ ) = (cid:101) V ( −I P )( σ ∗ ) . Proof.
Firstly, suppose dim σ = d and σ (cid:54) = κ ( d ) , the d -cell attached to the boundary. Suppose p ∈ I is the corresponding element of the image domain, so that I ( p ) = (cid:98) T ( I )( σ ). The dual cell σ ∗ ∈ V ( I P ) /∂V ( I P ) corresponds to the same voxel but is given the negative value (cid:101) V ( −I P )( σ ∗ ) = −I ( p ) = − (cid:98) T ( I )( σ ) . For the remaining d -cell κ ( d ) , with dual [ ∂V ( I P )] ∗ , the function values satisfy − (cid:98) T ( I )( κ ( d ) ) = − N = (cid:101) V ( −I P )([ ∂V ( I P )]) . Lastly, suppose σ ∈ T ( I ) (cid:116) ∂T ( I ) κ ( d ) and dim σ < d . By construction, it follows that − (cid:98) T ( I )( σ ) = − min τ ( d ) (cid:23) σ (cid:98) T ( I )( τ ( d ) ) = max τ ( d ) (cid:23) σ − (cid:98) T ( I )( τ ( d ) ) = max υ (0) (cid:22) σ ∗ (cid:101) V ( −I P )( υ (0) ) = (cid:101) V ( −I P )( σ ∗ )as required. 12ow define the functions (cid:101) T ( −I P ) on T ( I P ) /∂T ( I P ) and (cid:98) V ( I P ) on V ( I P ) (cid:116) ∂V ( I P ) κ ( d ) similarlyto those above. Lemma 4.9.
For each σ ∈ V ( I P ) (cid:116) ∂V ( I P ) κ ( d ) and dual cell σ ∗ ∈ T ( I P ) /∂T ( I P ) we have − (cid:98) V ( I P )( σ ) = (cid:101) T ( −I P )( σ ∗ ) . Proof.
Similar to Lemma 4.8 with the roles of V and T interchanged.
Corollary 4.10.
For a grayscale digital image I : I → R
1. The filtered complexes ( T ( I ) (cid:116) ∂T ( I ) κ ( d ) , (cid:98) T ( I )) and ( V ( I P ) /∂V ( I P ) , (cid:101) V ( −I P )) are dual.2. The filtered complexes ( V ( I P ) (cid:116) ∂V ( I P ) κ ( d ) , (cid:98) V ( I P )) and ( T ( I P ) /∂T ( I P ) , (cid:101) T ( −I P )) are dual.Proof. This follows directly from applying Lemma 4.8 for part (1) and Lemma 4.9 for part (2), thenProposition 2.4.
In the previous section, we showed that the T- and V-constructions built from an image can bemodified via padding, cell attachment, and taking quotients to become dual cell complexes of the d -sphere. Our next step is to examine the effect of such operations on the persistence module anddiagram.As in Section 4, suppose I : I → R is a grayscale digital image and N > max I . The specificoperations we study are1. Padding an image I : I → R with a outer shell of N -valued pixels, then forming the V- andT-constructions.2. Attaching a d -cell to the boundary of V ( I P ) or to T ( I ) with value N .3. Taking the quotient modulo the boundary in the negative padded filtration, i.e. changing from V ( −I P ) to (cid:101) V ( −I P ) and from T ( −I P ) to (cid:101) T ( −I P ).Of these, the first two have relatively transparent effects on the persistent homology of the filteredspaces. Padding the image as in (1) does not change the persistence diagrams; attaching a d -cell asin (2) simply creates an essential d -cycle with birth at N . We summarise these formally as follows. Proposition 5.1.
