arXiv:2110.01405v1 [physics.med-ph] 29 Sep 2021

AN INVERSION ALGORITHM FOR P´FUNCTIONS WITH APPLICATIONSTO MULTI-ENERGY CT

GUILLAUME BAL, RUOMING GONG, AND FATMA TERZIOGLU

Abstract. Multi-energy computed tomography (ME-CT) is an x-ray transmission imaging tech-nique that uses the energy dependence of x-ray photon attenuation to determine the elementalcomposition of an object of interest. Mathematically, forward ME-CT measurements are modeledby a nonlinear integral transform. In this paper, local conditions for global invertibility of the ME-CT transform are studied, and explicit stability estimates quantifying the error propagation frommeasurements to reconstructions are provided. Motivated from the inverse problem of image re-construction in ME-CT, an iterative inversion algorithm for the so-called P´functions is proposed.Numerical simulations for ME-CT, in two and three materials settings with an equal number ofenergy measurements, confirm the theoretical predictions.

1. Introduction

Multi-energy computed tomography (ME-CT) is a diagnostic imaging technique that uses x-raysfor identifying material properties of an examined object in a non-invasive manner. While standardCT is based on the simplifying assumption of mono-energetic radiation, ME-CT exploits the factthat the attenuation of x-ray photons depends on the energy of the x-ray photon in addition to thematerials present in the imaged object [1–5].

ME-CT employs several energy measurements acquired from either an energy integrating detectorthat uses different x-ray source energy spectra or a photon counting detector that can registerphotons in multiple energy windows [6–8]. Energy dependence of photon attenuation can be utilizedto distinguish between different materials in an imaged object based on their density or atomicnumbers [1]. As a result, ME-CT provides quantitative information about the material compositionof the object, whereas standard CT can only visualize its morphology. More information on thephysics and practical applications of ME-CT can be found, for example in [4, 9, 10].

Mathematically, ME-CT measurements are modeled by a nonlinear integral transform that mapsx-ray attenuation function (or coefficient) of the object to weighted integrals of its x-ray transformover photon energy. Let Ω P Rd, d “ 2, 3, be the spatial domain of the imaged object. For y P Ω andphoton energy E, we denote by µpy,Eq the x-ray attenuation coefficient of the object. For 1 ď i ď n,let wipEq be the spectral weight function of the i-th energy measurement. For example, in the caseof energy integrating detectors, wipEq is the product of the i´th x-ray source energy spectrumSipEq and the detector response function DpEq. These weights wipEq are known and assumed tobe compactly supported and normalized so that

ş8

0 wipEqdE “ 1. Then, the corresponding ME-CTmeasurements for a line l are given by the integrals

Iiplq “

ż 8

0wipEqe

´ş

l µpy,EqdydE, 1 ď i ď n.(1)

The x-ray attenuation coefficient µpE, yq is commonly expressed by a superposition of the (known)elemental x-ray attenuation functions µjpEq weighted by the (unknown) partial density of each

2020 Mathematics Subject Classification. Primary 65R32; Secondary 92C55.Key words and phrases. P-function, inversion, global, uniqueness, stability, multi-energy CT, spectral CT.

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2 GUILLAUME BAL, RUOMING GONG, AND FATMA TERZIOGLU

respective element ρjpyq [1, 9]:

µpE, yq “nÿ

j“1

µjpEqρjpyq.

Here, we assume as many energy measurements as the number of unknown material densities. Then,we can write

Iipxq “

ż 8

0wipEqe

´MpEq¨xplqdE,(2)

where MpEq “ pµjpEqq1ďjďn and xplq “ pxjplqq1ďjďn with xjplq “ş

l ρjdl denoting the x-raytransform of ρj along a line l. We assume that x “ xplq P R Ă Rn where R Ă Rn is a closedrectangle (a Cartesian product of closed intervals).

Figure 1. Left: Examples of x-ray source spectrum for varying tube potentials computed usingthe publicly available code SPEKTR 3.0 [11], and then normalized. Right: The x-ray attenuationcoefficients of bone, iodine and water as functions of x-ray energy in log-log scale. The raw datawas obtained from NIST [12].

Therefore, one way to perform image reconstruction is to perform first a nonlinear inversionreconstructing x “ xplq from Ipxq for each line l; and then a linear tomographic reconstruction torecover the material density maps ρi from their line integrals xiplq. Since the invertibility of theline integral transform is well-studied, here we focus on the map g : R Ă Rn Ñ Rn defined by

gpxplqq “ pgipxplqqq1ďiďn, gipxq “ ´ ln Iipxq.(3)

The invertibility of the maps gi and Ii are equivalent.The map g is smooth for wi compactly supported and µj bounded, and its Jacobian at x P R is

given by the matrix Jpxq with coefficients

Jijpxq “Bgipxq

Bxj“ egipxq

ż 8

0wipEqµjpEqe

´MpEq¨xdE, 1 ď i, j ď n.(4)

We note that µjpEq ě 0 and wipEq ě 0 for all 1 ď i, j ď n, and that all entries of the Jacobianmatrix are strictly positive.

By the inverse function theorem, if det Jpxq ‰ 0, then the map g is locally injective. For n “ 2,Alvarez [13] studied the invertibility of g by testing for zero values of the Jacobian. In a previouswork [14], we proved that local injectivity of g guarantees injectivity in the whole domain R. Thismay not be true for a general map, but it holds in our case because of the positivity of the entriesof the Jacobian matrix J . On the other hand, for n ě 3, nonvanishing of the Jacobian determinantis not sufficient, and thus we need to impose further conditions on the Jacobian matrix to ensureglobal injectivity. This is given in the following theorem.

AN INVERSION ALGORITHM FOR P´FUNCTIONS 3

Theorem 1.1 ( [15]). Let F : R Ă Rn Ñ Rn be differentiable on the closed rectangle R. If theJacobian Jpxq of F is a P´matrix for each x P R, then F is injective in R.

A matrix A is called a P´matrix if all principal minors of A are positive [16]. Principal minorsof an nˆ n matrix A are defined as follows. Let K be a subset of xny “ t1, . . . , nu. We denote byAK the submatrix of A formed by deleting the rows and columns with indices in K. The principalminor of A associated to K, denoted by rAsK , is the determinant of AK . We let rAsxny “ 1.

A map F : R Ă Rn Ñ Rn is called a P´function if for any x, y P R, x ‰ y, there exists an indexk “ kpx, yq such that

pxk ´ ykqpfkpxq ´ fkpyqq ą 0.

Here xk and fkpxq are the k-th components of x and F pxq, respectively [17].It is known that [17, theorem 5.2], a differentiable map F defined on a rectangle R Ă Rn is a

P´function if its Jacobian Jpxq is a P´matrix for all x P R. Thus, a P´function F is injective andhence invertible on its range F pRq. Moreover, the inverse is a P´function as well ( [17, theorem3.1]).

