The problem of cryo-electron microscopy (cryo-EM) asks for the following: Given a collection of noisy -dimensional (D) projected images, reconstruct the -dimensional (D) structure of the molecule that gave rise to these images. Viewed from a high level, it takes the form of an inverse problem similar to those in medical imaging [3, 4, 45], remote sensing [15, 5], or underwater acoustics [11, 38], except that for cryo-EM the data comes from an electron microscope instead of a CT scanner, radar, or sonar. However, when examined at a finer level of detail, one realizes that the cryo-EM problem possesses mathematical structures that are quite different from those of other classical inverse problems. It has inspired studies from the perspectives of representation theory [22, 23], differential geometry [49, 48], and is related to profound problems in computational complexity  and operator theory . This article examines the problem from an algebraic topological angle — we will show that the problem of cryo-EM is a problem of cohomology, or, more specifically, the Čech cohomology of a simplicial complex with coefficients in the Lie group and the discrete group , i.e., endowed with the discrete topology.
Despite its abstract appearance, the aforementioned cohomology framework is actually concrete and natural. The fact that cohomology has an important role to play in understanding D projections of D objects is already evident in simple examples like the Penrose tribar or Escher brick, as we will see in Section 2. Our analysis of discrete and continuous cryo-EM cocycles requires a more sophisticated type of cohomology but is essentially along the same lines. In fact, the same ideas that we use to study the cryo-EM problem also underlies the classical field theory of electromagnetism . The cohomology framework allows us to classify cryo-EM cocycles: Given two different collections of D projected images, are they equivalent in the sense that they will give us the same
D reconstruction? The insights gained also shed light on the denoising techniques: What are we really trying to achieve when we minimize a certain loss function to denoise cryo-EM images?
The technique of cryo-electron microscopy has been described in great detail in [18, 19] and more than adequately summarized in [22, 23, 40, 47, 49, 48, 50, 53, 54]. It suffices to provide a very brief review here. A more precise mathematical model, for the following high-level description will be given in Section 4. The basic idea is that one first immobilizes many identical copies of a molecule in ice and employs an electron microscope to produce D images of the molecule. As each copy of the molecule is frozen in some unknown orientation, each of the D images may be regarded as a projection of the molecule from an unknown viewing direction. The cryo-EM dataset is then the set of these D projected images. Such a D image shows not only the shape of the molecule in the plane of the viewing direction but also contains information about the density of the molecule, captured in the intensity of each pixel of the D image . The ultimate goal of cryo-EM is to construct the D structure of the molecule from a cryo-EM dataset. In practice, these D images are very noisy due to various issues ranging from the electron dosage of the microscope to the structure of the ice in which the molecule are frozen. Hence the main difficulty in cryo-EM reconstruction is to denoise these D images by determining the true viewing directions of these noisy D images so that one may take averages of nearby images. There has been much significant progress toward this goal in recent years [40, 47, 50, 53, 54].
Our article attempts to understand cryo-EM datasets of D images via Čech and singular cohomology groups. We will see that for a given molecule, the information extracted from its D cryo-EM images determines a cohomology class of a two-dimensional simplicial complex. Furthermore, each of these cohomology classes corresponds to an oriented circle bundle on this simplicial complex. We note that there are essentially two interpretations of cohomology: obstruction and moduli. On the one hand, a cohomology group quantifies the obstruction from local to global. For example, this is the sense in which cohomology is used when demonstrating the non-existence of an impossible figure  or in the solution of the Mittag-Leffler problem [21, p. 34]. On the other hand, a cohomology group may also be used to describe a collection of mathematical objects, i.e., it serves as a moduli space for these objects. For example, when we use a cohomology group to parameterize all divisors or all line bundles on an algebraic variety [24, p. 143], it is used in this latter sense.
The line bundles example is a special case of a more general statement: A cohomology group serves as the moduli space of principal bundles over a topological space. This forms the basis for our use of cohomology in the cryo-EM reconstruction problem — as a moduli space for all possible cryo-EM datasets. Obviously, such a classification of cryo-EM datasets comes under the implicit assumption that the D images in a dataset are noise-free. Our classification depends on a standard mathematical model for molecules in the context of cryo-electron microscopy under a noise-free assumption. Here the reader is reminded that a molecule is a physical notion and not a mathematical one. A mathematical answer to the question ‘What is a molecule?’ depends on the context. In one theory, a molecule may be a solution to a Schrödinger pde (e.g., quantum chemistry) whereas in another, it may be a path in a -dimensional phase space (e.g., molecular dynamics). In our model, a molecule is a real-valued function on representing potential. When our images are noisy, this model gives us a natural way, namely, the cocycle condition, to denoise them by fitting them to the model. Various methods for denoising cryo-EM images [47, 50] may be viewed as nonlinear regression for fitting the cocycle condition under additional assumptions.