For a grayscale digital image I : I → R Dgm ( V ( I P )) = Dgm ( V ( I )) and Dgm ( T ( I P )) = Dgm ( T ( I )) Dgm ( (cid:98) V ( I )) = Dgm ( V ( I )) ∪ { [ N, ∞ ) d } and Dgm ( (cid:98) T ( I )) = Dgm ( T ( I )) ∪ { [ N, ∞ ) d } The remaining operation to investigate is the third, namely the effect of taking the quotientmodulo the boundary. For this we need some machinery we will now introduce.13 .1 Long Exact Sequence of a Filtered Pair
To examine the effect of taking quotients on persistence diagrams, we use the description of persistencemodules as functors together with a long exact sequence (LES) for these. Given a pair of cell complexes(
X, A ) with A ⊆ X , we obtain a short exact sequence (SES) of cellular chain complexes inducing theLES . . . → H k ( A ) i −→ H k ( X ) p −→ H k ( X, A ) δ −→ H k − ( A ) → . . . which is a standard tool for analysing the homology of the pair, where H k ( X, A ) denotes the relativehomology of the pair.Similarly, suppose we have a filtered cell complex (
X, f ) and a sub-complex A ⊆ X . Then therestriction f | A : A → R induces a filtered sub-complex ( A, f | A ) and, at each index r ∈ R , we obtaina pair ( X r , A r ), where X r = f − ( −∞ , r ] and A r = f | − A ( −∞ , r ]. Theorem 4.5 in [22] states that the R -indexed collection of SESs of cellular chain complexes0 → C ∗ ( A r ) → C ∗ ( X r ) → C ∗ ( X r , A r ) → Theorem 5.2 (Long Exact Sequence for Relative Persistence Modules) . Given a monotonic function f : X → R and a sub-complex A ⊆ X , there is a long exact sequence of persistence modules . . . → H k ( f | A ) → H k ( f ) → H k ( f, f | A ) → H k − ( f | A ) → H k − ( f ) → . . . where H k ( f, f | A ) denotes the persistence module given by r (cid:55)→ H k ( X r , A r ) . Here the persistence modules are functors from the poset category ( R , ≤ ) to Vec Z / Z and themorphisms are natural transformations obtained from the standard connecting homomorphisms andthe linear transformations induced by inclusions and projections at each filtration index. The kernelsand cokernels of the morphisms are themselves persistence modules defined by taking the kernelor cokernel at each filtration index, with maps between corresponding vector spaces at differentfiltration indices induced by inclusions. This result is implicit in the recent work in [1, 18], where itfollows as corollary of the fact that persistence modules form an abelian category whereby the snakelemma holds. The remaining operation to investigate is the effect of taking the quotient of a padded image modulothe boundary filtered with the negative of the image function. We state the result in terms of aspace X homeomorphic to the d -dimensional closed disc D d , filtered by a function f : X → R takinga constant minimal value, min f = − N , on the boundary of X so that Lemma 5.3 applies to bothT- and V-constructions. Using the long exact sequence of a pair, we show that the ( d − − N, max f ) d − representing the boundary is removed while a d -cycle with interval[max f, ∞ ) d is added. Lemma 5.3.
Take a monotonic function f : X ∼ = D d → R with σ ∈ ∂X ⇒ f ( σ ) = − N = min f and induced quotient map (cid:101) f : X/∂X → R . Then Dgm ( (cid:101) f ) = (cid:0) Dgm ( f ) \ { [ − N, max f ) d − } (cid:1) ∪ { [max f, ∞ ) d } . roof. For pairs of cell complexes the relative homology groups are naturally isomorphic to thereduced homology groups ˜ H k ( ˜ f ) of the quotient [14, p.124]. Naturality implies that the result extendsto persistence modules and that the reduced persistence modules differ only by the essential interval I [ − N, ∞ ) in degree 0. To compute the reduced persistence modules ˜ H k ( ˜ f ) of the quotient, we thereforeconsider the LES of the filtered pair ( f, f | ∂X ) . . . → H k ( f | ∂X ) α k −−→ H k ( f ) → H k ( f, f | ∂X ) → H k − ( f | ∂X ) α k − −−−→ H k − ( f ) → . . . where α k is the map induced by the inclusion ∂X ⊆ X . Taking the cokernel of α k and the kernel of α k − the LES yields the SES0 → Coker ( α k ) → H k ( f, f | ∂X ) → Ker ( α k − ) → . First assume d > H k ( f | ∂X ) ∼ = (cid:40) I [ − N, ∞ ) for k = d − ,
00 otherwiseThus Im ( α k ) ∼ = Ker ( α k ) = 0 for k (cid:54) = d − ,
0. For α d − , the image of the essential ( d − X, f ) have been filtered at function value max f . Hence Im ( α d − ) ∼ = I [ − N, max f ) and Ker ( α d − ) ∼ = I [max f, ∞ ) As − N = min f we conclude α ( I [ − N, ∞ ) ) = I [ − N, ∞ ) , so that Im ( α ) ∼ = I [ − N, ∞ ) and Ker ( α ) = 0Since X is homeomorphic to a d -dimensional disc, H d ( f ) = 0. Hence Coker ( α d ) = 0 and, for k = d , the SES implies ˜ H d ( ˜ f ) ∼ = H d ( f, f | ∂X ) ∼ = Ker ( α d − ) ∼ = I [max f, ∞ ) . For 0 ≤ k < d the persistence module on the right of the SES is trivial. Thus˜ H k ( ˜ f ) ∼ = H k ( f, f | ∂X ) ∼ = Coker ( α k ) ∼ = H d − ( f ) / I [ − N, max f ) for k = d − H k ( f ) for 0 < k < d − H ( f ) / I [ − N, ∞ ) for k = 0and the result follows for d > d = 1 the only non-trivial persistence module of the boundary is H ( f | ∂X ) ∼ = I [ − N, ∞ ) ⊕ I [ − N, ∞ ) and we obtain Im ( α ) ∼ = I [ − N, ∞ ) ⊕ I [ − N, max f ) and Ker ( α ) ∼ = I [max f, ∞ ) For k = 1 we proceed as above and, for k = 0, the SES yields˜ H ( ˜ f ) ∼ = H ( f, f | ∂X ) ∼ = Coker ( α ) ∼ = H ( f ) / (cid:0) I [ − N, ∞ ) ⊕ I [ − N, max f ) (cid:1) As above we conclude H ( ˜ f ) ∼ = H ( f ) / I [ − N, max f ) . Corollary 5.4.