The class of P´matrices includes positive quasi-definite matrices as well as strictly diagonallydominant matrices with positive diagonal entries. According to our numerical experiments in [14],the Jacobians in ME-CT are often P´matrices for varying spectral weights, but they are neitherquasi-definite nor diagonally dominant. If the Jacobian matrix is quasi-definite or strictly diagonallydominant everywhere, then iterative algorithms such as Gauss-Seidel are guaranteed to convergeto the global inverse [18]. However, in the case of P´matrix Jacobians, we were not able to findan algorithm in the literature that is guaranteed to converge. In this paper, we propose such analgorithm and prove its convergence for which we need an estimate for the Lipschitz constant of theinverse map. Such a result was given in our previous paper [14, Theorem 9] under conditions on theLipschitz constant of the forward map that were not explicitly formulated. In this paper, we makethese conditions explicit and present new estimates.

The paper is organized as follows. Section 2 presents quantitative estimates for injectivity ofP´functions. In section 3, we propose a damped newton type algorithm for the inversion ofP´functions. Section 4 contains the application of our results to ME-CT in the two and threematerials settings with an equal number of measurements.

We use the following notation: xny “ t1, . . . , nu, I: the identity matrix, x: the Euclidean norm,x8 “ max

iPxny|xi|, ‖A‖ “ max

‖x‖“1‖Ax‖, and ~A~ “ max

i,jPxny|aij |.

2. Injectivity and its quantitative estimation

Let R Ă Rn be a closed rectangle. Suppose that F : RÑ Rn is a continuously differentiable mapwith Jacobian matrix Jpzq, z P R. The Lipschitz constant of F inR is given by L “ maxzPR ~Jpzq~.In this section, we prove that the inverse map F´1 is also Lipschitz continuous on F pRq, and providea bound for its Lipschitz constant. To this end, we extend the map F to Rn using F : Rn Ñ Rndefined by

F pxq “ F pP pxqq ` Lpx´ P pxqq,(5)

where P : Rn Ñ R is the orthogonal projection map given by P pxq “ arg minzPR x´z. Note thatP pxq “ x when x P R while P pxq P BR when x P RnzR. The above extension with L “ 1 was usedby Mas-Colell [19] in proving theorem 1.1 of Gale and Nikaido for polyhedral domains.

Since F and P are Lipschitz continuous, F is also Lipschitz continuous, and hence almost every-where differentiable (by Rademacher’s theorem). Moreover, for all x P Rn where F is differentiable,the Jacobian of F is given by

Jpxq “ JpP pxqqDP pxq ` LpI´DP pxqq,(6)


with DP pxq being an n ˆ n diagonal matrix with diagonal entries either 0 or 1. Observe that theLipschitz constant of F is also equal to L. We now state our quantitative injectivity estimate.

Theorem 2.1. Let R Ă Rn be a closed rectangle. Suppose that F : R Ñ Rn is a continuouslydifferentiable map with a P´matrix Jacobian Jpzq at every z P R. Then, for all x, y P Rn, for theextended map F , we have

F pxq ´ F pyq8 ě τx´ y8,(7)

where

τ “ n´12 pn´ 1q

1ń2 min

zPRminKĂxny

L|K|`1ńrJpzqsK .(8)

In particular, for all x, y P R,

F pxq ´ F pyq8 ě τx´ y8.(9)

The constant 1τ provides an upper bound for M “ maxxPRn

~Jpxq´1~.

The proof of theorem 2.1 will be given later in this section. Two alternative derivations of (9)with different constants τ are provided in an Appendix.

Remark 2.2. Different extensions than (5) may also be considered. For instance, when n “ 2 and

J ą 0, the constant Lmay be replaced by any matrix„

L ´m´m L

, with 0 ď m ď minzPR

mini,j“t1,2u

Jijpzq.

Then, F is also a P´function and ~Jpxq´1~ ď Ldet JpP pxqq . The extension in (5) was chosen to

optimize the bound on the norm of the inverse Jacobian.

2.1. Regularized extension and proof of theorem 2.1. Our proof of theorem 2.1 requires aregularized version of the extension (5). Without loss of generality, we consider R “ r0, 1sn. Wefirst assume that the Jacobian Jpxq of F pxq is a P´matrix at every x P R, so F is globally injectivein R. By continuous differentiability, there exists ε ą 0 such that Jpxq is a P´matrix at everyx P Rε “ r´ε2, 1` ε2s

n.Let pε : RÑ r´ε2, 1` ε2s be the function defined by

pεpxq “

$

’

’

’

’

’

’

’

’

&

’

’

’

’

’

’

’

’

%

´ε2, x ă ´ε,1

2

´

x`ε

πsin

´πx

ε

¯¯

, ´ε ă x ă 0,

x, 0 ď x ď 1,1

2

ˆ

x` 1`ε

πsin

ˆ

πp1´ xq

ε

˙˙

, 1 ď x ď 1` ε,

1` ε2, 1` ε ă x.

(10)

For ε “ 0.25, the plots of pε, and its first and second order derivatives are given in fig. 2.Let Lε be the Lipschitz constant of F in Rε, i.e., Lε “ max

zPRε

~Jpzq~. We define Fε : Rn Ñ Rn by

Fεpxq “ F pPεpxqq ` Lεpx´ Pεpxqq,(11)

where Pεpxq “ ppεpxiqqiPxny. We note that as ε Ñ 0, we have Pε Ñ P and thus Fε Ñ F pointwise.We then have the following result.

Proposition 2.3. Fε extends F from R to Rn, and is as smooth as F . Moreover, it is a P´function,and hence a diffeomorphism.

Proof. Since Pεpxq “ x for all x P R and ε ě 0, we have Fεˇ

ˇ

R “ F . By [17, theorem 5.2], we knowthat Fε is a P´function if its Jacobian Jεpxq is a P´matrix for all x P Rn. Observe that

Jεpxq “ JpPεpxqqDPεpxq ` LεpI´DPεpxqq,(12)


Figure 2. Plots of (a) pεpxq given in (10), and its (b) first and (c) second derivatives whenε “ 0.25.

with DPεpxq being an nˆ n diagonal matrix with diagonal entries p1εpxiq, i P xny.Therefore,

detpJεpxqq “ÿ

KĂxny

L|K|ε

´

ź

kPK

p1´ p1εpxkqq¯´

ź

kPxnyzK

p1εpxkq¯

rJpPεpxqqsK .(13)

Now since PεpRnq “ Rε and Jpxq is a P´matrix for all x P Rε, we have rJpPεpxqqsK ą 0 for allK Ă xny. Moreover, 0 ď p1ε ď 1. Thus, all terms in the sum (13) are nonnegative and do not vanishat the same time, which implies that detpJεpxqq ą 0 for all x P Rn. Applying the same argumentto principal submatrices, we obtain that all principal minors of Jεpxq are positive for all x P Rn.Hence, Jεpxq is a P´matrix at every x P Rn.

