Invertibility and Stability for A Generic Class of Radon Transform

Invertibility and Stability for A Generic Class of Radon Transforms with Application to Dynamic Inverse Operators

Siamak RabieniaHaratbar Department of Mathematics, Purdue University, West Lafayette, IN 47907
Abstract.

Let be an open subset of . We study the dynamic inverse operator, , integrating over a family of level curves in when the object changes between the measurement. We use analytic microlocal analysis to determine which singularities can be recovered by the data-set. Our results show that not all singularities can be recovered, as the object moves with speed lower than the X-ray source. We establish stability estimates and prove that the injectivity and stability are of a generic set if the dynamic inverse operator satisfies the visibility, no conjugate points, and local Bolker conditions. We also show this results can be implemented to Fan beam geometry.

Partly supported by NSF Grant DMS  1600327

1. Introduction

Tomography of moving objects has been attracting a growing interest recently, due to its wide range of applications in medical imaging, for example, X-ray of the heart or the lungs. Data acquisition and reconstruction of the object which changes its shape during the measurement is one of the challenges in computed tomography and dynamic inverse problems. The major difficulty in the reconstruction of images from the measurement sets is the fact that object changes between measurements but does not move fast enough compared to the speed of X-rays. This means that some singularities of the object might not be detectable even if the source fully rotates around the object. The application of known reconstruction methods (based on the inversion of the Radon transform) usually results in many motion artifacts within the reconstructed images if the motion is not taken into account. One extreme example will be the case when the object (or some small part of it) rotates with the same rate as the scanner. This leads to integration over the same family of rays, and therefore, one may not locally recover all the singularities.

Analytic techniques for reconstruction of dynamic objects, known as motion compensation, have been used widely for different types of motion, like affine deformation, see e.g. [2, 3, 9, 18]. In the case of non-affine deformations, there is no inversion formula. Iterative reconstructions, however, do exist in order to detect singularities by approximation of inversion formulas for the parallel and fan beam geometries [14], as well as cone beam geometry [15]. In a recent work, Hahn and Quinto [8] studied the dynamic inverse operator

(1.1)

with a smooth motion where the limited data case has been analyzed, and characterization of visible and added singularities have been investigated.

Our work in this paper is motivated by these dynamic measurements. We first show this dynamic problem can be reduced to an integral geometry problem integrating over level curves. By an appropriate change of variable (see section 2), can be written as

Therefore, we study the following general operator:

which allows us to study the original dynamic problem with a more general set of curves (see also [4],) and then transfer the result to a dynamic inverse operator given by (1.1).

The dynamic inverse operator formulated as above falls into the general microlocal framework studied by Beylkin [1] (see also [13]) which goes back to Guillemin and Sternberg [6, 7] who studied the integral geometry problems with a more general platform from the microlocal point of view. See also [4], where a weighted integral transform has been studied on a compact manifold with a boundary over a general set of curves (a smooth family of curves passing through every point in every direction).

The main novelty of our work, compared to previous works which are concentrated on the microlocal invertibility, is that for the dynamic problem, under some natural microlocal conditions, the actual uniqueness and stability results have been established. In fact, our imposed natural microlocal conditions guarantee that one can recover each singularity, and a functional analysis argument leads to stability results. We show that under these conditions, the dynamic inverse operator is stably invertible in a neighborhood of pairs in a generic set, and in particular, it is injective and stable for slow enough motion (which is not required to be a periodic motion model). This is the similar kind of stability result which has been studied in [13] for the generalized Radon transform and in [4] which coincide when the dimension is two. The data is cut (restricted) in a way to have the normal operator related to the localized dynamic inverse operator as a pseudodifferential operator (DO) near each singularity. We do not analyze the case where these conditions are not satisfied globally, but our analysis (see also [8]) shows that one may still recover the visible singularities in a stable way, and periodicity or non-periodicity plays no role in the reconstruction process. We also show that, due to the generality of our approach, our results can be implemented to other geometries, for instance, fan beam geometry.

This paper is organized as follow: Section one is an introduction. In section two, we state the definitions of , , - , and our main result. Some preliminary results have been stated in section three. Section four is devoted to analytic microlocal analysis approach which is used to show that the operator is a Fourier Integral Operator (FIO). Then the canonical relation is computed and it is shown that it is a four-dimensional non-degenerated conic submanifold of the conormal bundle. In section five, it is shown that a certain localized version of the normal operator is an elliptic pseudodifferential operator (DO) under the visibility, and the local and semi-global Bolker conditions. In section six, we study the operators and globally, and show that uniqueness and stability (injectivity) are of a generic set with the corresponding topology. In the last section, we implement our results for the initial dynamic problem of scanning a moving object while changing its shape. We also show that our results can be applied to fan beam geometry by an appropriate choice of phase function

2. Main Results

In this section, we first introduce the dynamic operator and then reduce it to an integral geometry problem integrating over level curves. After some necessary propositions, we state our main results.

