Francesco Amoroso

Martín Sombra
email: sombra@ub.edu

Factorization of bivariate sparse polynomials

Keywords: 2010 Mathematics Subject Classification, Primary 13P05; Secondary 12Y05, sparse polynomials, toric Bertini's theorem

Introduction

A polynomial is said to be sparse (or lacunary) if it has few terms compared with its degree. The factorization problem for sparse polynomials can be vaguely stated as the question of whether the irreducible factors of a sparse polynomial are, apart from obvious exceptions, also sparse. Aspects of this problem have been studied in various settings and for different formalizations of the notion of sparsenness, see for instance [Len99, Sch00, KK06, AKS07, FGS08, Gre16, ASZ17]. Many times, these studies were based on tools from Diophantine geometry like lower bounds for the height of points and subvarieties, and unlikely intersections of subvarieties and subgroups of a torus.

In this text, we consider families of bivariate Laurent polynomials given as the pullback of a fixed regular function on a torus by a varying 2-parameter monomial map. Precisely, let t, z be variables, x = (x 1 , . . . , x n ) a group of other n variables, and

F ∈ C[x ±1 , z ±1 ] = C[x ±1
1 , . . . , x ±1 n , z ±1 ] a Laurent polynomial. For each vector a = (a 1 , . . . , a n ) ∈ Z n , we consider the bivariate Laurent polynomial given as the pullback of F by the monomial map (t, z) → (t a , z) = (t a 1 , . . . , t an , z), that is (1.1)

F a = F (t a , z) ∈ C[t ±1 , z ±1 ].
The number of coefficients of each F a is bounded by those of F , and so these Laurent polynomials can be considered as sparse when F is fixed and a is large.

The following is our main result.

Theorem 1.1. Let F ∈ C[x ±1 , z ±1 ].
There is a finite set of matrices Ω ⊂ Z n×n satisfying the following property. For each a ∈ Z n , there are M ∈ Ω and b ∈ Z n with a = M b such that if P is an irreducible factor of F (x M , z), then P (t b , z) is, as an element of C(t)[z ±1 ], either a unit or an irreducible factor of F (t a , z).

This result shows that for each a ∈ Z n , there is a matrix M within the finite set Ω ⊂ Z n×n and a vector b ∈ Z n with a = M b such that, unless F x M , z = 0, the irreducible factorization (1.2) F x M , z = P P (x, z) e P yields the irreducible factorization

F a = γ P P (t b , z) e P ,
in the ring C(t)[z ±1 ] for F a as in (1.1), the product being over the irreducible factors P in (1.2) such that P (t b , z) is not a unit, and with γ ∈ C(t)[z ±1 ] × . Hence, the irreducible factorizations in C(t)[z ±1 ] of the F a 's can be obtained by specializing the irreducible factorizations of the Laurent polynomials F (x M , z) for a finite number of matrices M . These irreducible factors of the F a 's are sparse, in the sense that they are all represented as the pullback of a finite number of regular functions on the (n + 1)-dimensional torus G n+1 m by 2-parameter monomial maps. In particular, both the number of these irreducible factors and of their coefficients is bounded above independently of a.

As explained in Section 2, Theorem 1.1 is a consequence of our function field analogue of a conjecture of Schinzel (Theorem 2.4). The proof of Theorem 2.4 relies on a toric version of Bertini's theorem. The first toric Bertini's theorem was proved by Schinzel [Sch90a, Theorem 1] (see also [START_REF]Un critère d'irréductibilité de polynômes[END_REF]), but only over the rational field and with an additional assumption of non self-reciprocity. Then Zannier [Zan10, Theorem 3] remove both restrictions. Later the result was further precised and generalized by Fuchs, Mantova and Zannier [FMZ17, Theorem 1.5]. Roughly speaking, their result states that, for an irreducible quasiprojective variety W of dimension n ≥ 0 and a dominant map π : W -→ G n m of finite degree e and such that [e] * W is irreducible, the fiber of a generic coset is irreducible.

Here we need a variant of [FMZ17, Theorem 1.5]. We do not ask that [e] * W is irreducible. In this more general situation, the conclusion of that theorem does not necessarily hold. Instead, this conclusion is replaced by an alternative that "explains" the possibility that a fiber is reducible by its factorization through a reducible pullback of the variety W by an isogeny of G n m within a finite set. Theorem 1.2. Let W be an irreducible quasiprojective variety of dimension n and π : W → G n m a dominant map that is finite onto its image. There is a finite union E of proper subtori of G n m and a finite set Λ of isogenies of G n m such that, for a subtorus

T ⊂ G n m and a point p ∈ G n m (C) = (C × ) n , one of the next conditions holds: (1) T ⊆ E; (2) there is λ ∈ Λ with λ * W reducible and a subtorus T ⊂ G n m with λ inducing an isomorphism T → T ; (3) π -1 (p • T ) is irreducible.
We prove this theorem by reducing it to the previous toric Bertini's theorem, through a variation (Proposition 3.8) of a factorization result for rational maps from [START_REF] Zannier | Hilbert irreducibility above algebraic groups[END_REF].

Back to the factorization problem for sparse polynomials, it is natural to consider the more general setting of pullbacks of regular functions on G n m by arbitrary monomial maps, instead of only those appearing in (1.1). Let y = (y 1 , . . . , y n ) and t = (t 1 , . . . , t k ) be groups of n and k variables, respectively. For a Laurent polynomial G ∈ C[y ±1 ], consider the family of k-variate Laurent polynomials given by the pullback of G by the monomial map

G k m → G n m defined by t → t A for a matrix A ∈ Z n×k , that is G A = G(t A ) ∈ C[t ±1 ].
Denote by S the multiplicative subset of C[t ±1 ] generated by the Laurent polynomials of the form

f (t d ) for f ∈ C[z ±1 ] and d ∈ Z k .
We propose the following conjecture which, as explained in Remark 4.2, partially generalizes Theorem 1.1.

Conjecture 1.3. Let G ∈ C[y ±1
] and k ≥ 2. There is a finite set of matrices Ω ⊂ Z n×n satisfying the following property. For each A ∈ Z n×k , there are N ∈ Ω and B ∈ Z n×k with A = N B such that if P is an irreducible factor of G(y N ), then P (t B ) is, as an element of C[t ±1 ] S , either a unit or an irreducible factor of G(t A ).

The validity of this conjecture would imply that the irreducible factors of the G A 's that truly depend on more than one variable, are also the pullback of a finite number of regular functions on G n m by k-parameter monomial maps. The possible univariate irreducible factors of the G A 's split completely, and so they cannot be accounted from a finite number of such regular functions.

A conjecture of Schinzel and its function field analogue

In [START_REF] Schinzel | On the reducibility of polynomials and in particular of trinomials[END_REF], Schinzel proposed the conjecture below on the factorization of univariate polynomials over Q.

An element of Q[x ±1 ] is cyclotomic if it is an irreducible factor of a binomial of the form x a -1 for a nonzero vector a ∈ Z n . The cyclotomic part of a Laurent polynomial F ∈ Q[x ±1 ] \ {0}, denoted by cyc(F ), is defined as a maximal factor of F that is a product of cyclotomic Laurent polynomials. This cyclotomic part is well-defined up to a unit of Q

[x ±1 ].
For a, b ∈ Z n , we denote by a, b = n i=1 a i b i their scalar product. Conjecture 2.1. Let F ∈ Q[x ±1 ] be a non-cyclotomic irreducible Laurent polynomial. There are finite sets Ω 0 ⊂ Z n×n of nonsingular matrices and Γ ⊂ Z n of nonzero vectors satisfying the following property. Let a ∈ Z n ; then one of the next conditions holds:

(1) there is c ∈ Γ verifying c, a = 0;

(2) there are M ∈ Ω 0 and b ∈ Z n with a = M b such that if

F (x M ) = P P e P
is the irreducible factorization of F (x M ), then

F (t a ) cyc(F (t a ) = P P (t b ) cyc(P (t b )) e P
is the irreducible factorization of F (t a )/cyc(F (t a )).

For the validity of this statement, in its condition (2) it is necessary to take out the cyclotomic part of F (t a ) and of the P (t b )'s, as shown by the example below.