We now present a quantitative injectivity estimate for Fε.

Theorem 2.4. For all x, y P Rn,

Fεpxq ´ Fεpyq8 ě τεx´ y8,(14)

where

τε :“ n´12 pn´ 1q

1ń2 min

zPRε

minKĂxny

L|K|`1ńε rJpzqsK .(15)

Proof. Let x, y P Rn. Then Fεpxq “ u and Fεpyq “ v for some u, v P Rn. Applying the Mean Valuetheorem to F´1ε , then using Cauchy-Schwarz inequality and the equivalence of norms, we obtain

x´ y8 “ F´1ε puq ´ F´1ε pvq8 ď

?nMεu´ v8,

where Mε :“ maxxPRn

~Jεpxq´1~.

We now need to find an upper bound for

pJεpxq´1qij “

p´1qi`jrJεpxqstju,tiu

det Jεpxq,

for all x P Rn and i, j P xny. By Hadamard inequality (see e.g. [20, p. 1077]), we have

rJεpxqstiu,tju ď pn´ 1qn´12 ~Jεpxq~

n´1.

Moreover,

~Jεpxq~ “ maxi,jPxny

JijpPεpxqqp1εpxjq, JjjpPεpxqqp

1εpxjq ` Lεp1´ p

1εpxjqq

(

ď Lε.


We also have

detpJεpxqq “ÿ

KĂxny

L|K|ε

´

ź

kPK

p1´ p1εpxkqq¯´

ź

kPxnyzK

p1εpxkq¯

rJpPεpxqqsK

ě

”

ÿ

KĂxny

´

ź

kPK

p1´ p1εpxkqq¯´

ź

kPxnyzK

p1εpxkq¯ı

minKĂxny

L|K|ε rJpPεpxqqsK(16)

“ minKĂxny

L|K|ε rJpPεpxqqsK ,

as the sum in square brackets is equal to 1. Hence,

~Jεpxq´1~ ď

pn´ 1qn´12 Ln´1ε

minKĂxny L|K|ε rJpPεpxqqsK

.(17)

Since PεpRnq “ Rε, by maximizing over all x P Rn, one obtains that Mε ď1

?nτε

, which yields

(14).

Proof of Theorem 2.1. For ε Ñ 0, we have Fε Ñ F and τε Ñ τ . Thus, the result follows fromtheorem 2.4. ˝

The following estimates will be used in section 3.

Proposition 2.5. Let DJ , DJε, and DJ´1ε denote the Hessian of F , Fε and F´1ε , respectively.Then,

~DJ´1ε ~ ď 2nτ´2ε ~DJε~,(18)

and

~DJε~ ď ~DJ~ `π

εLε.(19)

Proof. We continue using the notation ~A~ “ maxi,j,kPxny

|aijk|. For i, j, k P xny, and x P Rn,

pDJεpxqkqij “B2pFεqkpxq

BxjBxi“B2FkpPεpxqq

BxjBxip1εpxiqp

1εpxjq `

ˆ

BFkpPεpxqq

Bxi´ Lεδki

˙

δijp2εpxiq

“ p1εpxiqp1εpxjqpDJpPεpxqqkqij ` δijp

2εpxiq ppJpPεpxqqki ´ Lεδkiq .

Since |p1ε| ď 1 and |p2ε| ďπ

2ε, the estimate (19) holds. We note that since PεpRnq “ Rε and F is

smooth in Rε, DJ is bounded, hence so is DJε.For the estimate (18), we differentiate

pJεpxq´1qij “

p´1qi`jrJεpxqstju,tiu

det Jεpxq,

to obtain

B

BxkpJεpxq

´1qij “p´1qi`j B

BxkrJεpxqstju,tiu ´ pJεpxq

´1qijBBxk

det Jεpxq

det Jεpxq.

For a matrix Apxq “ paijpxqqi,jPxny, by differentiating detA “nÿ

i,j“1

p´1qi`jaijrAstiu,tju, we have

BpdetApxqq

Bxk“

nÿ

i,j“1

BpdetApxqq

Baij

Baijpxq

Bxk“

nÿ

i,j“1

p´1qi`jrApxqstiu,tjuBaijpxq

Bxk.


Thus,

B

Bxkpdet Jεpxqq “

nÿ

i,j“1

p´1qi`jrJεpxqstiu,tjuB2pFεqipxq

BxkBxj,

and

B

BxkrJεpxqstju,tiu “

n´1ÿ

l,m“1

p´1ql`mrJεpxqstj,l1u,ti,m1uB2pFεql1pxq

BxkBxm1,

where l1 “ l ` δjďl and m1 “ m` δiďm with δjďl “ 1 if j ď l; and 0, otherwise.By Hadamard’s inequality, we have

rJεpxqstiu,tju ď pn´ 1qn´12 ~Jεpxq~

n´1,

andrJεpxqsti,l1u,tj,m1u ď pn´ 2q

n´22 ~Jεpxq~

n´2.

Now using the above estimates in

ˇ

ˇ

ˇ

ˇ

B

BxkpJεpxq

´1qij

ˇ

ˇ

ˇ

ˇ

ď

ˇ

ˇ

ˇ

BBxkrJεpxqstju,tiu

ˇ

ˇ

ˇ`

ˇ

ˇ

ˇJεpxq

´1qij

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

BBxk

det Jεpxqˇ

ˇ

ˇ

det Jεpxq,(20)

and then maximizing over all x P Rn, in view of (15) and (17), we obtain (18).

3. An inversion algorithm for P´functions

In section 2, we presented stability estimates for a P´function F and its extension Fε. Weare now interested in finding an algorithm to solve the inverse problem F pxq “ y where F is aP´function defined on a rectangular region. We did not find any standard iterative algorithm thatis guaranteed to converge to the unique attractor in the P´function setting. In this section, we willpropose an algorithm taking the form of damped Newton’s method which is guaranteed to convergefor a smooth extension F :“ Fε at a fixed value of ε.

3.1. Existence of periodic orbit for Newton’s method. Consider the inverse problem F pxq “y˚ and the standard iterative Newton’s method F pxnq `DF pxnqpxn`1 ´ xnq “ y˚, or equivalently,xn`1 “ xn ` DF´1pxnqpy

˚ ´ F pxnqq. We now show that such an algorithm is not guaranteed toconverge when F is a P´function. Indeed, let A be a P´matrix and F pxq “ Ax linear in thequadrant x ą 0 (i.e., each coordinate positive).

Let F pxq be the extension (5) (with ε “ 0) to Rn. The Jacobian of F is then piecewise-constantand equal to Aq for q labeling the 2n quadrants (e.g. equal to identity in the quadrant x ă 0 andto A in the quadrant x ą 0).

By linearity, using q “ qpxq the quadrant label to which x belongs, we observe that

xn`1 “ xn À´1q pxnqpy

˚ ´ F pxnqq “ xn À´1q pxnqpy

˚ Áqpxnqxnq “ A´1q pxnqy˚.