2 Cohomology and D projections of D objects
The idea that cohomology arises whenever one attempts to analyze D projections of D objects was first pointed out by Roger Penrose, who proposed in  a cohomological argument to analyze Escher-type optical illusions. In the following, we present Penrose’s elegantly simple example since it illustrates some of the same principles that underly our more complicated use of cohomology in cryo-EM.
We follow the spirit of Penrose’s arguments in  but we will deviate slightly to be more in-line with our discussions of cryo-EM and to obtain a proof for the nonexistence of Penrose tribar. The few unavoidable topological jargons are defined in Section 3 but they are used in such a way that one could grasp the intuitive ideas involved even without knowledge of the jargons. To be clear, a D object is one that can be embedded in by an injective map such that whenever , , are points in this object, and .
The Penrose tribar is defined to be a fictitious D object — fictitious as it does not exist in — obtained by gluing three rectangular solid cuboids (i.e., bars) in as follows: is glued to by identifying a cubical portion at one end of with a cubical portion at one end of as depicted in Figure 1(b), .
The tribar is more commonly shown in its D projected form as in Figure 1(a). Let be the triangular D object in Figure 1(a), which appears to be the projection of the Penrose tribar, should it exist, onto a plane . Indeed, there are (infinitely) many D objects that, when projected onto a plane , gives as an image. An example is the object in Figure 2, as we explain below.
Note that the object in Figure 2 is an abstraction of the sculpture in Figure 3, which depicts how it projects to give when viewed from an appropriate angle. The plane in this case is either the viewer’s retina or the camera’s photographic film.
be a hyperplane, which partitionsinto two half-spaces. Let be an arbitrary point in one half-space and the three bars be in the other. The reader should think of as the position of the viewer and the viewing direction as a normal to . Now we are going to arrange in such a way that their projections onto give us . This is clearly possible; for example, the D object in Figure 2, upon an appropriate rotation dependent on and , would give as a projection.
Define to be the distance from to the center of and , . Let be the matrix of cross ratios
Then is a matrix with and for all .
The matrix is a function of the positions of the bars
, or, to be precise, a function of the centroids of these rigid bodies. These bars have a certain degree of freedom: We maymove each of them independently along the viewing direction and this would keep their projections in invariant, always forming . This movement is a similarity transform that preserves the direction of the bar, with no rotation. Moving in the viewing direction results in a rescaling of the distance by a factor for all , i.e., if denotes the new distance upon moving ’s along viewing directions, then , for all . Let be the new matrix of cross ratios upon moving ’s along viewing direction. Then we have
Suppose that we could eventually move to form the tribar in . Then, in this final position, the centers of and coincide and so for all , and thus for all . In other words, the matrix must be a coboundary, i.e.,
for some , .
In summary, what we have shown is that if could be moved into place to form a tribar, then for in any positions that form upon projection onto , the corresponding matrix must be a coboundary, i.e., it satisfies (2), or equivalently, is the identity element in the cohomology group . With this observation, we will next derive a contradiction showing that the tribar does not exist. Let be arranged as in Figure 2 and recall that their projections onto give . In this case, the matrix is
If the tribar exists, then is a coboundary, i.e., (2) has a solution for some , , and so
implying . However, as is evident from Figure 2, does not even intersect and so , a contradiction.
Although the tribar does not exist as a D object, i.e., it cannot be embedded in , it clearly exists as an abstract geometrical object (a cubical complex) defined by the gluing procedure described earlier — we will call this the intrinsic tribar to distinguish it from the nonexistent D object. In fact, the intrinsic tribar can be embedded in a three-dimensional manifold , a quotient space of under a certain action of the discrete group related to Figure 2 (see  for details).
We emphasize that a tribar is a geometrical object, not a topological one. It may be tempting to draw a parallel between the non-embeddability of the intrinsic tribar in with the non-embeddability of the Möbius strip in or the Klein bottle in . But these are different phenomena. As a topological object, a Möbius strip is only defined up to homotopy, i.e., we may freely deform a Möbius strip continuously. However the definition of the tribar does not afford this flexibility, i.e., a tribar is not homotopy invariant. For instance, we are not allowed to twist or bend the bars. In fact, had we allowed such continuous deformation, the intrinsic tribar is homotopy equivalent to a torus and therefore trivially embeddable in . This is much like our study of cryo-EM, where the goal is to reconstruct the D structure of a molecule precisely, and not just up to homotopy.
The discussions above also apply to other impossible objects in . For example, the Escher brick, defined as the (nonexistent) D object obtained by gluing four bars as in Figure 4. If the Escher brick exists in , then whenever projects onto to form Figure 4(a), the matrix is necessarily a coboundary, i.e., satisfies for some , . We may construct an analogue of Figure 2 whereby we glue three of the four ends in Figure 4(b). This D object projects onto to form Figure 4(a) but its corresponding matrix is not a coboundary. Hence the Escher brick does not exist in .