Example 2.2.

Set F = x 1 +x 2 -2 ∈ Q[x ±1 1 , x ±1 2 ]
. Let a ∈ Z 2 and choose a nonsingular matrix M ∈ Z 2×2 and a vector b ∈ Z 2 with a = M b. We have that

F (x M ) = x m 1,1 1 x m 1,2 2 + x m 2,1 1 x m 2,2 2 -2
is irreducible, and so P := F (x M ) is the only irreducible factor of this Laurent polynomial. However, t -1 divides F (t a ) = P (t b ), and so these univariate Laurent polynomials are not irreducible, unless we divide them by this cyclotomic factor.

Schinzel proved this conjecture when n = 1 in loc. cit. and, under a restrictive hypothesis (non-symmetry), when n ≥ 2 [START_REF]Reducibility of lacunary polynomials. I[END_REF], see also [Sch00, §6.2]. The general case when n ≥ 2 remains open. In this paper, we prove a function field analogue for Laurent polynomials over the field C(z), from which we deduce Theorem1.1.

An element of

C(z)[x ±1 ] is constant if it lies in C[x ±1
], up to a scalar factor in C(z) × . The constant part of a Laurent polynomial F ∈ C(z)[x ±1 ] \ {0}, denoted by ct(F ), is defined as its maximal constant factor. This constant part is well-defined up to a unit of C(z)[x ±1 ].

Remark 2.3. The analogy between cyclotomic Laurent polynomials over Q and irreducible constant Laurent polynomials over C(z) stems from height theory. Let K denote either Q or C(z), and h the canonical height function on subvarieties of the torus G n m,K , induced by the standard inclusion

G n m,K → P n K . Let F ∈ K[x ±1 ] be an irreducible Laurent polynomial defining a hypersurface V (F ) of G n m .
Then the condition that h(V (F )) = 0 is equivalent to the fact that F is cyclotomic when K = Q, and to the fact that F is constant when

K = C(z). Theorem 2.4. Let F ∈ C(z)[x ±1
] be a non-constant irreducible Laurent polynomial. There are finite sets Ω 0 ⊂ Z n×n of nonsingular matrices and Γ ⊂ Z n of nonzero vectors satisfying the following property. Let a ∈ Z n ; then one of the next conditions holds:

(1) there is c ∈ Γ verifying c, a = 0;

(2) there are M ∈ Ω 0 and b ∈ Z n with a = M b such that if

F (x M ) = P P e P
is the irreducible factorization of F (x M ), then

F (t a ) ct(F (t a )) = P P (t b ) ct(P (t b )) e P
is the irreducible factorization of F (t a )/ct(F (t a )).

Similarly as for Conjecture 2.1, for the validity this statement it is is necessary to take out in its condition (2) the constant part of F (t a ) and of the P (t b )'s.

Example 2.5.

Set F = x 1 + zx 2 -z -1 ∈ C(z)[x ±1 1 , x ±1 2 ]. Let a ∈ Z 2 and choose M ∈ Z 2×2 nonsingular and b ∈ Z 2 with a = M b. We have that F (x M ) = x m 1,1 1 x m 1,2 2 + z x m 2,1 1 x m 2,2 2 -z -1
is irreducible, and so P := F (x M ) is its only irreducible factor. Again, t -1 divides F (t a ) = P (t b ), and so these univariate Laurent polynomials are not irreducible, unless we divide them by a suitable constant factor.

The proof of Theorem 2.4 relies on the toric Bertini's theorem 1.2 or, more precisely, on its polynomial version in Theorem 3.15. In Section 3 we prove these latter results, while in Section 4 we explain how to deduce Theorem 2.4 from Theorem 3.15, and how to deduce Theorem 1.1 from Theorem 2.4.

A variant of Zannier's toric Bertini's theorem

Building on [START_REF] Dvornicich | Cyclotomic Diophantine problems (Hilbert irreducibility and invariant sets for polynomial maps)[END_REF], Zannier proved in [START_REF] Zannier | Hilbert irreducibility above algebraic groups[END_REF] a toric version of Hilbert irreducibility theorem and then deduce from it an analogue of Bertini's theorem for covers, where the subtori of G n m replaced the linear subspaces in the classical version of this theorem. This result was precised and generalized (with a completely different proof) by Fuchs, Mantova and Zannier to include fibers of arbitrary cosets of subtori [FMZ17, Theorem 1.5] and to obtain a more uniform result.

As before, let x = (x 1 , . . . , x n ) be a group of n variables and denote by

G n m = Spec(C[x ±1 ]) the n-dimensional torus over C.
Let W be a variety, that is, a reduced separated scheme of finite type over C. We assume that W is irreducible and quasiprojective of dimension n ≥ 0, and equipped with a dominant map

π : W -→ G n m
of degree e ≥ 1 that is finite onto its image. Given an isogeny λ of G n m , that is, an endomorphism of G n m with finite kernel, we denote by λ * W the fibered product G n m × λ,π W , and by

(3.1) λ * W λ / / π W π G n m λ / / G n m
the corresponding fibered product square. The aforementioned result by Fuchs, Mantova and Zannier can be stated as follows.

Theorem 3.2. Let W be an irreducible quasiprojective variety of dimension n and π : W → G n m a dominant map that is finite onto its image and that satisfies the property PB. There is a finite union E of proper subtori of G n m such that, for every subtorus

T ⊂ G n m not contained in E and every point p ∈ G n m (C) = (C × ) n , we have that π -1 (p • T ) is an irreducible subvariety of W .
When the property PB is not verified, the conclusion of this theorem does not necessarily hold, as already pointed out in [START_REF] Zannier | Hilbert irreducibility above algebraic groups[END_REF] since the map π factors through a nontrivial isogeny.

Example 3.3. Let F = z 2 -x 1 x 2 2 ∈ C[x ±1 1 , x ±1 2 , z ±1 ], set W = V (F ) ⊂ G 3 m and consider the map π : W -→ G 2 m defined by π(x 1 , x 2 , z) = (x 1 , x 2 ) for (x 1 , x 2 , z) ∈ W .
Since F is irreducible, the variety W is irreducible and, since F is monic in z, the map π is finite. However, it does not satisfy the property PB, since for the isogeny λ of G 2 m defined by λ(

x 1 , x 2 ) = (x 2 1 , x 2 ), λ * W V (z -x 1 x 2 ) ∪ V (z + x 1 x 2 )
and so this pullback is reducible. Indeed, this map does neither satisfy the conclusion of Theorem 3.2: given (a 1 , a 2 ) ∈ Z 2 \{(0, 0)} with a 1 even, let T ⊂ G 2 m be the 1-dimensional subtorus given as the image of the map t → (t a 1 , t a 2 ). Then

π -1 (T ) = V (z -t a 1 /2 t a 2 ) ∪ V (z + t a 1 /2 t a 2 ),
and so this fiber is not irreducible. Theorem 1.2 is a variant of this result which holds when the property PB is not necessarily verified. Instead, the conclusion of Theorem 3.2 is replaced by an alternative that "explains" the possibility that a fiber is reducible by a factorization of this fiber through a reducible pullback of the variety W by an isogeny of G n m in a certain finite set. We repeat the statement of Theorem 1.2, for the convenience of the reader.

Theorem 3.4. Let W be an irreducible quasiprojective variety of dimension n and π : W → G n m a dominant map that is finite onto its image. There is a finite union E of proper subtori of G n m and a finite set Λ of isogenies of G n m such that, for a subtorus

T ⊂ G n m and a point p ∈ G n m (C) = (C × ) n , one of the next conditions holds: (1) T ⊆ E; (2) there is λ ∈ Λ with λ * W reducible and a subtorus T ⊂ G n m with λ inducing an isomorphism T → T ; (3) π -1 (p • T ) is irreducible. Remark 3.5. When the condition (2) above is satisfied, there is a diagram π -1 (T ) / / λ * W λ / / π W π T ι / / G n m λ / / G n m
with λ * W reducible and λ : T → T an isomorphism, and where ι denotes the inclusion of the subtorus T into G n m . Both inner squares in this diagram are fibered products, and so is the outer square. This implies that the fibers π -1 (T ) and π -1 (T ) are isomorphic. Thus π -1 (T ) can be identified with the fiber of a subtorus for the reducible cover π : λ * W → G n m , and so this fiber is expected to be reducible as well.