For a fixed y˚, the above right-hand side thus takes a maximum of 2n values. As soon as xn`1belongs to the same quadrant as xn, then xn`j “ xn`1 for all j ě 1 and the algorithm converges.However, it turns out that cycles in the xk are quite possible and thus prevent the algorithm fromconverging to the unique solution x of F pxq “ y˚ (which does not belong to the cycle).

As a concrete example in dimension n “ 3, consider

A “

»

–

6 1 77 3 11 8 3

fi

fl , y˚ “

»

–

´9´4´6

fi

fl .(21)


Here, F pxq “ Ax is a P´function for x ą 0 since A is a P´matrix. Incidentally, A has positiveentries (as do Jacobian matrix for ME-CT models). We extend F pxq using Mas-Colell’s extension(5). For suitable choices of the initial point x0, we observe a periodic trajectory in Figure 3 (a).

The points involved in the above cyclic trajectories all live away from the hyperplanes separatingquadrants. Therefore, for ε ą 0 sufficiently small, F and Fε coincide in the vicinity of the abovenumerical trajectories. This shows that the Newton algorithm would also fail to converge for thesmooth P´function Fε, or as a matter of fact for any possibly C8 P´function equal to the aboveF in the vicinity of the trajectories. We have not seen such obstructions to the convergence of theNewton algorithm with smooth functionals in the literature.

To avoid the above periodic orbits in the iterative algorithm, let us consider the following dampedNewton’s method,

xn`1 “ xn ´DF pxnq´1pF pxnq ´ y

˚qh,(22)

where h ď 1 is a constant step size. When DF is not defined at xn belonging to (the closure of)more than one quadrant, we choose DF pxnq “ DFxn´1pxnq, with Fxn´1 the C2 function equal to Fon the rectangular sector where xn´1 lives.

In the piecewise linear example with A defined in (21), we actually observe the persistence ofperiodic orbits for values of h ă 1, for instance h “ 0.8 in Fig.3(b). For h “ 0.7 however, we observethat the discrete trajectory converges to the target point F´1py˚q as in Fig.3(c). In our choice ofF and y˚, we have F´1py˚q “ y˚.

(a) h “ 1 (b) h “ 0.8 (c) h “ 0.7

Figure 3. Plots of 3D discrete trajectories of the damped Newton algorithm with step sizes (a)h=1; (b) h=0.8; and (c) h=0.7. x0 is the initial point and y˚ “ F´1

py˚q the target point. Thedotted line in (c) represents a few steps in the discrete trajectory that are not presented.

The above example provide counterexamples to the convergence of the damped algorithm whenh is not sufficiently small. We now show that for smooth P´functionals such as Fεpxq, the dampedNewton algorithm with h sufficiently small does indeed converge to the unique solution Fεpxq “ y˚.

3.2. First-order evolution ODE for injective functions. Recall that we are interested in solv-ing the inverse problem F pxq “ y, where F : R Ă Rn Ñ Rn is a continuously differentiableP´function defined on rectangle region R.

We showed in section 2 that the P´function F defined on a rectangle as well as its extension Fεgiven in (11) were injective. We define the associated dynamical system

Ψ : Rn ˆ R` Ñ Rn, Ψpy, tq “ y˚ ` e´tpy ´ y˚q,(23)

parametrizing the segment between any y in image space and the target y˚. It is known [21] thatfor any injective continuous function F : Ω Ă Rn Ñ Rn defined on a connected domain Ω, thereexists a flow Φ : Ω Ă Rn Ñ Ω satisfying

F pΦpx, tqq “ ΨpF pxq, tq,(24)


for all x P Ω, t ě 0. When F is differentiable, with fixed initial point x0 (for convenience, we writeΦpx0, tq “ x), we have

dF pxq

dt“ DF pxq 9x,

dΨpF px0q, tq

dt“drF px˚q ` e´tpF px0q ´ F px

˚qqs

dt“ ´rF pxq ´ F px˚qs.

Since F pxq “ ΨpF px0q, tq, we have DF pxq 9x “ ´rF pxq ´ F px˚qs, which implies

9x “ ´pDF pxqq´1rF pxq ´ F px˚qs “ ´pDF pxqq´1rF pxq ´ y˚s.(25)

This yields the following result.

Theorem 3.1 ( [21]). Let F : Ω Ñ Rn, where Ω Ă Rn is open and connected, be a local diffeomor-phism with convex range, and x˚ P Ω, y˚ “ F px˚q . The mapping Φ defined in (24) is C1 and it isthe flow of the following differential equation

9x “ ´pDF pxqq´1rF pxq ´ y˚s :“ Gpxq.(26)

That is, with xp0q “ x0,

Φpx0, tq “ xptq.(27)

Equation (26) provides a continuous version of the damped Newton algorithm in the limit hÑ 0.For any choice of the initial condition x0, the solution xptq of the dynamical system converges to thedesired point x˚ (since F pxptqq converges to y˚) provided F is an injective differentiable functionwith convex range. This shows the usefulness of the extension Fε since the P´function F definedon a rectangle may not have (and indeed does not have in several ME-CT cases) convex range.

3.3. Convergence of the algorithm for smooth extension. The damped Newton algorithm,which may be seen as a first-order discretization of (26), is given by

xn`1 “ xn `Gpxnqh “ xn ´DFεpxnq´1pFεpxnq ´ y

˚qh,(28)

with h ą 0 the step size. We now show that for h sufficiently small, the above algorithm convergeswith linear rate of convergence. More precisely, we have:

Theorem 3.2. Let F : R Ă Rn Ñ Rn be a C2 P´function defined on rectangle region R. Let Lεbe the Lipschitz constant of Fε and define c1 “ max

xPRn

∥∥∥DFεpxq´1∥∥∥, c2 “ maxxPRn,‖v‖“1

∥∥∥DpDFεpxq´1qv∥∥∥,and c3 “

∥∥∥Fεpx0q ´ y˚∥∥∥. Then there exists r ą 0, such that for all h ă H and i,∥∥∥Fεpxi`1q ´ y˚∥∥∥∥∥∥Fεpxiq ´ y˚∥∥∥ ă 1´ rh,(29)

where H “ Opεq. In particular, F pxiq converges to y˚ with linear rate.

The proof follows from the error analysis of the Euler method for the above dynamical system.

The constant for the linear convergence rate r may be chosen as, e.g., r “1

8and the step size H is

linear in ε; see the proof for a more explicit dependency.We first prove the following lemma.

Lemma 3.3. Let x1 be the solution to the discretized ODE (28) starting at x0. Let xptq be thesolution to the ODE (26) with initial condition xp0q “ x0. Let t1 “ h. Then, we have

‖xpt1q ´ x1‖ ď‖DG‖ ‖G‖

2h2

where ‖DG‖ “ maxtPr0,t1s,‖v‖“1

‖DGpxptqqv‖ and ‖G‖ “ maxtPr0,t1s

‖Gpxptqq‖.