Definition 2.1.

Let be a fixed open set in and be the open sets of lines determined by in . For , the operator of the dynamic inverse problem is defined by

where and the function is a non-vanishing smooth weight changing with respect to the variable and the position .

Here is a diffeomorphism in , which is identity outside , smoothly depending on the variable , and is the euclidean measure restricted to the lines parametrized by . Notice that each point (position) , lies on the lines in parametrized by .

The operator can be written in the following format:

Since is a diffeomorphism, by performing a change of variable , we get and therefore, we have

From now on, we do most of our analysis on the following general operator:

(2.1)

where is a new positive weight and the map

is real-valued. Here is the Euclidean measure of the level curves of function , defined as

We, first, need to show for any time and point , there exists a curve passing through the point with direction .

Proposition 2.1.

Let be the level curves of . Then, locally near and near a fixed the followings are equivalent.
i) The map from the variable to the unit normal vector of the level curves :

(2.2)

is a local diffeomorphism, where .
ii) The Local Bolker Condition:

(2.3)

holds locally near and near .

Remark 2.1.

i) The proof of Proposition 2.1 is postponed to the next section. In our setting, the equation (2.3) is the generalization of what it is known as a Bolker condition in [Theorem  3.4 (2), [8]].
ii) One may always rotate the unit normal vector by (at a fixed point on the curve) to get the tangent vector at that fixed point. Now the first part in Proposition 2.1 implies that the map from the variable to the tangent vector at point on the level curve , is also a local diffeomorphism.
iii) We work locally near and a fixed on the level curve. Let denote the unit tangent (normal) vector at . By the first part, for any unit tangent vector in some small neighborhood of ( is some perturbation of ), the map from the variable to the unit tangent vector at a fixed point is a local diffeomorphism. Now the Implicit Function Theorem implies that for any given , there exists a curve passing through the fixed point with a tangent vector . This indeed is what to expect if we want the level curves to behave like the geodesic curves.
iv) The local Bolker condition requires that when the object moves by time, the curve changes its direction. The counterexample when the local Bolker condition does not hold is the case where an object and the scanner move with the same rate. In this situation, the object can be considered stationary where it is being scanned with stationary parallel rays. The above proposition guarantees that locally and microlocally this situation will not happen and the parameter changes the angle if we keep the object stationary. (i.e the movement is not going to be synchronized with the scanner)
v) Proposition 3.1 in the next section, shows that one may connect the local Bolker condition to Fourier Integral Operator (FIO) theory by extending the function to a homogeneous function of order one (see [1]), and therefore one may use the condition (2.3) for the analysis.

For main results, we first state the following definitions.

Definition 2.2.

The function satisfies the Visibility condition at if there exists a pair with property such that Here is the cotangent bundle of .

The visibility condition requires that at a point and co-direction , locally, there exists a unique curve passing through which is conormal to . As we pointed out in Remark 2.1, this property is a natural property of level curves as are expected to behave like geodesic curves. It also means that each singularity can be probed locally.

Definition 2.3.

The function satisfies the Semi-Global Bolker Condition in a neighborhood of a fixed if

(2.4)

The first equation in (2.4) implies that at instance , the measurement will not distinguish between points and , when they are both on the same level curve. The second equation implies that the new measurement, due to perturbation in the variable , still cannot distinguish between these two points as they both belong to the same perturbed level curve. Note that, we only need our fixed point to have no conjugate points along the curve passing through it with conormal . Also, note that we do not require any restriction on and it can vary freely in that neighborhood.

We now are ready to state our main result for the operator given by (2.1).

Theorem 2.1.

Consider the operator with a nowhere vanishing smooth weight . Let be a set of all possible pairs which are smooth in some -topology with an arbitrary large natural number. Assume that for any , (i) the visibility condition holds and (ii) the local and semi-global Bolker conditions are satisfied for some given by the visibility condition.
Then within , there exists a dense and open (generic) set of pairs of such that locally near any pair in , the uniqueness results and therefore stability (injectivity) estimates given by Proposition 6.2 hold.

To formulate above result for the dynamic inverse operator given by (1.1), we first state the visibility, and the local and semi-global Bolker conditions for .

Visibility. This condition implies that for the map

(2.5)

is locally surjective. Here the point lies on the level curve

Local Bolker Condition. This condition (see Proposition 4.1) implies that

(2.6)

Semi-global Bolker condition (No conjugate points condition). By condition (2.4), semi-global Bolker condition holds if the map

(2.7)

is one-to-one.

Now for the dynamic inverse operator given by (1.1), we have the following result:

Theorem 2.2.