Example 3.6. We keep the notation from Example 3.3. In particular,

F = z 2 - x 1 x 2 2 ∈ C[x ±1 1 , x ±1 2 , z ±1 ], W = V (F ) ⊂ G 3 m , and π : W → G 2 m the map defined by π(x 1 , x 2 , z) = (x 1 , x 2 ).
Let (a 1 , a 2 ) ∈ Z 2 \ {(0, 0)} with a 1 even, and set T ⊂ G 2 m for the 1-dimensional subtorus given as the image of the map t → (t a 1 , t a 2 ). These vectors satisfy the condition (2) in Theorem 3.4 for the isogeny λ :

G 2 m → G 2 m defined by λ(x 1 , x 2 ) = (x 2
1 , x 2 ). Indeed, λ * W is reducible, and this isogeny induces an isomorphism T → T with the subtorus T ⊂ G 2 m given as the image of the map t → (t a 1 /2 , t a 2 ). We give the proof of this theorem after some auxiliary results. We first study the reducibility of pullbacks of varieties with respect to isogenies of tori.

Lemma 3.7. Let π : W → X be a map of varieties and λ : X → X an étale map. Then X × λ,π W is a variety.

In particular, for a map π : W → G n m and an isogeny λ of G n m , we have that λ * W is a variety. Proof. Since λ : X → X is étale, the map .4], this implies that, for every closed point q ∈ X × λ,π W and p := λ(q) ∈ W , the induced map of completed local rings

(3.2) λ : X × λ,π W -→ W is also étale, because
(3.3) O p -→ O q
is an isomorphism. Since W is a variety, the local ring O p is reduced and, by a theorem of Chevalley [ZS75, §8.13], the completion O p is reduced too. By the isomorphism in (3.3), the completed ring O q is reduced. Since this is the completion of a ring with respect to a maximal ideal, the map O q → O q is injective, and so the local ring O q is also reduced. Since the condition of being reduced is local, this implies that X × λ,π W is a variety.

The last statement comes from the fact that the isogenies of algebraic groups over C are étale maps.

Thanks to this result, λ * W can be identified with its underlying algebraic subset in the Cartesian product

G n m (C) × W (C), namely (3.4) λ * W = {(p, w) ∈ G n m (C) × W (C) | λ(p) = π(w)}.
Hence, λ * W is irreducible if and only if this algebraic subset is irreducible. In particular, the map π satisfies the property PB if and only if for every isogeny λ of G n m , the pullback λ * W has a single irreducible component.

The following proposition is implicit in the proof of [Zan10, Proposition 2.1].

Proposition 3.8. Let π : W → G n m be a map from an irreducible variety W , and λ an isogeny of G n m . The following conditions are equivalent: (1) the pullback λ * W is reducible;

(2) there is a factorization λ = µ • τ with µ, τ isogenies of G n m such that µ is not an isomorphism, and a map ρ : W → G n m such that π = µ • ρ. In other terms, the condition (2) in the proposition above amounts to the existence of the commutative diagram extending (3.1) of the form

λ * W λ / / π W π ρ n n G n m λ / / τ " " G n m G n m µ = =
Proof. Suppose that the condition (2) holds. In this case, for p ∈ G n m (C) and w ∈ W (C), the fact that λ(p) = π(w) is equivalent to µ(τ (p)) = µ(ρ(w)), and so this holds if and only if there is ζ ∈ ker(µ) with τ (p) = ζ • ρ(w). From (3.4), the pullback decomposes into disjoints subvarieties as

λ * W = ζ∈ker(µ) G n m × τ,ζ•ρ W.
Since µ is not an isomorphism, this pullback is reducible, giving the condition (1).

Conversely, suppose that the condition (1) holds. Then λ * W has a decomposition into irreducible components

λ * W = k i=1 U i with k ≥ 2.
Similarly as in (3.2), the map λ * W → W is étale, and so the U i 's are disjoint. Since λ is an isogeny, the map λ * W → W is also finite.

The finite subgroup ker(λ) of G n m (C) acts on λ * W via the maps (p, w) → (ζ • p, w) for ζ ∈ ker(λ), and this action respects the fibers of λ. The action is transitive on the fibers, and so it is also transitive on the U i 's.

Let H ⊂ ker(λ) be the stabilizer of the irreducible component U 1 , and U 1 /H the quotient variety. We have that H acts on U 1 transitively on the fibers and without fixed points. The induced map U 1 /H -→ W is a finite étale map of degree 1, and so it is an isomorphism [Mum88, §III.10, Proposition 2].

Then we define the map ρ : W → G n m as the map obtained from the quotient map

U 1 /H → G n m /H and the identifications U 1 /H W and G n m /H G n m . In concrete terms and identifying G n m /H G n m , this map is defined, for w ∈ W , as ρ(w) = τ (p • H) for any p ∈ G n m such that (p, w) ∈ U 1 . Both G n
m /H and G n m / ker(λ) are isomorphic to G n m , and so there is a factorization λ = µ • τ, with τ and µ corresponding to the projections

G n m → G n m /H and G n m /H → G n m / ker(λ), respectively. For w ∈ W and (p, w) ∈ U 1 , we have that µ • ρ(w) = µ • τ (p) = π(w).
Since the action of ker(λ) on the U i 's is transitive and k ≥ 2, we have that H = ker(λ) and so µ is not an isomorphism, giving the condition (2). Remark 3.9. By this proof, if λ * W is reducible, then the number of its irreducible components is equal to the maximum of the quantity deg(µ) = [ker(λ) : H] over all possible maps ρ as in the condition (2).

The next result allows to factorize the dominant map π : W → G n m as a map satisfying the property PB followed by an isogeny. It is a variant of [Zan10, Proposition 2.1], that states a similar property for dominant rational maps.

Corollary 3.10. Let W be an irreducible variety of dimension n and π : W → G n m a dominant map. There are a map ρ : W → G n m satisfying the property PB and an isogeny λ of G n m with π = λ • ρ. Proof. Choose ρ as a map W → G n m of minimal degree among those that give a factorization of the form π = λ • ρ with λ an isogeny of G n m . Suppose that there is a further isogeny ν such that ν * W = G n m × ν,ρ W is reducible. By Proposition 3.8, there would be an isogeny µ that is not an isomorphism and a map ρ :

W → G n m with ρ = µ • ρ . Hence π = λ • ρ = (λ • µ) • ρ and deg(ρ) = # ker(µ) • deg(ρ ) > deg(ρ ).
By construction, this is not possible. Hence ν * W is irreducible for every isogeny ν of G n m , and so ρ satisfies the property PB.

The next result gives a criterion to detect if the inclusion of a subtorus can be factored through a given isogeny as in Proposition 3.8(2). Lemma 3.11. Let T ⊂ G n m be a subtorus and λ an isogeny of G n m . The following conditions are equivalent:

(1) there is a subtorus T ⊂ G n m such that λ induces an isomorphism T → T ;

(2) λ -1 (T ) is the union of deg(λ) distinct torsion cosets.

Proof. First suppose that λ -1 (T ) is the union of deg(λ) = # ker(λ) distinct torsion cosets, and denote by T the one that contains the neutral element. Then T is a subtorus and T ∩ ker(λ) = {1}. It follows that λ| T : T → T is an isogeny of degree 1 and hence an isomorphism, giving the first condition.

Conversely, let T ⊂ G n m be a subtorus such that λ| T : T → T is an isomorphism. Then λ -1 (T ) = ker(λ) • T .

Since T ∩ ker(λ) = {1}, this fiber is the union of # ker(λ) = deg(λ) distinct torsion cosets, giving the second condition.