Proof. Using Taylor’s formula, we expand xptq at time t0 and evaluate at t1 “ h to obtain

xpt1q “ xpt0q ` h 9xpt0q `h2

2:xpτq “ xpt0q ` hGpxpt0qq `

h2

2DGpxpτqqGpxpτqq.(30)

By definition of x1 “ x0 ` hGpx0q,

x1 ´ xpt1q “ x0 ` hGpx0q ´ xpt1q “ ´h2

2pDGpxpτqqGpxpτqqq.

Therefore, ‖xpt1q ´ x1‖ ďh2

2‖DGpxpτqqGpxpτqq‖ ď ‖DG‖ ‖G‖

2h2.

Proof. (Theorem 3.2). We fix i and define ∆i “

∥∥∥Fεpxiq ´ y˚∥∥∥. Let xptq “ Φpxi, tq where Φpx, tq isdefined in (27) so that xp0q “ xi. Define Ei “ ‖xi`1 ´ xphq‖ for convenience.

Since Fεpxphqq ´ y˚ “ e´hpFεpxiq ´ y˚q as we see from (23) and (24), and e´h ă 1 ´1

4h for h

small (in fact, it is true for 0<h<1), we obtain using the Lipschitz continuity of Fε that∥∥∥Fεpxi`1q ´ y˚∥∥∥∥∥∥Fεpxiq ´ y˚∥∥∥ ď

∥∥∥Fεpxi`1q ´ Fεpxphqq∥∥∥` ∥∥∥Fεpxphqq ´ y˚∥∥∥∆i

ďLεEi ` p1´

14hq∆i

∆i.(31)

Note that∥∥∥Fεpxptqq ´ y˚∥∥∥ ď ∆i since F pxptqq lives on a straight line. We use induction to show

∆k ď c3 for any k. For k “ 0, we have ∆0 “ c3, which provides the base case. We assume afterstep k (that is we start with xk) that ∆k ď c3, so that

‖Gpxptqq‖ ď∥∥∥DF´1ε

∥∥∥∥∥∥Fεpxptqq ´ y˚∥∥∥ ď c1∆k

‖DGpxptqq‖ “∥∥∥DpDFεpxptqq´1qpFεpxptqq ´ y˚q ` I

∥∥∥ ď c2∆k ` 1 ď c2c3 ` 1.

By Lemma 3.3 we have

Ek ď‖G‖ ‖DG‖

2h2 ď

c1∆kpc2c3 ` 1q

2h2(32)

By (31) and (32) ∥∥∥Fεpxk`1q ´ y˚∥∥∥∥∥∥Fεpxkq ´ y˚∥∥∥ ďLεEk ` p1´

h4 q∆k

∆kď 1´

1

8h.(33)

The inequality (33) holds with r “1

8when h sufficiently small. In fact, we can choose h “

1

4

1

Lεc1pc2c3 ` 1q. By Proposition 2.5, c2 “ Op1

εq, so h “ Opεq necessarily. This shows that

∆k`1 ă ∆k ă c3. By induction, we conclude ∆k ă c3 for any k.With ∆i ă c3, the result follows from (32) and (33) by replacing k with i.

Remark 3.4 (Quadratic rate of convergence). Theorem 3.2 shows that the algorithm (28) convergesto the attractor x˚ “ F´1ε py˚q with linear convergence rate. When close enough to x˚, we may infact switch to a standard Newton’s method

xn`1 “ xn ´DFεpxnq´1pFεpxnq ´ y

˚q,

to achieve quadratic convergence rate. The radius of convergence of Newton’s method [22] is ρ ă2

3βγwhere β “

∥∥∥DFεpx˚q´1∥∥∥ and γ satisfies∥∥∥DFεpyq ´DFεpxq∥∥∥ ď γ ‖y ´ x‖ ,@x, y P R.


(a) (b)

Figure 4. The plots of discrete trajectories (blue) of the algorithm in (a) physical space and (b)image space with k “ 5, h “ 0.1. x0 is the initial point of the algorithm and y˚ is the target pointin image space.

We may therefore choose T so that1

τε‖ypT q ´ y˚‖ ă ρ, where τε is defined by (15). In particular,

T ě log3βγ ‖y0 ´ y˚‖

2τε.(34)

The above results therefore provide an algorithm with overall quadratic rate of convergence. Wesolve the damped Newton algorithm for a finite time T as given in (34) (i.e., for a finite number ofsteps given by T h) to obtain an approximation ypT q sufficiently close to y˚. We then switch to astandard Newton algorithm (with h “ 1) whereby obtaining a quadratic rate of convergence to thefixed point x˚.

3.4. Remarks on the smoothness of the extension F . The above results show the uncondi-tional convergence of the damped Newton algorithm when F is extended to F and F is sufficientlysmooth. When F “ Fε a smooth extension of F , then h needs to be chosen of order ε. This is nota major constraint in practice since ε may in fact be chosen reasonably large without significantlymodifying the stability of the inversion procedure.

Consider the two-dimensional P´function

F px, yq “ px, kx2 ` yq,

defined on rectangle region R : r´1, 1s ˆ r´1, 1s. The extension F on R2 is

F px, yq “

#

px, y ` kq x ă ´1, x ą 1

px, kx2 ` yq ´1 ď x ď 1

In Fig.4(a), we display the trajectory associated with the damped Newton algorithm for the ex-tension F and for a very small value of h. We observe that this trajectory mostly lives outside ofthe initial rectangular domain. This shows that the trajectories of the damped Newton algorithmleave the original domain R and thus require F to be extended first. There may still be differentalgorithms allowing us to converge to F´1py˚q with discrete dynamics staying inside R but this isnot what we considered here.

It would be interesting to see how the algorithm behaves when F is the unregularized extension(5), or whether h may be chosen independently of ε if the latter is small. We do not have afull theoretical understanding of this case, and in particular do not know if the damped Newtonalgorithm always converges for the extensions in (5).


Here, we provide a family of examples displaying the difficulty to obtain such a convergence result.In particular, we show that the algorithm is not necessarily a contraction at each step in the imagespace no matter how small h is chosen. This is in sharp contrast with the proofs of convergence ofiterative methods for functions with positive definite Jacobians [22].

Consider the linear map

F px, yq “ px` ky, yq, k ą 0,

defined on rectangular domain r0, 1s ˆ r0, 1s. The Jacobian matrix of F is a P´matrix everywhereinside the domain, so F is a P´function. The extension F on R2 is given by

F px, yq “

$

’

&

’

%

px, yq y ă 0

px` ky, yq 0 ď y ď 1

px` k, yq y ą 1.