3 Singular Cohomology and Čech Cohomology
This article is primarily intended for an applied and computational mathematics readership. For readers unfamiliar with algebraic topology, this section provides in one place all the required definitions and background material, kept to a bare minimum of just what we need for this article.
We will define two types of cohomology groups associated to a topological space and a topological group that will be useful for our study of the cryo-EM problem: , the singular cohomology group with coefficients in ; and , the Čech cohomology group with coefficients in . For a given , these cohomology groups are in general different; but they would always be isomorphic for the space that we construct from a given collection of cryo-EM images (see Section 4). The reason we need both of them is that they are good for different purposes: the cohomology of cryo-EM is most naturally formulated in terms of Čech cohomology; but singular cohomology is more readily computable and facilitates our explicit calculations.
Our descriptions in the next few subsections are highly condensed, but in principle complete and self-contained. While this material is standard, our goal here is to make them accessible to practitioners by limiting the prerequisite to a few rudimentary definitions in point set topology and group theory. We provide pointers to standard sources at the beginning of each subsection.
We use to denote isomorphism if are groups, homotopy equivalence if are topological spaces, and bundle isomorphism if are bundles. We use to denote homeomorphism of topological spaces.
3.1 Singular cohomology
The standard -simplex for , is the set
is the convex hull of its vertices,
The standard -simplex is a point, the standard -simplex is a line, the standard -simplex is a triangle, and the standard -simplex is a tetrahedron.
For , the convex hull of any vertices of , where , is called a face of and denoted by .
Let be a topological space and . A continuous map is called a singular simplicial simplex on . We denote by the free abelian group generated by all singular simplicial simplices on . The boundary maps are homomorphisms of abelian groups
defined respectively by the linear extensions of
Here denotes the restriction of to the face of , denotes the restriction of to the face of , and denotes the restriction of to the face of . We set to be the zero map.
The sequence of homomorphisms of abelian groups
forms a chain complex, i.e., it has the property that
which are easy to verify. For , let be the subgroup of -cycles and be the subgroup of -boundaries. It follows from (4) that . The quotient group
is called the th singular homology group of , .
For , define , the set of all group homomorphisms from to . is clearly an abelian group itself under addition of homomorphisms. The map induced by the boundary map is defined as
for any and . The sequence of homomorphisms of abelian groups
forms a cochain complex, i.e., it has the property that
which follows from (4). For , let be the subgroup of -cocycles and be the subgroup of -coboundaries. The quotient group
is called the th singular cohomology group of , . More generally, let be a group then one can define the th singular cohomology group with coefficient of to be the cohomology groups of the cochain complex
where , is the map induced by , and
Note that when , , , , .
For the purpose of this paper, would take the form of a finite simplicial complex, a collection of finitely many simplices such that
every face of a simplex in is also contained in ;
the intersection of two simplices in is a face of both and .
We denote the union of simplices in by . We also say that a topological space is a finite simplicial complex if can be realized as for some finite simplicial complex . For example, spheres and tori are finite simplicial complexes in this more general sense.
For the purpose of this paper, readers only need to know that
and that if is a simplicial complex of dimension , then for all .
A topological space is contractible if there is a point and a continuous map such that
Roughly speaking, this means that can be continuously shrunk to a point . For example, an open/closed/half-open-half-closed line segment is contractible, as is an open/closed disk or a disk with an arc on the boundary. The following is the only fact about contractible spaces that we need for this article.
If is contractible and is an abelian group, then for all and .
3.2 Principal bundles and classifying spaces
Let be a group with multiplication map , and inversion map , . If is also a topological space such that and are continous then together with this topology is called a topological group. Every group is a topological group if we put the discrete topology on ; we will denote such a topological group by (unless the natural topology is the discrete topology, in which case we will just write ). For example, with its natural discrete topology is a topological group. In this article, we are primarily interested in the case where is the group of real orthogonal matrices. When endowed with the manifold topology, this is , the special orthogonal group in dimension two and is homeomorphic to the unit circle as a topological space. On the other hand, is just a discrete uncountable collection of orthogonal matrices. Both and will be of interest to us.
Let be topological spaces. We say that is a fiber bundle with fiber and base space if is a continuous surjection and every point of has a neighborhood such that is homoeomorphic to .
In particular, for all .
A principal -bundle is a tuple where is a fiber bundle with fiber and is a group action such that
is a continuous map;
for any ;
if for some , then is the identity element in ;
For any and , there is a such that .
We will often say ‘ is a principal -bundle on ’ to mean the above, without specifying and . A principal -bundle is called an oriented circle bundle and a principal -bundle is called a flat oriented circle bundle. We will have more to say about these in Sections 4 and 5.