Proof of Theorem 3.4. By Corollary 3.10, there are a map ρ : W → G n m satisfying the property PB and an isogeny λ of G n m with π = λ • ρ. For each subgroup H of ker(λ), both G n m /H and G n m / ker(λ) are isomorphic to G n m , and we consider then a factorization

(3.5) λ = µ H • τ H with τ H and µ H corresponding to the projections G n m → G n m /H and G n m /H → G n
m / ker(λ), respectively. We set Λ as the finite set of isogenies of G n m of the form µ H as above, for a proper subgroup H of ker(λ).

Since ρ : W → G n m satisfies the property PB, by [FMZ17, Theorem 1.5] there is a finite union E of proper subtori of G n m such that, for every subtorus T of G n m not contained in E and every point p ∈ G n m (C), the fiber

ρ -1 (p • T ) is irreducible. Set E = λ(E ).
We next show that the pair (Λ, E) satisfies the requirements of Theorem 3.4. Let T be a subtorus of G n m that is not contained in E and write λ -1 (T ) = k i=1 T i as a disjoint union of torsion cosets

T i of G n m . When k = 1, we have that λ -1 (T ) = T 1 is a subtorus of G n m that is not contained in E. Hence, π -1 (T ) = ρ -1 (T 1 ) is irreducible.
Otherwise, k ≥ 2. Let H ⊂ ker(λ) be the stabilizer of the (unique) subtori in this decomposition, say T 1 . This is a proper subgroup, because ker(λ) acts transitively on this collection of torsion cosets and k ≥ 2.

Consider the factorization λ = µ H • τ H as in (3.5). Then µ H ∈ Λ and µ -1 H (T ) splits as an union of k = [ker(λ) : H] = deg(µ H ) distinct torsion cosets. By Lemma 3.11, µ H induces an isomorphism between a subtorus T of G n m and T . Moreover, Proposition 3.8(2) applied to the map τ H • ρ and the isogeny µ H shows that the pullback µ * H W is reducible, completing the proof.

We next prove a polynomial variant of this result. To state it, we first introduce some further notation and auxiliary results.

Let t = (t 1 , . . . , t k ) be a group of k variables. A matrix A = (a i,j ) i,j ∈ Z n×k defines the family of n monomials in the variables t given by

t A = k j=1 t a 1,j j , . . . , k j=1 t a n,j j . The rule t → t A defines a k-parameter monomial map G k m → G n m
. This is a group morphism and indeed, every group morphism from G k m to G n m is of this form. The isogenies of G n m correspond to the nonsingular matrices of Z n×n . Given a = (a 1 , . . . , a n ) ∈ Z n , we can consider it as a row vector, that is, as a matrix in Z 1×n . In this case,

x a = n j=1 x a j j is an n-variate monomial. Row vectors give characters of G n m , that is, group morphisms G n m → G m .
When a is primitive, the kernel of its associated character is a subtorus of G n m of codimension 1, and every such subtorus arises in this way. Else, we can consider a as a column vector, that is, as a matrix in Z n×1 . Then

t a = (t a 1 , . . . , t an )
is a collection of n univariate monomials in a variable t. Column vectors give group morphisms G m → G n m . When a = 0, the image of such a morphims is a subtorus of G n m of dimension 1, that we denote by T a . When a is primitive, the associated group morphism G m → G n m gives an isomorphism between G m and T a . For subvarieties of tori, fibered products like those in (3.1) can be expressed in more concrete terms. The next lemma gives such an expression for the case of hypersurfaces.

Lemma 3.12. Let F ∈ C[x ±1 , z ±1 ], G ∈ C[x ±1 ] \ {0}, and A ∈ Z n×k . Let W be the hypersurface of G n+1 m \ V (G) defined by F , π : W → G n m the map defined by π(x, z) = x, and λ : G k m → G n m the group morphism defined by λ(t) = t A . Then G k m × λ,π W is isomorphic to the subscheme of G k+1 m \ V (G(t A
)) defined by F (t A , z). Proof. The maps π and λ correspond to the morphisms of C-algebras

C[x ±1 ] -→ C[x ±1 , z ±1 ] G /F and C[x ±1 ] -→ C[t ±1 ] C[x ±1 , t ±1 ]/(x -t A ),
and the fibered product G k m × λ,π W is the scheme associated to the tensor product

C[x ±1 , z ±1 ] G /F ⊗ C[x ±1 ] C[x ±1 , t ±1 ]/(x -t A ).
This tensor product is isomorphic to the C-algebra

C[x ±1 , z ±1 , t ±1 ] G /(F, x -t A ) C[z ±1 , t ±1 ] G(t A ) /(F (t A , z)),
which gives the statement.

Lemma 3.13. Let f ∈ C(t)[z] be an irreducible polynomial of degree d ≥ 1, and such that f (t m , z) is reducible for some m ∈ N. There is e ∈ N dividing gcd(m, d) such that f (t e , z) is also reducible.

Proof. The proof relies on the action of torsion points on irreducible factors as in [Zan10, Proposition 2.1]. By Lemma 3.12, the subscheme of G 2 m defined by f (t m , z) is isomorphic to the pullback [m] * V (f ), with [m] the m-th multiplication map of G m . By Lemma 3.7, this pullback is reduced, and so f (t m , z) is separable. Consider its decomposition into distinct irreducible factors

(3.6) f (t m , z) = k i=1 p i , with k ≥ 2.
The group µ m of m-th roots of the unity acts on the set of these irreducible factors by p i (t, z) → p i (ζ • t, z), i = 1, . . . , k, for ζ ∈ µ m . Let P ⊂ {p 1 , . . . , p k } be a nonempty orbit of this action. The polynomial p∈P p is invariant under the action of µ m , and so it is of the form g(t m , z) with g ∈ C(t)[z]. This product is a nontrivial factor of f (t m , z), and so g coincides with f up to a scalar. It follows that P = {p 1 , . . . , p k } and so the action is transitive. In particular, all the p i 's have the same degree in the variable z, and so this degree is positive and k|d.

The stabilizer of an irreducible factor p i is a subgroup of µ m , hence it is of the form µ l with l|m. Since the action is transitive and µ m is abelian, this subgroup does not depend on the choice of p i . Moreover, m/l is equal to k, the number of irreducible factors of f (t m , z), also because of the transitivity of the action.

By the invariance of each p i under the action of µ l , there is

q i ∈ C(t)[z] \ C(t) with p i = q i (t l , z). It follows from (3.6) that f (t e , z) = k i=1 q i (t, z),
with e = m/l. Clearly e|m and as explained, e = k, and so this quantity also divides d, completing the proof.

Lemma 3.14. Let F ∈ C[x, z] be an irreducible polynomial of degree d ≥ 1 in the variable z, and G ∈ C[x] \ {0} its leading coefficient. (1) Let W = V (F ) \ V (G) ⊂ G n+1 m and π : W → G n m the map defined by π(x, z) = x. The image of π is the open subset G n m \ V (G) of G n m ,

and this map is finite onto this open subset.

(2) There is a finite subset ∆ F of Z n such that for a ∈ Z n with c, a = 0 for all c ∈ ∆ F , the polynomial F (t a , z) has degree d in the variable z.

(3) If A ∈ Z n×n is nonsingular, then F (x A , z) has no nontrivial factors in C[x ±1 ] G .
Proof. For the first statement, the image of the map π is contained in the open set

U = G n m \V (G). The induced map W → U corresponds to the morphism of C-algebras C[x ±1 ] G -→ C[x ±1 , z] G /(F ).
This morphism is an integral extension because the leading term G is invertible in C[x ±1 ] G , and so the map W → U is finite and, a fortiori, surjective.

For the second statement, write G = r j=1 G j x c j with G j ∈ C × and c j ∈ N n , j = 1, . . . , r, and consider the finite subset of Z n given by

∆ F = {c j -c 1 | j = 2, . . . , r}.
For a ∈ Z n with c, a = 0 for all c ∈ ∆ F , we have that G(t a ) = 0 and so deg z (F (t a , z)) = d.

As for Lemma 3.13, the proof of the last assertion relies on the action of torsion points on irreducible factors, and so we only sketch it. Using Lemmas 3.12 and 3.7, we show that F (x A , z) is separable. Let

F (x A , z) = k i=1 P i
the decomposition of this Laurent polynomial into distinct irreducible factors. The action of the finite group {x ∈ G n m | x A = 1} on the the sets of these irreducible factors is transitive, and so the P i 's have the same degree with respect to the variable z. Hence for i = 1, . . . , k, we have that k deg z (P i ) = d ≥ 1. In particular, deg z (P i ) ≥ 1, proving the statement.