For concreteness, choose a initial point x0 “ p3132 ,3132q, y

˚ “ F px0q ` p116 ,

116q, and k “ 20. Let di

be the distance between F pxiq and y˚ in the image space. Then, for h “ 0.1, we find d6 “ 0.0470,d7 “ 0.0644, d8 “ 0.0579, d9 “ 0.0522, and d10 “ 0.0469. The increase of the distance at step 7 isdue to the crossing of the discontinuity of the Jacobian at the boundary y “ 1. We easily verify thatthe distance increases when crossing such a singular interface no matter how small h is chosen. Thealgorithm still converges eventually. We were able to prove (details not shown) for two-dimensionalextensions F that the algorithm is always contracting after m steps, i.e., F pxmnq ´ y˚ decreaseswith n for an appropriately chosen m (equal to 3 in the above example). The damped Newtonalgorithm is therefore convergent. By appropriately choosing k, we can force m to be as large as wewant. We were not able to extend the derivation to higher dimensions because of the complexity ofthe discrete dynamics in the vicinity of the singularities of the Jacobian of the extension F .

4. Application to Multi-energy CT

In this section, we present numerical experiments for multi energy CT transforms with two andthree commonly used materials and an equal number of energy measurements. The configurationof parameters was done as follows.

‚ The diagnostic energy range 10´ 150 keV was considered.‚ The energy spectra Si, i “ 1, . . . n, corresponding to tube potentials tpi were computed us-ing the publicly available code SPEKTR 3.0 [11]. For practical purposes, only integer valuedtube potentials ranging from 40-150 kVp were considered. We denote tp “ ptp1, . . . , tpnq.We assume that the detectors have linear sensitivity, i.e., DpEq “ E, which is the case forenergy integrating detectors.

‚ The domain of the transform I is chosen as

R “

#

px1, . . . , xnq P Rn` : 0 ď xj ď16

max10ďEď150

MjpEq

+

,

where MjpEq denotes the energy-dependent mass-attenuation of the j-th material, andMpEq “ pMjpEqq1ďjďn. We note that then e´MpEq¨x ě e´16, which is more conservativethan necessary in practice.

4.1. DE-CT: Two materials - two measurements. In the following, we consider three differ-ent material pairs bone-water, iodine-water, and bone-iodine in the said order, and present thecorresponding plots of

M “ maxzPR

~Jpzq´1~ “ maxzPR

~Jpzq~

|det Jpzq|,(35)


and its estimate (see theorem 2.1)

Mest “L

minzPR | det Jpzq|.(36)

For bone-water material pair, both M and Mest attained their minimum at ptp1, tp2q “ p40, 150q,which are 9.03 and 15.95, respectively. A not so good choice for the tube potentials is ptp1, tp2q “p135, 150q where M equals 117.4 (see fig. 5). One would expect the choice ptp1, tp2q “ p40, 150q tolead a better posed problem.

(a) (b) (c) (d)

Figure 5. The plots of (a) M and (b) Mest as a function of tube potentials for the materialpair (bone,water) in the rectangle R “ r0, 0.3s ˆ r0, 3s. The plots of ~Jpxq´1

~ on R when (c)tp “ p40, 150q and (d) tp “ p135, 150q.

For iodine-water material pair, M attained its minimum at ptp1, tp2q “ p40, 68q, which is 8.15.Mest attained its minimum at ptp1, tp2q “ p40, 66q, and is 11.3. We note that Mp40, 66q “ 8.2. Onthe other hand, a poor choice for the tube potentials would be ptp1, tp2q “ p55, 82q where M equals78865 (see fig. 6). We will demonstrate in section 4.3 that the former indeed leads to considerablybetter reconstructions.

(a) (b) (c) (d)

Figure 6. The plots of (a) M and (b) Mest as a function of tube potentials for the materialpair (iodine,water) in the rectangle R “ r0, 0.02s ˆ r0, 3s. The plots of ~Jpxq´1


For bone-iodine material pair, M attained its minimum at ptp1, tp2q “ p40, 74q, which is 0.68.Mest attained its minimum at ptp1, tp2q “ p40, 67q, which is 0.99. We note that Mp40, 67q “ 0.71.A poor choice for the tube potentials would be ptp1, tp2q “ p79, 132q where M equals 30536 (see fig.7).

The above examples suggest that M and Mest are well-aligned in the case of the two-materials.

4.2. ME-CT: Three materials - three measurements. We now consider the materials bone,iodine, and water, in the said order. The considered rectangle is R “ r0, 0.3s ˆ r0, 0.02s ˆ r0, 3s. In


(a) (b) (c) (d)

Figure 7. The plots of (a) M and (b) Mest as a function of tube potentials for the materialpair (bone,Iodine) in the rectangle R “ r0, 0.3s ˆ r0, 0.02s. The plots of ~Jpxq´1


this case, we have

M “ maxxPR3

~Jpxq´1~ “ maxzPR

maxiPx3y

t~Jpzq´1i ~,~Jpzq´1~u,(37)

where Jpzqi denotes the principal submatrix of the Jacobian Jpzq obtained by deleting the i-th rowand column. Using positivity of the Jacobian matrix everywhere, one can estimate M , accordingto theorem 2.1, by

Mest “L2

miniPx3y

tLrJpzqsi, det Jpzqu.(38)

The plots for M and Mest are given in fig. 8. The minimum value of M , which is 23.49, isattained at ptp1, tp2, tp3q “ p40, 74, 150q. However, it can also get very large, for example M equals13235 at ptp1, tp2, tp3q “ p70, 120, 150q.

The minimum value for Mest, which is equal to 1005, is attained at ptp1, tp2, tp3q “ p40, 67, 150q.We note that Mp40, 67, 150q “ 24.43. We observed that although M and Mest display differentoverall behaviors, they attain their minimum at nearby points.

(a) (b) (c) (d)

Figure 8. The plots of M and Mest as functions of tube potentials when (a,b) tp3 “ 150 kVp andwhen (c,d) tp1 “ 40 kVp for the material pair (bone,iodine,water) in the rectangle R “ r0, 0.3s ˆr0, 0.02s ˆ r0, 3s.

4.3. Sinogram Reconstructions from DE-CT measurements. In this section, we present theresults of the numerical implementation of our inversion algorithm using MATLAB. We focus onreconstructing the line integrals (sinograms) from the DE-CT measurements. The second step,reconstructing material density maps from their sinograms can be obtained by standard Radontransform inversion, which is not presented here.


In the experiments, we considered a two-material (iodine and water) phantom supported inr´2, 2s ˆ r´2, 2s. The material density maps for iodine and water are given by

ρ1 “ 0.05pχD2 ´χD1q and ρ2 “ χD3 ´

χD2 `χD1 ,

respectively. Here, χDi , i “ 1, 2, 3, denotes the characteristic function of the disk centered at theorigin and having radii 0.3, 0.5, and 1.5, respectively (see fig. 9(a)).

The sinograms of the material densities ρi, i “ 1, 2, were analytically computed at 257 uniformlysampled nodes in r´2

?2, 2?