Let and be two principal -bundles on . We say that is isomorphic to , denoted , if there is a homeomorphism compatible with the group actions , and the projection maps , in the following sense:
Here is the identity map. Let be an open covering of such that via some isomorphism for all . A transition function corresponding to is a map , defined for all such that . It may be regarded as a -valued function . Transition functions are important because one may construct a principal -bundle entirely from its transition functions .
For , transition functions of an oriented circle bundle are continuous -valued functions on open sets . For , transition functions of a flat oriented circle bundle are continuous -valued functions on open sets but since has the discrete topology, this means that are locally constant -valued functions on . In particular, if is connected, then are constant -valued functions on . In our case, the covering that we choose (see (13)) will have connected ’s and so we may regard
In other words, flat oriented circle bundles are just oriented circle bundles whose transition functions are constant-valued.
Let be topological spaces. Two maps are homotopic if there is a continuous function such that
Homotopy is an equivalence relation and the set of homotopy equivalent classes of maps from to is denoted by . Let be the -sphere. We say that a topological space is weakly contractible if contains only the equivalence class of the trivial map, i.e., the map that sends all points in into a fixed point of . The classifying space of a topological group is a topological space together with a principal -bundle on such that is weakly contractible.
For any topological space and topological group , there is a one-to-one correspondence between the following two sets:
given by , the principal -bundle on whose fiber over is the fiber of over .
For the purpose of this paper, readers only need to know that the classifying space of the unitary group is , the Grassmannian of -planes in . In particular, if , since , we have
Let be an abelian group with identity . We write for the set of all homomorphisms from to . An element is a torsion element if it has finite order, i.e., for some . The subgroup of all torsion elements in is called its torsion subgroup and denoted . For example, every element in is a torsion element whereas is the only torsion element in . For an abelian group , we also denote its torsion subgroup as
The reason for including this alternative notation is that it is very standard — a special case of groups for defined more generally [25, 26]. We now state some routine relations  that we will need for our calculations. Let and be abelian groups. Then
Singular homology and singular cohomology are related via and in the following well-known theorem.
Theorem 1 (Universal coefficient theorem)
Let be a topological space. Then we have a natural short exact sequence
In particular we have an isomorphism,
where is the second Betti number of and is the torsion subgroup of .
The second Betti number of counts the number of -dimensional ‘voids’ in . In the case of interest to us, where is a finite two-dimensional simplicial complex, the second Betti number counts the number of -spheres (by which we meant the boundary of a -simplex, which is homeomorphic to ) contained in .
We will also need the following alternative characterization [34, Chapter 22] of .
Let be a topological space. Then we have
3.3 Čech cohomology
Let be a topological abelian group and let be a topological space. For any open subset of we define an assignment
for all open subset . By definition, if is a discrete group and is any connected open subset of , then . If then we have a restriction map
defined by the restriction of -valued continuous functions on to .
Let be a topological space and be a topological abelian group on . Let be an open covering of . We may associate a cochain complex to , , and as follows:
To be precise, we have
It is easy to check that and so (8) indeed forms a cochain complex.
As in the case of singular cohomology, and are the groups of Čech -coboundaries and Čech -cocycles respectively. Again we have . The first Čech cohomology group associated to with coefficients in is then defined to be the quotient group
Explicitly, we have
We have in fact already encountered this notion in Section 2, , the Čech cohomology group of the plane with coefficients in the group has appeared implicitly in our discussion.
By its definition, depends on the choice of open covering of . To obtain a Čech cohomology group of independent of open covering, we take the direct limit over all possible open coverings of . The first Čech cohomology group of with coefficients in is defined to be the direct limit
with running through all open coverings of .
For those unfamiliar with the notion of direct limit, may be defined explicitly using an equivalence relation:
where denotes the disjoint union of for all possible open coverings of . The equivalence relation is given as follows: For and , iff
there is an open covering such that every open set is contained in for some and ;
there is an element such that the restriction of and the restriction of are both equal to .
The term “restriction” needs elaboration. Let , be open covers of such that for any , there is some with . Fix a map such that . There is a natural restriction map induced by where
The image of is called the restriction of to . It does not depend on the choice of .
As the reader can guess, calculating the Čech cohomology group using such a definition would in general be difficult. Fortunately, the following theorem (really a special case of Leray’s theorem ) allows us to simplify the calculation in all cases of interest to us in this article.
Theorem 3 (Leray’s theorem)
Let be a topological space and be an topological abelian group. Let be an open cover of such that for all . Then we have
Furthermore, we will often be able to reduce calculation of Čech cohomology to calculation of singular cohomology since they are equal in the case when is a finite simplicial complex .
If is a finite simplicial complex and is an abelian group, then