For a d -dimensional image I : I → R Dgm ( (cid:101) V ( −I P )) = Dgm ( V ( −I P )) \ { [ − N, − min I ) d − } ∪ { [ − min I , ∞ ) d } and Dgm ( (cid:101) T ( −I P )) = Dgm ( T ( −I P )) \ { [ − N, − min I ) d − } ∪ { [ − min I , ∞ ) d } Proof.
This follows from Lemma 5.3 applied to f = V ( −I P ) and f = T ( −I P ) respectively, usingmax V ( −I P ) = − min I and max T ( −I P ) = − min I .15 Duality Results for Images
In this section, we explicitly describe the relationship between the diagrams of both the T- andV-constructions. Software to compute persistent homology of an image I : I → R typically buildsone of the two constructions implicitly so the results in this section provide a solution to the problemof how to use software based on the V-construction to compute a persistence diagram with respectto the T-construction, and vice versa.For the algorithms that we define in this section we assume the following sub-routines given agrayscale digital image I .1. Pad ( I , N ): returns the image padded with an outer shell of N -valued voxels.2. Neg ( I ): multiplies every gray value by − I ), min( I ): returns the maximum and minimum voxel values of I respectively.4. Vcon ( I ), Tcon ( I ): returns the persistence diagrams Dgm ( V ( I )) and Dgm ( T ( I )) of the V-and T-construction of the image respectively. Suppose we have software that computes the persistent homology of a d -dimensional grayscale digitalimage I using the V-construction. The following theorem states that the persistence diagram of theT-construction for I can be calculated directly from the pairs in that of the V-construction of thenegative padded image. Theorem 6.1 (T from V) . For a grayscale digital image I : I → R the diagrams of the V- andT-constructions satisfy Dgm F ( T ( I )) = { [ − q, − p ) d − k − | [ p, q ) k ∈ Dgm F ( V ( −I P )) } \ { [min I , N ) } and Dgm ∞ ( T ( I )) = { [min I , ∞ ) } . Proof.
That
Dgm ∞ ( T ( I )) = { [min I , ∞ ) } follows from the fact that T ( I ) ∼ = D ( d ) and the first cellin the filtration occurs at time min T ( I ) = min I . For the finite case: Dgm F ( T ( I )) = Dgm F ( (cid:98) T ( I ))= { [ − q, − p ) d − k − | [ p, q ) k ∈ Dgm F ( (cid:101) V ( −I P )) } = { [ − q, − p ) d − k − | [ p, q ) k ∈ Dgm F ( V ( −I P )) \ { [ − N, − min I ) d − } } = { [ − q, − p ) d − k − | [ p, q ) k ∈ Dgm F ( V ( −I P )) } \ { [min I , N ) } where the equalities follow from Proposition 5.1, Theorem 3.3 and Corollary 5.4 respectively.The structure of the algorithm follows immediately from the theorem and is summarised below:16 lgorithm 1 Computing the T-construction persistence diagram with V-construction software.
Require:
An image I and the sub-routine Vcon . Dgm ( T ( I )) ← { [min( I ) , ∞ ) } N ← max( I ) + C (cid:46) choose C to ensure N (cid:29) max( I ) −I P ← Neg ( Pad ( I , N )) Dgm ( V ( −I P )) ← Vcon ( −I P ) (cid:46) Apply V-construction software. for [ p, q ) k in Dgm ( V ( −I P )) with p (cid:54) = − N do Dgm ( T ( I )) ← Dgm ( T ( I )) ∪ { [ − q, − p ) d − k − } return Dgm ( T ( I )) (cid:46) Output T-construction persistence diagram.
In the other direction, suppose we have software that computes the persistent homology of a d -dimensional grayscale digital image I : I → R using the T-construction. The following theoremstates that a persistence diagram for the V-construction of I can be calculated directly from thepairs computed for the negative padded image using the T-construction. Theorem 6.2 (V from T) . For a grayscale digital image I : I → R the diagrams of the V- andT-constructions satisfy Dgm F ( V ( I )) = { [ − q, − p ) d − k − | [ p, q ) k ∈ Dgm F ( T ( −I P )) } \ { [min I , N ) } and Dgm ∞ ( V ( I )) = { [min I , ∞ ) } . Proof.