Theorem 3.15 (Polynomial toric Bertini's theorem). Let F ∈ C[x ±1 , z ±1 ] \ C[x ±1
] be an irreducible Laurent polynomial, and G ∈ C[x ±1 ] the coefficient of the term of highest degree in the variable z. There are finite subsets Φ ⊂ Z n×n of nonsingular matrices and Σ ⊂ Z n of nonzero vectors such that, for a ∈ Z n , one of the next alternatives hold:

(1) there is c ∈ Σ such that c, a = 0;

(2) there is M ∈ Φ such that a ∈ im(M ) and F (x M , z) is reducible;

(3) the Laurent polynomial

F (t a , z) ∈ C[t ±1 , z ±1 ] is irreducible in C[t ±1 , z ±1 ] G(t a ) .
Proof. This statement, restricted to primitive vectors a ∈ Z n , is a specialization of Theorem 3.4. To see this, first reduce, multiplying by a suitable monomial, to the case when

F is an irreducible polynomial in C[x, z] of degree d ≥ 1 in the variable z. Set W = V (F ) \ V (G)
U = G n m \ V (G) of G n m . Let Λ be a finite subset of isogenies of G n
m and E a finite union of proper subtori of G n m satisfying the conclusion of Theorem 3.4 applied to this map. Set then Φ 1 for the finite subset of nonsingular matrices in Z n×n corresponding to the isogenies in Λ, and Σ 1 for a finite subset of nonzero vectors of Z n such that

(3.7) E ⊂ c∈Σ 1 V (x c -1).
For a primitive vector a ∈ Z n , set T a for the 1-dimensional subtorus defined as the image of the group morphism G m → G n m . This map gives an isomorphism between G m and T a . By Lemma 3.12, the fiber π -1 (T a ) is isomorphic to the subscheme of G 2 m \ V (G(t a )) defined by F (t a , z). For the isogeny λ associated to a nonsingular matrix M ∈ Φ 1 , the same result shows that λ * W is isomorphic to the subscheme of G n+1 m \ V (G) defined by F (x M , z).

The three alternatives from Theorem 3.4 applied to the map π, the subtorus T a and the point p = (1, . . . , 1) ∈ G n m (C), then boil down to those in the theorem under examination, as explained below.

(1) Suppose that T a ⊂ E. By (3.7), there is c ∈ Σ 1 with c, a = 0.

(2) Else suppose that there is an isogeny λ ∈ Λ with λ * W reducible and a subtorus T of G n m with λ inducing an isomorphism between T and T a . For M ∈ Φ 1 the nonsingular matrix associated to λ, we have that a ∈ im(B) and, by Lemma 3.12, F (x M , z) is reducible.

(3) Else suppose that π -1 (T a ) is irreducible in G 2 m \ V (G). By Lemma 3.12, this implies that

F (t a , z) is irreducible in C(t)[z ±1 ].
We next enlarge these finite sets to cover the rest of the cases. Let d ≥ 1 be the degree of F in the variable z, and let e be a divisor of d. If F (x e , z) is irreducible, we respectively denote by Φ e and Σ e the finite subsets of nonsingular matrices in Z n×n and of nonzero vectors of Z n given by the application of Theorem 3.4 to this polynomial. Otherwise, we set Φ e = {I n } with I n the identity matrix of Z n×n , and Σ e = ∅. Set also

f (t e , z) = F (t eb , z) is irreducible in C(t)[z]
. By Lemma 3.13, we have that

F a = f (t m , z) is irreducible in C(t)[z],
giving the condition (3) for a and concluding the proof.

Remark 3.16. Using the toric Bertini's theorem 3.4 for cosets of arbitrary dimension, the present polynomial version in Theorem 3.15 might be extended to k-parameter monomial maps for any k, and also to arbitrary translates of these monomial maps.

We have kept the present more restricted statement for the sake of simplicity, and also because it is sufficient for our application.

Factorization of sparse polynomials

Here we prove the results on the factorization of Laurent polynomials announced in the introduction and in Section 2. Theorem 2.4 is easily seen to be implied by the following statement. ]. There are finite sets Ω 0 ⊂ Z n×n of nonsingular matrices and Γ ⊂ Z n of nonzero vectors satisfying the property that, for a ∈ Z n \ {0}, one of the next alternatives holds:

(1) there is c ∈ Γ with c, a = 0;

(2) there are M ∈ Ω 0 and b ∈ Z n with a = M b such that if

F (x M , z) = P P e P is the irreducible factorization of F (x M , z) in C[x ±1 , z ±1 ], then F (t a , z) = P P (t b , z) e P is the irreducible factorization of F (t a , z) in C(t)[z ±1 ].
Proof of Theorem 4.1. We proceed by induction on deg z (F ). When deg z (F ) = 0, the statement is trivial, and so we assume that deg z (F ) ≥ 1.

If F is irreducible, we respectively denote by Φ and Σ the finite sets of nonsingular matrices in Z n×n and of nonzero vectors in Z n from Theorem 4.1 applied to this Laurent polynomial. If F is reducible, we set Φ = {I n } and Σ = ∅.

Let a ∈ Z n . When F is irreducible, if the condition (1) in Theorem 3.15 holds, then the condition (1) in Theorem 4.1 also holds by taking Γ as any finite set containing Σ. Still in the irreducible case, if the condition (3) in Theorem 3.15 holds, the Laurent polynomial F (t a , z) is irreducible, and the condition (1) in Theorem 4.1 holds provided that Ω 0 contains I n .

Else, suppose that the condition (2) in Theorem 3.15 holds, that is, there are M ∈ Φ and b ∈ Z n with a = M b and F (x M , z) is reducible. Let

F (x M , z) = F 1 F 2 be a nontrivial factorization. By Lemma 3.14(3), F (x M , z) has no factors in C[x ±1 ]. Hence deg z (F i ) < deg z (F ), i = 1,
2, and by induction, Theorem 4.1 holds for these Laurent polynomials. Let Ω 0 i and Γ i respectively denote the finite sets of nonsingular matrices in Z n×n and of nonzero vectors in Z n whose existence is assured by this theorem.

By construction, either there is a vector c ∈ Γ 1 ∪ Γ 2 with c, b = 0, or we can find

M i ∈ Ω i and b i ∈ Z n with b = M i b i and a decomposition F i (x M i , z) = k i j=1 F i,j with F i,j (t b i , z) irreducible in C(t)[s ±1 ] for all i, j. Set (4.1) Γ = {adj(M )c | M ∈ Φ, c ∈ Γ 1 ∪ Γ 2 } with adj(M ) the adjoint matrix of M . If c , a = 0 for all c ∈ Γ, then c, b = 0 for all c ∈ Γ 1 ∪ Γ 2 and so F i,j (t b i , z) is irreducible in C(t)[s ±1
] for all i, j.

Consider the lattices K i = im(M i ), i = 1, 2, and set K = K 1 ∩ K 2 . Since K is also a lattice, there is a nonsingular matrix M ∈ Z n×n with K = im(M ) and, since K ⊆ K i , there are nonsingular matrices N

i , i = 1, 2, with M = M i N i . Furthermore, b ∈ K implies that there is b ∈ Z n with b = M b = M i N i b . Hence b i = M -1 i b = N i b and F (x M M , z) = F 1 (x M 1 N 1 , z)F 2 (x M 2 N 2 , z) = 2 i=1 r i j=1 F i,j (x N i , z).
Set M = M M , G i,j = F i,j (x B i , z) and consider the decomposition

F (x M , z) = 2 i=1 r i j=1 G i,j .
We have a = M b = M b and G i,j (t b , z) = F i,j (t B i b , z) = F i,j (t b i , z)

is irreducible in C(t)[s ±1 ] for all i, j. The statement follows by taking Ω 0 as any finite set containing all the matrices of the form M M for M ∈ Φ, and Γ as in (4.1).

We conclude by giving the proof of our main result.