Consider the dynamic inverse operator with a nowhere vanishing smooth weight . Let be a set of all possible pairs which are smooth in some -topology with an arbitrary large natural number. Assume that for any , (i) the visibility condition (2.5) holds and (ii) the local and semi-global Bolker conditions given by (2.6) and (2.7) are satisfied for some given by the visibility condition.
Then within , there exists a dense and open (generic) set of pairs of such that locally near any pair in , the uniqueness results and therefore stability (injectivity) estimates hold.

Corollary 2.1.

In particular, for a small perturbation of where there is no motion or the motion is small enough , we have the actual injectivity and invertibility as the set of pairs of is included in

Remark 2.2.

The Corollary 2.1 follows from the fact that the stationary Radon transform is analytic and for a small perturbation of phase function, the invertibility and injectivity still hold.

3. Preliminary Results

In this section, we first prove Proposition 2.1 and then connect the local Bolker condition (2.3) to Fourier Integral Operator theory. At the end, we state some definitions which will be used in the following sections.

Definition 3.1.

A set is conic, if then for all .

Proof of Proposition 2.1.

i) ii) Fix and let . We work on some neighborhood of and . Since , the map (2.2) is well-defined and there exists a tangent at a fixed time when varies. The map (2.2) is a local diffeomorphism, therefore and its inverse exists with non-zero derivative in a conic neighborhood.

Assume now that . Then there exists a non-zero constant such that

(3.1)

Plugging (3.1) into :

we get , which is a contradiction. Therefore

ii) i) Assume that (2.3) is true. This in particular implies that and are non-zero and linearly independent. For any , let denotes the unit normal at a fixed point on the curve. To show the map in (2.2) is a local diffeomorphism, we need to show in a conic neighborhood. Note that this map is well-defined as . Assume that . Then

which implies that

This contradicts with the fact that and are linearly independent. Now by Inverse Function Theorem, the map (2.2) is a local diffeomorphism as it is smooth and its Jacobian is nowhere vanishing. ∎

One may extend the function to a homogeneous function of order one as follow:

(3.2)

As we pointed out above, we work locally in a conic neighborhood of and . This guarantees that function is single-valued. To connect the local Bolker condition to Fourier Integral Operator theory, we have the following proposition.

Proposition 3.1.

For the function defined by in (3.2), the local Bolker condition (2.3) holds if and only if

Proof.

Since , we have

and

where . Assume first that and are linearly independent. We show that columns in the matrix are linearly independent for . So let

Then we have

Since and are linearly independent, we have

which simply implies that , and therefore are linearly independent for .

Assume now that are linearly independent for . We show that and are linearly independent. We first rewrite and as follow:

and

Adding the last two equations we get

Consider

and

Adding the last two equations, we have

Now assume that

In a similar way as we showed above and using the fact that are linearly independent for , we conclude that . This proves the proposition. ∎

In principle, Proposition 2.2 implies that we may use our analysis with (2.3), see [1].

Definition 3.2.

We say that is not in the Wave Front Set of , , if there exists with so that for any , there exists such that

for in some conic neighborhood of . This definition is independent of the choice of .

Definition 3.3.

For the case of a scalar-valued distribution, one may define the Analytic Wave Front Set, , as the complement of all such that

with some and equal to 1 near . We recall that, there are three equivalent definitions of Analytic Wave Front Set in the literature.

Definition 3.4.

We call a position singularity if . We call a visible singularity if satisfies the visibility condition by Definition 2.2. Similarly, we call a measurement singularity if .

4. Microlocal Analyticity

In this section, we study the microlocal analyticity of operator for a given . We first compute the adjoint operator.

Adjoint Operator . Let be given, where is embedded in an open set . We extend our function to be zero on . Consider now the one-dimensional level curves

with Euclidean measure induced by the volume form in the domain . There exists a non-vanishing and smooth function such that

Therefore,

where and . In the second equality above, we used the fact that the double integral equals to an integral over , by Fubini’s Theorem. Thus, the adjoint of in is

where is supported in . In fact, the adjoint is localized in and is equal to the average of over all lines determined by passing through the point .

Schwartz Kernel. Now we compute the Schwartz kernel of the operator .

Lemma 4.1.

The Schwartz kernel of is

where .

Proof.

Let . By (2.1) we have

Since and when , by Theorem (6.1.5) Hörmander, we have

Here is pullback with . The second integral above can be written as

Therefore, the Schwartz kernel of is

Remark 4.1.

One may compute the Schwartz kernel of and :

The following lemma shows that the operator is an elliptic Fourier Integral Operator (FIO).

Lemma 4.2.

Let Then the operator is an elliptic FIO of order associated with the conormal bundle of :

where and are the coordinates on and , respectively.

Proof.