That
Dgm ( V ( I )) = { [min I , ∞ ) } follows from the fact that V ( I ) ∼ = D ( d ) and the first cellin the filtration occurs at time min I . For the finite case, we have that Dgm F ( V ( I )) = Dgm F ( V ( I P ))= Dgm F ( (cid:98) V ( I P ))= { [ − q, − p ) d − k − | [ p, q ) k ∈ Dgm F ( (cid:101) T ( −I P )) } = { [ − q, − p ) d − k − | [ p, q ) k ∈ Dgm F ( T ( −I P )) \ { [ − N, − min I ) d − } } = { [ − q, − p ) d − k − | [ p, q ) k ∈ Dgm F ( T ( −I P )) } \ { [min I , N ) } where the equalities follow from Proposition 5.1, Theorem 3.3 and Corollary 5.4 respectively.The structure of the algorithm follows immediately from the theorem and is summarised below: Example.
Suppose we are working with the two dimensional digital grayscale image given in Figure1 and have only the software to compute the T-construction. We depict the filtration of T ( −I P ) inFigure , and the corresponding intervals in the persistence module. Similarly, we show the filteredV-construction V ( I ) in Figure 4. The reader may confirm that the correspondence between theintervals is accurately described by Theorem 6.2. lgorithm 2 Computing the V-construction persistence diagram with T-construction software.
Require:
An image I and the sub-routine Tcon . Dgm ( V ( I )) ← { [min( I ) , ∞ ) } N ← max( I ) + C (cid:46) choose C to ensure N (cid:29) max( I ) −I P ← Neg ( Pad ( I , N )) Dgm ( T ( −I P )) ← Tcon ( −I P ) (cid:46) Apply T-construction software. for [ p, q ) k in Dgm ( V ( −I P )) with p (cid:54) = − N do Dgm ( V ( I )) ← Dgm ( V ( I )) ∪ { [ − q, − p ) d − k − } return Dgm ( V ( I )) (cid:46) Output V-construction persistence diagram.Figure 3: The filtration T ( −I P ) and intervals of the persistence diagram Dgm ( T ( −I P )) for theimage I : I → R of Figure 1.Figure 4: The filtration V ( I ) and intervals of the persistence diagram Dgm ( V ( I )) for the image I : I → R of Figure 1. Our results clarify the relationship between the two cubical complex constructions commonly usedin digital image analysis software and provide a simple method to use software that implementsone construction to compute a persistence diagram for the other. This permits a user’s choice of18djacency type for their images to depend on that appropriate to the application rather than on thetype of construction used in available efficient persistence software. In addition to facilitating thisapplication, the results of Sections 3 and 5 may be of independent interest for the following reasons.Theorem 3.3 is a new interpretation of a duality relationship that manifests in many contextssuch as the correspondence between persistent homology and persistent relative cohomology [5],symmetries in extended persistence diagrams [3], and a discrete Helmoltz-Hodge decomposition [11].In [13], we show that the filtered discrete Morse chain complexes also exhibit this duality. Furtherinvestigations in this area may reveal other interpretations of this relationship.The results of Section 5 are formulated specifically for the case of an image with its domainhomeomorphic to a closed ball, but could be extended to spaces with more interesting topology. Weanticipate that the long exact sequence of a pair can be used to derive a relationship between filteredcell complexes that satisfy conditions for duality if their boundaries can be capped or quotiented asin Section 4 to obtain a manifold.Further applications of the relationship between the T- and V-constructions can be found. Inparticular, a discrete gradient vector field built on V ( I ) is easily transformed into a dual one on T ( I ) [13]. If the gradient vector field is built to be consistent with the grayscale image I accordingto the algorithm of [20], then the dual gradient vector field on T ( I ) will be consistent with −I . Thismeans the skeletonisation and partitioning algorithms of [7] can now be adapted to work on imageswhere the T-construction is preferred.There are also interesting questions about algorithm performance to explore. The results ofSection 6 suggest that persistence diagram computation from grayscale images should have thesame average run time independent of the choice of T- and V-construction. If the T-constructionexecutes faster on a particular image, then the V-construction should execute faster on the negativeof the image. To answer this question fully requires a careful analysis of the effects of taking theanti-transpose of the boundary matrix on the run time of the matrix reduction algorithm and theextra cells added when padding the image. This project started during the Women in Computational Topology workshop held in Canberra in Julyof 2019. All authors are very grateful for its organisation and the financial support for the workshopfrom the Mathematical Sciences Institute at ANU, the NSF, AMSI and AWM. AG is supported bythe Swiss National Science Foundation, grant No.
CRSII
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