2s and 400 uniformly sampled angles in r´π, πs (see fig. 10)(a).Noise is considered in the measurements as follows. For each line l and energy E, the number

of measured photons in our model is N0SipEqe´MpEq¨xplq with N0 the number of emitted photons

independent of energy and line by normalization of our spectra. Poisson (shot) noise is then includedin each such quantity. The number N0 of photons characterizing the Poisson distribution is chosenso that the relative L2-errors between noisy and noiseless DE-CT measurements of l Ñ Iiplq withtp “ r40, 68s for the low (i “ 1) and high (i “ 2) energies are approximately 0.6% and 0.3%,respectively. When tp “ r55, 82s, the corresponding errors were approximately 0.4% and 0.2%,respectively. In the experiments, we considered roughly 105 emitted photons per energy bin (1keV ) per mm2 (detector bin area) per mAs (milliampere-seconds), and hence to a relatively noisysituation in practice [23]. This value was chosen to display small but sizeable errors for the stabletube profile. They generated very large errors for the unstable tube profile.

The plots of DE-CT measurements corresponding to the two pairs of tube potentials are shownin fig. 9(b,c)).

We tested our inversion algorithm on noisy DE-CT measurements by considering both the originalDE-CT map and its extension. Reconstructions of the sinograms, obtained by using the extendedmap, from noisy DE-CT measurements for the two pairs of tube potentials are shown in fig. 10(b,c).Fig. 11 contains the central horizontal slices of the reconstructions shown in fig. 10(b,c). The rela-tive L2-errors between exact sinograms (denoted by X1 and X2) and their reconstructions are givenin table 1. As we expected, using the tube potentials minimizing the inverse Jacobian led to consid-erably better reconstructions. In all cases, using the extended map instead of the original enhancedthe quality of the reconstructions. When we considered the DE-CT measurements correspondingto the tube potentials tp “ p55, 82q, and performed inversion using the original map, the algorithmfailed to converge for approximately 17% of the lines considered. The relative L2-errors computedby excluding these lines is given in the last column of table 1.

(a) (b) (c)

Figure 9. (a)Iodine (top) and water (bottom) density plots of the two-material phantom, DE-CTmeasurements corresponding to tube potentials (b) tp “ p40, 68q and (c) tp “ p55, 82q.


(a) (b) (c)

Figure 10. (a) The exact sinograms of iodine (top) and water (bottom) density maps of thephantom given in Fig. 9, and their reconstructions from noisy DE-CT measurements correspondingto tube potentials (b) tp “ p40, 68q and (c) tp “ p55, 82q.

(a) (b)

Figure 11. Comparison of central profiles of the exact sinograms and their reconstructions fromnoisy DE-CT measurements corresponding to tube potentials (a) tp “ p40, 68q and (b) tp “ p55, 82q.

Extension Original maptp “ p40, 68q tp “ p55, 82q tp “ p40, 68q tp “ p55, 82q

L2-error for X1 0.0483 0.3408 0.0591 0.4792˚

L2-error for X2 0.0312 0.2280 0.0380 0.3105˚

Table 1. Normalized L2-errors of sinogram reconstructions using our inversion algorithm.˚Computed excluding the lines for which the algorithm failed to converge and hence shown forillustrative purposes.


5. Conclusions

Finding local criteria for the global injectivity of maps from a subset of Rn to Rn is a challengingproblem. One such criterion is based on the notion of P´functions we considered in this paper. Fol-lowing [14], many Multi-Energy Computed Tomography (ME-CT) problems were shown to satisfythe latter criterion.

Following [19], we obtained in section 2 several extensions of a given P´function to possiblysmooth injective maps from Rn to Rn and presented explicit stability estimates controlling thepropagation of measurement errors to reconstruction errors.

While stability is guaranteed for extended P´functions, standard algorithms such as those basedon Newton or Gauss-Seidel methods are not guaranteed to converge to the unique (fixed point)solution. We propose in section 3 an algorithm, taking the form of a damped Newton method, thatis guaranteed to converge to the fixed point, first with linear rate of convergence, and then withquadratic rate of convergence using a standard Newton method when the discrete dynamics aresufficiently close to the fixed point. The algorithm was shown to converge for (sufficiently) smoothextensions of the original problem. The algorithm also most likely converges (with h sufficientlysmall) for the piecewise-smooth Mas-Colell extension (5) but we do not have a complete proof inthat case.

However, we showed that choosing sufficiently small values of the damping parameter h wasnecessary to obtain a convergent algorithm. We presented examples of non-convergent discretecyclic trajectories for smooth P´functions with Jacobians with positive entries (as is the case inME-CT applications) when h “ 1 (standard Newton algorithm) as well as h close to 1.

We finally considered ME-CT inversions in section 4. We focused on the first step, namely thereconstruction of line integrals of absorption densities from ME-CT measurements. The second step,reconstructing spatially varying functions from their line integrals amounts to a standard inverseRadon transform procedure, which is not presented here. We considered two- (n “ 2) and three-(n “ 3) dimensional settings. We showed that the errors in the reconstructions strongly dependedon the choice of energy measurements (e.g., tube potentials). While not perfect, we also showedthat the selection of tube potentials based on the stability estimates of section 2, as opposed to thenumerical evaluation of the inverse Jacobian, also proved reasonable.

Numerical reconstructions of two-material line integrals (sinograms) from DE-CT measurementsusing the proposed damped Newton algorithm confirmed the theoretical predictions. Reconstruc-tions were shown to be significantly more stable for optimized choices of the tube potentials. More-over, for less stable tube potentials, we observed that using the extended map significantly improvedthe reconstructions, with the damped Newton algorithm failing to converge in unstable cases andin the presence of sufficiently large noise.

Acknowledgment

The authors thank Emil Sidky for useful discussions and references. This research was partiallysupported by the National Science Foundation, Grants DMS-1908736 and EFMA-1641100 and bythe Office of Naval Research, Grant N00014-17-1-2096.

Appendix A.

In this section, we provide two alternatives to the constant τ given in (8).

A.1. An alternative estimate. If A is a P´matrix, then the set of positive eigenvalues of allprincipal submatrices of A, denoted by ΛA, contains at least the diagonal entries of A, and hence itis nonempty. The constant

µA :“ minpΛAq,

can be seen as a characteristic quantity for P´matrices.


Proposition A.1 ( [14]). If A is a P´matrix, then A´ λI is a P´matrix for all 0 ď λ ă µ “ µA,and A´ µI is a P0´matrix.

The constant µ is particularly useful in obtaining a lower bound for the determinant of aP´matrix.

Proposition A.2. Let A be a n ˆ n P´matrix with k, 0 ď k ď n, real eigenvalues and µ “ µA.Then,

detA ě´

sinπ

n

¯n´kµn.(39)

Proof. Since A is a P´matrix, any real, hence positive, eigenvalue of A is bounded below by µ.Also, the eigenvalues of A are given by µ` λ with λ being an eigenvalue of B “ A´ µI which is aP0´matrix. By Kellogg’s theorem [24], | arg λ| ď πp1 ´ 1

nq, and thus |µ ` λ| ě µ sinπ

n. Since the

determinant of a matrix is equal to the product of its eigenvalues, we obtain (39).