Proof of Theorem 1.1. We proceed by induction on n. When n = 0 the statement is trivial, and so we assume that n ≥ 1. Let F ∈ C[x ±1 , z ±1 ] and write Otherwise, suppose that there is c ∈ Γ ∪ ∆ with c, a = 0. If C(t a , z) = 0, we add to the finite set Ω the matrix M ∈ Z n×n made by adding to n -1 zero columns to the vector a. Otherwise, choose a matrix L ∈ Z n×(n-1) defining a linear map Z n-1 → Z n whose image is the submodule c ⊥ ∩ Z n , and a vector d ∈ Z n-1 with a = Ld. Let u = (u 1 , . . . , u n-1 ) be a group of n -1 variables and set

G = F (u L ) ∈ C[u ±1 , z ±1 ].
By the inductive hypothesis, there is a finite subset Ω c ⊂ Z (n-1)×(n-1) satisfying the statement of Theorem 1.1 applied to this Laurent polynomial. In particular, there are N ∈ Ω c and e ∈ Z n-1 with d = N e such that, for an irreducible factor Q of G(u N , z), we have that Q(t e , z) is, as a Laurent polynomial in C(t)[z ±1 ], either a unit or an irreducible factor of G(t d , z).

We have that G(u N , z) = F (u LN , z), and so Q is an irreducible factor of this latter Laurent polynomial. Moreover, a = LN e. Enlarging the matrix LN ∈ Z n×(n-1) to a matrix M ∈ Z n×n by adding to it a zero column at the end, and similarly enlarging the vector e to a vector b ∈ Z n by adding to it a zero entry at the end, the previous equalities are preserved with M and b in the place of LN and e. Hence, a = M b and, if Q(x 1 , . . . , x n-1 ) is an irreducible factor of F (x M , z), then Q(t e , z) = Q(t b , z) is, as a Laurent polynomial in C(t)[z ±1 ], either a unit or an irreducible factor of G(t d , z) = F (t a , z).

The statement then follows by also also adding to Ω all the matrices M ∈ Z n×n constructed in this way.

Remark 4.2. In the setting of Theorem 1.1, the bivariate Laurent polynomials F a can be defined as the pullback of the multivariate Laurent polynomial F by the 2-parameter monomial map (t, z) → (t, z) A given by the matrix In Conjecture 1.3 applied to F and k = 2, one can consider all matrices in Z (n+1)×2 , and so its setting is more general than that of Theorem 1.1. On the other hand, the conclusion of Conjecture 1.3 in this situation is slightly weaker than that of Theorem 1.1, since it does not give the irreducible factorization of the F a in C(t)[z ±1 ], but rather its irreducible factorization modulo the Laurent polynomials of the form f (t d 1 z d 2 ) for a univariate f and d 1 , d 2 ∈ Z.

Definition 3. 1 .

 1 The map π satisfies the property PB (pullback) if, for every isogeny λ of G n m , we have that λ * W is an irreducible variety. By [Zan10, Proposition 2.1], it is enough to test this condition for λ = [e], the multiplication map of G n m by the integer e = deg(π).

  of the invariance of this property under base change [Har77, Chapter IV, Proposition 10.1(b)]. By [Har77, Chapter IV, Exercise 10

  and consider the mapπ : W -→ G n m defined by π(x, z) = x for (x, z) ∈ W .The quasi-projective variety W is irreducible and, by Lemma 3.14(1), this map is dominant and finite onto its image, the open subset

∆

  for the finite subset of nonzero vectors in Z n associated to F by Lemma 3.14(2). Set then Φ = e|d e Φ e and Σ = ∆ ∪ e|d Σ e . By Theorem 3.4 and the previous analysis, the statement holds for all vectors of the form e b with b ∈ Z n primitive and e|d. Given an arbitrary vector a ∈ Z n , write a = mb with m ∈ N and b ∈ Z n primitive, and set f = F (t b , z) ∈ C[t ±1 , z]. Suppose that neither (1) nor (2) hold for a. Let e ∈ N be a common divisor of d and m. A fortiori, these conditions do neither hold for e b and, as explained before,

Theorem 4. 1 .

 1 Let F ∈ C[x ±1 , z ±1 ] without nontrivial factors in C[x ±1

F

  = CF with C ∈ C[x ±1 ] and F ∈ C[x ±1 , z ±1 ] without nontrivial factors in C[x ±1 ]. By Lemma 3.14(2), there is a finite subset ∆ ⊂ Z n such that C(t b ) = 0 for all b ∈ Z n with c, b = 0 for all c ∈ ∆. Let also Ω 0 ⊂ Z n×n and Γ ∈ Z n be the finite subsets given by Theorem 4.1 applied to F .Let a ∈ Z n . When c, a = 0 for all c ∈ Γ ∪ ∆, Theorem 4.1(2) implies the statement, provided that we choose any finite subset Ω ⊂ Z n×n containing Ω 0 .

Acknowledgments. We thank Pietro Corvaja, Qing Liu, Vincenzo Mantova, Juan Carlos Naranjo and Umberto Zannier for useful conversations. Part of this work was done while the authors met the Universitat de Barcelona and the Université de Caen. We thank these institutions for their hospitality.

Amoroso was partially supported by the CNRS research project PICS 6381 "Diophantine geometry and computer algebra". Sombra was partially supported by the MINECO research project MTM2015-65361-P.


Christine Azevedo Coste

Milos R Popovic

Winfried Mayr

Functional Electrical Stimulation

Guest Editorial Functional Electrical Stimulation

Christine Azevedo Coste1 , Milos Popovic2 , and Winfried Mayr 3In 2016, the International Functional Electrical Stimulation Society (IFESS) celebrated its 20th anniversary. IFESS is promoting research, applications, and understanding of electrical stimulation. This special issue is based on a selection of extended versions of articles that have been presented during the anniversary annual conference in June 2016 in Montpellier/La Grande-Motte, France. They are intended to illustrate the multiple facets of functional electrical stimulation (FES) and recent advances in the domain. Engineers and clinicians demonstrate here the importance and potential of this very active research field. We have also invited Prof. Dejan Popovic and Dr. Thierry Keller to contribute with an article about IFESS history. The clinical applications presented in this issue cover various contexts: post-stroke hemiplegic upper limb therapy (Malesevic et Overall the contributions to this special issue demonstrate how versatile FES can act as a beneficial tool for a variety of clinical applications. We are confident that this collection will be interesting for a wide community of researchers and practitioners and hope it will promote awareness for the numerous beneficial options the use of FES in diagnostic and therapeutic applications can provide.

  al. (1), Irimia et al. (2)), complete spinal cord injuries (SCI) assisted cycling (Fonseca et al. (3), Tefertiller and Gerber (4), Andrews et al. (5)), obesity treatment (Lonys et al. (6)), Parkinson's disease (PD) gait assistance (Sijobert et al. (7)), and cerebral palsy (CP) gait assistance (Rose et al. (8)). In their article, Tefertiller and Gerber present a walking rehabilitation program dedicated to SCI patients. This very comprehensive protocol associates electrical stimulation and ergometry to enhance neurological recovery. Rose et al. discuss the use of multichannel electrical stimulation as a very promising assistive technology to help children with spastic CP achieve a more upright and functional gait. Andrews et al. present their long experience of FES-rowing in SCI patients. The article from Malesevic et al., investigates surface motor activation zones using a multi-pad functional electrical stimulation system to produce selective wrist, finger, and thumb extension movements in therapy sessions involving hemiplegic stroke patients. In their article, Fonseca et al. present an approach for cadence tracking and disturbance rejection in FES cycling in complete paraplegic subjects. Sijobert et al. report a new strategy to assist PD gait through somatosensory cueing based on electrical stimulation of the arch foot in PD gait. Irimia et al., introduce a brain-computer interface to monitor movement imagery to real-time control FES and bar feedback during post stroke hemiplegic patients upper limb training. Some articles of this issue are concerned with invasive approaches of FES. Lonys et al. present an in vivo validation of a less invasive procedure to stimulate the stomach with a gastrointestinal stimulator implant to activate the loss of weight in obese patients.

