SIMULTANEOUS RECONSTRUCTION AND SEPARATION IN A SPECTRAL CT FRAMEWORK

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SIMULTANEOUS RECONSTRUCTION AND SEPARATION IN A SPECTRAL CT FRAMEWORK

S. Tairi, S. Anthoine, C. Morel, Y Boursier

To cite this version:

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S

IMULTANEOUS RECONSTRUCTION AND SEPARATION

IN A SPECTRAL CT FRAMEWORK

S. T

AIRI

1

, S. A

NTHOINE

2

, C. M

OREL

1

AND

Y. B

OURSIER

1

AIX MARSEILLE UNIVERSITY, CNRS/IN2P3, CPPM UMR 7346, 13288, MARSEILLE, FRANCE

2

AIX MARSEILLE UNIVERSITY, CNRS, CENTRALE MARSEILLE, I2M UMR 7373, 13453, MARSEILLE, FRANCE

I

NTRODUCTION

The advent of new X-ray detection technology by hybrid pixel cameras working in a photon-counting mode paves the way to the development

of spectral CT.

This allows to simultaneously separate and recon-struct the physical components of an object and finds applications in many areas of imaging.

Our framework allows to iteratively reconstruct an image from a spectral CT data by setting a

poly-chromatic model that encompasses constraints

(positivity, sparsity) and solving an ill-posed

in-verse and non-convex problem.

Preliminary results are obtained on simulated im-ages containing four elements (water, Iodine,

Yt-trium and Silver).

Figure 1: From acquisition to resolution.

F

ORWARD MODEL

Acquisition

A Computerized-Tomography (CT) scan is ob-tained by shining a X-Ray light modulated by metallic filters through a rotating object. In the spectral setting, one explicitly exploits the spec-tral polychromaticity of the source attenuated

through different metallic filters.

The Beer-Lambert law governs the measurements: ym =

Z

R+

fm(E)e− RLp µ(l,E)dldE (1)

where f m(E) = I₀(E)F iq(E)Dr(E) _{denotes the} total spectral inputs:

• µ(l, E)_: _{absorption coefficients of the object}

for l on the line of sight Lp (to be found),

• I₀_{: X-ray source’s intensity energy spectrum,} • F i: filter’s attenuation energy spectrum,

• D: detector’s efficiency energy spectrum.

Absorption maps model

The absorption maps are naturally the sums of the contributions of each of their components; more-over the spectral signature is physically

indepen-dent of the spatial location of a component:

µ(l, E) =

K

X

k=1

ak(l)σk(E), ₍₂₎

with ak(l) _{the concentration of component k at} point l, and σk(E) _{its interaction cross section.} Eq. (1) now reads:

ym = Z R+ fm(E)e− PKk=1 σ k (E) R_Lm ak(l)dl_dE (3)

Discretized forward model

The energy E is discretized in N bins, and the 3D-volume where the object lives in D voxels. The forward discretized model reads:

Y = (F e−SAΣ)1_N , ₍₄₎

• Y ∈ RM : discretized measured data;

• F ∈ RM ×N : dictionary of energy modulat-ing filters;

• S ∈ RM ×D: X-ray Transform operator;

• A ∈ RD×K: concentration matrix: A[d, k] = a_k(v_d)_;

• Σ = (σ₁(.), . . . , σ_K(.))T ∈ RK×N _: dictio-nary of the interaction cross sections of the K _{components: Σ[k, n] = σ}_k(E_n)_.

(1N = (1, ..., 1)T and is the Hadamard product.)

M

ETHOD

The measurements Y are noisy realizations of the perfect measurement described by Eq. (4). The inverse problem of recovering A, the matrix con-taining the concentration coefficient maps of the K _{components of the object, is solved by} minimiz-ing:

J (A) = D(Y, (F e−SAΣ)1_N ) + R(A) ₍₅₎

with

• D(Y, Z) _{a discrepancy measure (negative}

log-likelihood for Gaussian or Poisson noise),

• R(A) _{a regularization term that models our}

a priori on A.

With R(A) the non-negativity constraint, Eq. (5) is minimized with a classical trust-region algorithm

Trust-Region Algorithm Require: A0, ∆0 for k = 1, 2, 3.. do m_k(p) = J (A_k) + g_kT p + 1₂ pT B_kp . Quadratic model. p_k ← ArgM in kpk<∆k m_k(p) . Step calculation. R_K ← f (Xk)−f(Xk+pk)

mk(0)−mk(pk) . Ratio actual/predicted reduction.

if Rkacceptable then A_k+1 ← A_k + p_k U pdate∆_k+1 else Reduce∆_k+1 end if end for return Ak

R

ESULTS

We have generated a contrast phantom made of one large cylinder filled with water and six smaller tubes filled with contrast agents. Three tubes tain Yttrium at different concentrations, two con-tain Silver and one concon-tains Iodine (K = 4).

We have simulated a set of tomographic scans

smeared by Gaussian noise with 5 different

metallic filters F i_{, which are ideal pass-band}

fil-ters around the discontinuities specifying the con-trast agents. The discretized sizes are M = 1440, D = 256_{, and N = 43.}

We have then minimized Eq. (5) with the non-negativity constraint using a trust-region algo-rithm and report the results obtained for 10 realizations of noise. The figures show that the low-rank model allows to reconstruct sim-ple maps from non-linear polychromatic measure-ments. The output SNR evolves linearly with the input SNR.

C

ONCLUSION

We have established a new flexible model for spec-tral CT reconstruction. Results obtained with a classical Trust-Region approach on simulated data