By Lemma 3.1 the Schwartz kernel has singularities conormal to the manifold . Since , the Schwartz kernel is conormal type in the class , see (Section 18.2, [11]). This shows that the operator is an elliptic FIO of order associated with the conormal bundle Note that is a one-dimensional non-zero variable. ∎

We now compute the canonical relation and show it is a four-dimensional non-degenerated conic submanifold of parametrized by . Note that is a Lagrangian submanifold of .

Proposition 4.1.

Let be the canonical relation associated with . Then

Furthermore, the canonical relation is a local canonical graph if and only if for any the map

(4.1)

is locally injective and local Bolker condition (2.3) holds.

Proof.

The twisted conormal bundle of M:

gives the canonical relation associated with . We first calculate the differential of the function . We have

Therefore, the canonical relation is given by

Now consider the microlocal version of double fibration:

{tikzcd}

[] & C \arrowdl[swap]Π_Y\arrowdrΠ_X &

T^*(Y) && T^*(X)

where

Our goal is to find out when the Bolker condition (locally) holds for , that is, is an injective immersion. We first compute its differential:

If has rank equal to four, then the Bolker condition is locally satisfied. Indeed, this is true, as has rank equal to four if and only if the condition (2.3) holds. This implies that . Since the map in (4.1) is one-to-one, the projection is an injective immersion. Hence, is a local diffeomorphism. ∎

The following lemma states whether position singularities and measurement singularities can affect each other. We refer the reader to Definitions 3.3 and 3.4, for position and measurement singularities.

Lemma 4.3.

Let be a fixed open set in and be the open sets of lines determined by in . Then, the map

is a local diffeomorphism.

Proof.

Consider the map . We show that for a given , one can determine . Since is non-zero ( and are both non-zero,) for a given there exists a tangent vector to each level curve . By Remark 2.1, one can find a non-zero normal vector on each level curve, and therefore . On each level curve , we have . Since , the Implicit Function Theorem implies that the variable determines . Hence, the map is a local diffeomorphism.

Now consider the map . Our goal is to determine , for a given . By Proposition 2.1, the map

is a local diffeomorphism for a fixed point provided that the condition (2.3) holds. Thus, determines the variable . In particular, for a given this implies that one can identify the level curve , as determines , and therefore (on each level curve we have .) Since with , one can determine . To determine the last variable , it is enough to take the partial derivative of with respect to the variable . Thus, the map is a local diffeomorphism. We remind that the above argument is valid when the condition (2.3) is satisfied.

Now since dim=dim and and are local diffeomorphisms, the map

will be a local diffeomorphism. ∎

Remark 4.2.

i) Note that, by Proposition 4.1.4 (Hörmander [12]), if we show one of the maps or is a local diffeomorphism, then the other map is also a local diffeomorphism as dim=dim. We, however, in above lemma have shown that both maps are local diffeomorphisms, as the proof reveals whether each map will be a global diffeomorphism or not. In fact, for a fixed there might be more than one curve which resolves the same singularity.
ii) The map being a local diffeomorphism implies that one can always track the position singularities by having the measurement singularities .
iii) From the geometrical point of view, the map being a local diffeomorphism means that for any fixed position and covector , there exists a curve (NOT necessarily unique) passing through perpendicular to This means singularities in data, i.e. , can affect the measurement singularities, i.e. .
iv) Proposition 4.1 and Lemma 4.3 show the local surjectivity of the map

Note that if the visibility condition holds, then we have the global surjectivity on

5. Global Bolker Condition

In this section, we study the normal operator to prove a stability estimate. It is known that the normal operator is a DO if the projection is an injective immersion (see Proposition  8.2, [5]). For our analysis, we assume that the Semi Global Bolker Condition holds which is similar to the No Conjugate Points assumption for the geodesics ray transform studied in ([4, 17]).

Remark 5.1.

There might be some points which do not satisfy the visibility, semi-global and local Bolker conditions. A point for which all above three conditions are satisfied is called a point.

Our goal is to show that normal operator is a DO of order -1. From now on, we fix a covector and work in a small conic neighborhood of this covector.

Theorem 5.1.

The normal operator is a classical DO of order with principal symbol

near . Here the functions and are defined as

where is well-defined locally by Lemma 4.3.

Proof.

For the proof we mainly follow (Lemma  2, [13]) Considering the Schwartz kernel of , we split the integration over into and . So we get

where and are the Schwartz kernels of the operators and with . We first consider . Note that is localized as the function priori satisfies the local Bolker condition (2.3). By semi-global Bolker condition (2.4), we have

Now a stationary phase method implies that is smooth away from the diagonal . Since , for a fixed there exists a neighborhood on which we have normal vectors. We work on normal coordinates as coordinates on , with . In these local coordinates, we may expand the phase function near the diagonal . Let

(5.1)

where is defined by the map