INRIA, LIRMM, Montpellier, France. E-mail: christine.azevedo@inria.fr

Toronto Rehabilitation Institute -University Health Network, Toronto, Ontario, Canada

Medical University Vienna, Center for Medical Physics and Biomedical Engineering, Vienna, Austria


Yang Chen

Senior Member, IEEE L U Zhang

Wenlong Yuan

Guanyu Yang

Jian Yang

Tianjie Xu

Huazhong Shu

Xuetong Zhai

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 1

Keywords: Model observer, PCJO, detection-localization, CT images, hypersignals, hyposignals. I . denoising, interpolation, compression, enhancement…)

The anthropomorphic model observer (MO) plays an important role in the assessment and optimization of medical imaging systems. The MO is a task-based approach. While the abnormality can appear as a hypersignal or a hyposignal for different imaging modalities, sequences or organs, no MO has been proposed for the hyposignals detection-localization task in the literature. To improve the clinical relevance of the existing MOs, we propose an anthropomorphic MO that can also deal with hyposignals in this paper. In a previous study, we reported a perceptually relevant channelized joint observer (PCJO) for detecting and localizing multiple signals with unknown amplitude, orientation, size and location. Here, we extend it mathematically to hyposignals task. A free-response study (close to the real diagnostic procedure) for both hypersignals and hyposignals in cerebral and abdominal CT images was conducted with four radiologists. The equally weighted alternative free-response operating characteristic (wAFROC1) was used as the figure of merit (FOM). Statistical analyses show that the extended PCJO approaches the experts' performances with no significant difference in the studied tasks. The results demonstrate that the extended PCJO is an alternative to replace radiologists for the evaluation and comparison of different medical image processing algorithms. The PCJO has been originally proposed on Magnetic Resonance Imaging (MRI) but tested on computerized tomography (CT) here; the coherent results show that the PCJO can be generalized to another modality -CT. We also provide in this paper the reference values of all the parameters in the PCJO to facilitate its future application on MR or CT images.

evaluation method is still an open question. For example, the simple numerical metrics, such as the normalized mean-square error (NMSE) and the peak signal-to-noise-ratio (PSNR), were still frequently used [START_REF] Chen | Performance evaluation of functional medical imaging compression via optimal sampling schedule designs and cluster analysis[END_REF] [2] [START_REF] Mok | Interpolated average CT for attenuation correction in PET-A simulation study[END_REF], while it has been widely admitted that they correlate poorly with human perception. The image quality actually has a close relationship with the intended purpose the image serves [START_REF] Jg Brankov | Learning a channelized observer for image quality assessment[END_REF]. Since medical images are used for diagnosis, researchers are prone to define the medical image quality in terms of observer performance for a given diagnostic task. Human is the end-user of medical images, consequently the most correct and the ultimate way of image quality evaluation is to use human observers [START_REF] Lalush | An observer study methodology for evaluating detection of motion abnormalities in gated myocardial perfusion SPECT[END_REF]. However, this method is costly and time-consuming. Nowadays, it has been largely recognized that medical image quality can be objectively assessed by the anthropomorphic model observer (MO) as the surrogate of the human observer [START_REF] Barrett | Foundations of image science[END_REF], and the MO performance should be measured by its capacity of predicting human observer performance on the specific task [START_REF] He | Model observers in medical imaging research[END_REF]. The task should be clinically relevant, such as detecting the presence or absence of an abnormality, locating lesions' positions, etc.

One of the first anthropomorphic MOs may be the nonprewhitening matched filter (NPWMF) [START_REF] Barrett | Model observers for the assessment of image quality[END_REF], considering that the human eyes cannot perform the pre-whitening operation. Then, the channelization mechanism was adopted by MOs to model the frequency and orientation selectivity of the human visual system (HVS), e.g. the channelized ideal observer (CIO) [START_REF] Myers | Addition of a channel mechanism to the ideal-observer model[END_REF] and the channelized Hotelling observer (CHO) [START_REF] Yao | Predicting human performance by a channelized Hotelling observer model[END_REF]. The above anthropomorphic MOs have successfully predicted human performance in signal-known-exactly (SKE) experiments [START_REF] Hernandez-Giron | Comparison between human and model observer performance in low-contrast detection tasks in CT images: application to images reconstructed with filtered back projection and iterative algorithms[END_REF] [START_REF] Elshahaby | Estimating model observer performance with small image ensembles[END_REF], where the entire signal attributes are fixed and a priori known by observers. However, in clinical routine radiologists are uncertain about the abnormality attributes before looking at the images; the experiments thus have to be signal-known-statistically (SKS) to approach the real diagnostic procedure. An SKS MO could detect or localize one or multiple abnormalities without knowledge about their size, shape, amplitude or orientation, etc. A detailed review on the SKS MOs can be found in [START_REF] Zhang | An overview of model observers[END_REF]. Later developments on the SKS MOs include the MO proposed by Georges Acharian et al. that models the sequence of eye fixations that human observers make when exploring an image to detect an object [START_REF] Acharian | A model observer based on human perception to quantify the detectability[END_REF]. Lee et al. replaced the 2D spatial channels in the CHO by a set of 4D space-time channels to develop a 4D MO for the detection of motion defects in 4D gated medical images [START_REF] Ts Lee | Development of 4D mathematical observer models for the task-based evaluation of gated myocardial perfusion SPECT[END_REF]. Gifford et al. continued to improve the visual-search (VS) observer's prediction performance by integrating the eye tracking data or a support-vector machine (SVM) [16] [17]. Diaz et al. extended the CHO and the NPWE to the Filtered Channel Observer (FCO) based on convolution channels [START_REF] Li | Group-sparse representation with dictionary learning for medical image denoising and fusion[END_REF]. The FCO can be implemented as a template with an internal noise component which renders the performance calculation very simple even though it could not match all the human performances with a single internal noise component for all signal shape, size and contrast conditions even for one signal. Note that our previously proposed SKS MO, the perceptually relevant channelized joint observer (PCJO) [START_REF] Zhang | Channelized model observer for the detection and estimation of signals with unknown amplitude, orientation, and size[END_REF], still remains one of the most clinically relevant MOs: (1) it can not only perform the detection task, but also the localization task; (2) it covers a large variety of unknown signal parameters (amplitude/contrast, orientation, position, size and the number of signals); (3) the number of signals per image can be more than one; (4) it has been validated using simulated multiple sclerosis (MS) lesions and real medical background (cerebral MR images) on which there is no significant difference between the task performance of experts and that of the PCJO.

Each medical imaging modality has its own specific features corresponding to the studied physical and physiological phenomena; the pixel or voxel values depend on the chemical and physical characteristics of the studied tissues [START_REF] Cavaro-Ménard | Diagnostic quality assessment of medical images: Challenges and trends[END_REF]. In radiological terminology, an abnormality appears as a hypersignal or a hyposignal, respectively indicating that the abnormality has a higher or lower intensity than its surrounding tissues. For example, the MS lesions appear as hypersignals on a cerebral CT image (as shown by red arrows in Fig. 1 (a)) while the hepatic carcinoma abnormalities appear as hyposignals on an abdominal CT image (refer to Fig. 1 (b)); the traumatic lesions of hematoma appear as hypersignals on a cerebral MR image (see Fig. 2 (a)) while the pancreatic cancer lesions appear as hyposignals on an abdominal MR image (as illustrated in Fig. 2 (b)). As human observer, the radiologist has to be able to interpret both kinds of signals to achieve a precise diagnosis. However, existing MOs are limited to mimic human observers in hypersignal related tasks. To the best of our knowledge, no MO has been proposed or tested for hypo-signals, which is however frequently encountered by radiologists in clinical routine.

In this paper, we try to further improve the clinical relevancy of the SKS MO by incorporating the hyposignal profile into the mathematical model. Our main contributions are: 1) a mathematical derivation based on the PCJO that allows it to detect and localize not only hypersignals but also hyposignals with unknown amplitude, orientation, size and number of signals; 2) an application of the extended PCJO on the cerebral and abdominal CT images, along with a free-response experiment using human observers, not only for the performance evaluation of the extended PCJO with regard to hyposignals, but also for the validation of the PCJO on another imaging modality to verify its generality; 3) the analysis of the reference values of all the parameters in the PCJO to facilitate its future application on the MR or CT images.