Proposition A.3. Let Jε be given as in (12). Then, for all ε ě 0 and x P Rn, we have

µJεpxq ě µJpPεpxqq,

with equality if ε “ 0. Thus, in view of proposition A.2, we have

det Jεpxq ě´

sinπ

n

¯n´1µnJpPεpxqq

,(40)

and

~Jεpxq´1~ ď

pn´ 1qn´12 Ln´1ε

`

sin πn

˘n´1µnJpPεpxqq

.(41)

Proof. Observe that Jεpxq ´ λI “ pJpPεpxqq ´ λIqDPεpxq ` pLε ´ λqpI´DPεpxqq. Then, using theproperties of the determinant, we obtain

detpJεpxq ´ λIq “ÿ

KĂxny

cKpxqpLε ´ λq|K|rJpPεpxqq ´ λIsK ,(42)

where

cKpxq “´

ź

kPK

p1´ p1εpxkqq¯´

ź

kPxnyzK

p1εpxkq¯

.

We note that, since 0 ď p1ε ď 1, we have 0 ď cKpxq ď 1 for all x P Rn and K Ă xny. Moreover, forall x P Rn, there is K Ă xny such that cKpxq ą 0.

Now let x P Rn be arbitrary, and suppose that λ is an eigenvalue of Jεpxq. If λ ă µJpPεpxqq, thenrJpPεpxqq ´ λIsK ą 0 for all K Ă xny by proposition A.1. Also, since µJpPεpxqq ď JpPεpxqqii ď Lε,we obtain that the right hand side of (42) is positive while the left hand side is zero, which is acontradiction. Hence, we must have λ ě µJpPεpxqq. Applying the same argument to the eigenvaluesof the principal submatrices of Jεpxq, we obtain that µJεpxq ě µJpPεpxqq.

For ε “ 0, we observe that cK “ 0 for all but one K 1 Ă xny, for which cK1 “ 1. This impliesthat the eigenvalues of submatrices of Jεpxq are either equal Lε or coincide with an eigenvalue of asubmatrix of JpPεpxqq. Thus, µJpxq “ µJpP pxqq.

Then, the estimate (40) follows from proposition A.2. Finally, proceeding as in the proof oftheorem 2.4 and using (40) in estimating the determinant of Jε, we obtain (41).


Definition A.4. Let F be a continuously differentiable map on a closed rectangle R with a P´matrixJacobian Jpxq at every x P R. The quantity

µF :“ minxPR

µJpxq,(43)

is called the injectivity constant of F .

Theorem A.5. Let F be given as in (5). Then, for all x, y P Rn,

F pxq ´ F pyq8 ě γx´ y8,(44)

where

γ :“

`

sin πn

˘n´1

n12 pn´ 1q

n´12

µnFLn´1

.(45)

Proof. Proceeding as in the proof of theorem 2.4, using (41) in estimating the determinant of~Jεpxq

´1~, and finally letting εÑ 0, we obtain (44).

The comparison of the estimates (7) and (44) is given in the following proposition.

Proposition A.6. Let A be a nˆ n P´matrix. Then, for any K Ă xny, we have

~A~|K|rAsK ě´

sinπ

n

¯n´1µnA,(46)

and thus τ ě γ where τ and γ are given in (8) and (45), respectively.

Proof. Let K Ă xny be arbitrary. By definition of µA, we have ~A~ ě aii ě µA for all i P xny,which implies the result if K “ xny. For otherwise, we apply proposition A.2 to AK to obtain

rAsK ě

ˆ

sinπ

n´ |K|

˙n´|K|´1

µn´|K|AK

ě

´

sinπ

n

¯n´1µn´|K|A .

Therefore,

~A~|K|rAsK ě µ|K|A

´

sinπ

n

¯n´1µn´|K|A “

´

sinπ

n

¯n´1µnA,

which can be used to obtain the inequality τ ě γ.

A.2. An estimate without using an extension. In this section, we derive an estimate that doesnot require any extension. We start with the following geometric property of P´matrices.

Theorem A.7 ( [15, 16]). An nˆ n matrix A is a P´matrix if and only if A reverses the sign ofno vector except zero, that is for every nonzero vector v P Rn, there is an index i P xny such thatvipAvqi ą 0.

Theorem A.7 was used in [25] to obtain another characteristic quantity for P´matrices. Evidently,A is a P´matrix if and only if

αA :“ minv8“1

maxiPxny

vipAvqi ą 0.(47)

Consequently, for all v P Rn (see also [17, Lemma 3.12]),

maxiPxny

vipAvqi ě αAv28.(48)

We note that αA ď µA. Indeed, since A ´ µAI is no longer a P´matrix, we must have 0 ěαpA´µAIq ě αA ´ µA. We can now obtain the following quantitative estimate of injectivity.


Theorem A.8. Let R Ă Rn be a closed rectangle. Suppose that F : R Ñ Rn is a continuouslydifferentiable map with a P´matrix Jacobian Jpzq at every z P R. We define

α :“ minzPR

αJpzq.(49)

Then, for al x, y P R,

F pxq ´ F pyq8 ě αx´ y8.(50)

Proof. Let x, y P R. If x “ y, we are done, so we assume that x ‰ y. For each i P xny, we define

gi : r0, 1s Ñ R, giptq “ Fiptx` p1´ tqyq.

Then, by the Mean Value Theorem, there exists ti P p0, 1q such that gip1q´gip0q “ g1iptiq. Observingthat gip1q “ Fipxq, gip0q “ Fipyq, and g1iptq “

řnj“1

BFiBxjpptx` p1´ tqyqpxj ´ yjq, we obtain

Fipxq ´ Fipyq “ pJpziqpx´ yqqi,(51)

where zi “ tix` p1´ tiqy for some ti P p0, 1q.Since the Jacobian Jpzq is a P´matrix at every z P R, we have

maxiPxny

vipJpzqvqi ě αJpzqv28 ě αv28,(52)

for all v P Rn. Thus, we obtain

x´ y8F pxq ´ F pyq8 ě maxiPxny

pxi ´ yiqpFipxq ´ Fipyqq

“ maxiPxny

pxi ´ yiqpJpziqpx´ yqqi ě αx´ y28.

Finally, since x ‰ y, we can divide both sides by x´ y8 and obtain (50).

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Departments of Statistics and Mathematics and CCAM, University of Chicago, Chicago, IL 60637Email address: [email protected]

Department of Engineering Science and Applied Mathematics, Northwestern University, Evanston,IL 60201

Email address: [email protected]

Department of Mathematics, North Carolina State University, Raleigh, NC 27695Email address, Corresponding author: [email protected]

arXiv:2110.01405v1 [physics.med-ph] 29 Sep 2021

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