The rest of this paper is organized as follows: in section 2, we introduce the mathematical model of background and signal, redefine the signal detection model for hyposignals, and deduce the new SKS MO for hyposignals; section 3 introduces the selected experimental protocol; section 4 shows and discusses the results; section 5 concludes the paper.

II. EXTENDED PCJO FOR HYPOSIGNALS

In this work, we use the same background and signal models as in the previous studies [START_REF] Zhang | Channelized model observer for the detection and estimation of signals with unknown amplitude, orientation, and size[END_REF] and [START_REF] Zhang | A perceptually relevant channelized joint observer (PCJO) for the detection-localization of parametric signals[END_REF] [START_REF] Zhang | Channelized model observer for the detection and estimation of signals with unknown amplitude, orientation, and size[END_REF] on each candidate blocks to calculate its test statistic (related to detection decision). For more details on the PCJO, please refer to [START_REF] Zhang | A perceptually relevant channelized joint observer (PCJO) for the detection-localization of parametric signals[END_REF].

Since the VDP can always provide a detection probability map (between a reference image and a distorted image) even when the abnormality has a lower intensity than its surroundings, we have just to adapt the CJO to the hyposignal. The Candidates Localization part in our extended PCJO remains the same as in [START_REF] Zhang | A perceptually relevant channelized joint observer (PCJO) for the detection-localization of parametric signals[END_REF]. Our extension concerns only the CJO Detection Decision part. The mathematical derivations of the CJO for hypersignals were introduced in [START_REF] Zhang | A perceptually relevant channelized joint observer (PCJO) for the detection-localization of parametric signals[END_REF], in the following we will only give a brief introduction of the CJO for hyposignals.

Since the hyposignal intensity is lower than its surrounding tissues, we can model the image g by subtracting the signal x Λ (Λ is the signal parameters vector) from the background b. The hyposignal detection task can then be seen as the validation of one of two exclusive hypotheses:

: , 0,1 k H k k = - + = Λ g x b (1)
where the binary variable k controls the hyposignal absence (k = 0) or presence (k = 1), and g is an 1 M × vector for an image with M pixels.

Based on the joint detection and estimation theory, the signal parameter Λ and the hypothesis k H are estimated together to maximize the joint posterior probability ( )

k P H Λ, | g : ( , ) argmax ( ) k k k H P H Λ,Η Λ = Λ, |g (2)
Similar to the derivations in [START_REF] Zhang | A perceptually relevant channelized joint observer (PCJO) for the detection-localization of parametric signals[END_REF], considering the statistical independence of Λ and k H , the monotonicity of the logarithmic function, the constancy of ( ) P g and the CGB with b Σ as its covariance matrix, Eq. ( 2) can be substituted by:

1 , 1 ( , ) arg max{ln ( )+ln ( ) ( )} 2 k t k k H PH P k k - Η = - + Λ b Λ Λ Λ Λ x Σ g x (3)
In order to estimate the parameter Λ , we can maximize the posterior probability ( | )

P Λ g : ˆ=arg max ( | ) P Λ Λ Λg (4)
Two assumptions (the parameter Λ is uniformly distributed over feasible region ( ) ℜ Λ and the presence of the signal or not is equiprobable) lead Eq.( 4) to:

1 1 1 1 =arg max{ ( )}, ( ) 0 ˆ2 2 any value in ( ), t t e otherwise - - ⎧ - + - + > ⎪ = ⎨ ⎪ ℜ ⎩ Λ b Λ Λ b Λ Λ Λ xΣ g x xΣ g x Λ Λ
(5) Considering the classical probabilistic decision model, we can set a decision rule as follows: when

0 1 ˆ( | , ) ( | , ) P H P H > Λ g
Λ g we choose 0 H , which means that the hyposignal is absent; otherwise, the hyposignal is present.

Then the following decision equation can be derived:

1 0 1 0 ˆˆ1 ( ) 1 ( ) l n 2 ( ) H t H P H P H λ - > = - + < b Λ Λ x Σ g x ( 6
)
where λ is the test statistic for deciding the presence or absence of the hyposignal.

As in [START_REF] Zhang | A perceptually relevant channelized joint observer (PCJO) for the detection-localization of parametric signals[END_REF], we solve the dimensionality problem by introducing the channel matrix Λ U to firstly obtain the channelized image

' g : ' = Λ g U g (7)
The matrix Λ U is built in the same way as in [20] to be capable of searching the optimal parameter in the channel domain rather than in the spatial domain, without loss of accuracy: ' ( )

t t = = Λ Λ Λ U A U U A (8)
where Λ

A is used to map the parametric signal Λ x to a reference signal 0

x whose parameters are known:

0 = Λ Λ A x x ; ' Λ A serves to map the channelized parametric signal ' Λ x to a channelized reference signal ' 0 x : ' ' ' 0 = Λ Λ A x x . The matrices

U and '

Λ

A remain the same as in [20], where U is constructed by K steerable channels and J scale-shiftable channels. We can finally get the following equations for the parameters estimation and the detection decision: [START_REF] Yao | Predicting human performance by a channelized Hotelling observer model[END_REF] For the hyposignal, the expression of

' t ' 1 ' ' ' 0 0 ' t 2 F 1 1 = argmax ( ) ( ) ( + ) || ( ) || 2 e - - b Λ Λ Λ Λ x Σ A g x U A (9) ' ' 1 ' ' ' 0 0 ' 2 1 1 ( ) ( ) ( + ) || ( ) || 2 t t F λ - - = b Λ Λ x Σ A g x U A
' b Σ is: ( )( ) ( ) ( ) ( ) ( ) ' ' ' ' ' 0 0 0 ' ' ' ' ' ' ' ' 1 1 1 1 ˆ2 1 2 t t H H H H H H ∧ ∧ ∧ ∧ = - - + ⎡ ⎤ ⎡ ⎤ + - + + - + ⎣ ⎦ ⎣ ⎦ b Σ g g g g g x g x g x g x (11)

III. EXPERIMENTAL PROTOCOL

A. Experimental images

With the extension on the CJO Detection Decision part, the new version of the PCJO can not only work on hypersignals, but also on hyposignals.

To study the PCJO task performance on hyposignals, we selected the hepatic carcinoma (cf. Fig. 3) as the target pathology and CT as the imaging modality. To further verify its generality over modalities, we also selected the MS lesions appearing as hypersignals under CT (cf. Fig. 3).

From a retrospective database of the Nanjing First Hospital, we extracted 10 healthy subjects' abdominal and cerebral CT image. We then chose 100 independent slices for the abdominal images and 80 independent slices for the cerebral images as the reference image-slices.

We then inserted (i) one to three subtractive lesions with an amplitude value between 20 and 35 on the abdominal CT image-slices to simulate the hepatic carcinoma lesions (hyposignals), and (ii) one to three additive lesions with an amplitude value between 40 and 55 on the cerebral CT image to simulate the MS lesions (hypersignals). The proprieties of the simulated lesions' amplitude, shape, size, position and orientation had been verified and confirmed by a consultant CT expert who did not participate in the subjective experiments. According to the consultant expert, these additive and subtractive lesions are sufficiently representative and mimic well the real abnormalities. Note that for the detection-localization task, the lesions must not be too conspicuous or too difficult. Otherwise, observers (radiologists or MOs) could always detect and localize the abnormalities easily, or they would never detect and localize any of them. In these extreme cases, different imaging systems cannot be adequately differentiated through different task performances of observers. The subtlety of the simulated lesions in our experiments was well controlled to make the task not too easy nor too difficult.

B. Human Observer studies

In order to objectively assess radiologists' performances, we conducted the same free-response study as in [START_REF] Zhang | A perceptually relevant channelized joint observer (PCJO) for the detection-localization of parametric signals[END_REF] with four radiologists. The four radiologists are hereafter referred to as E1, E2, R1 and R2, where E1 and E2 are CT experts, with respectively 12 and 10 years' abdominal and cerebral imaging experience and R1 and R2 have respectively 2 and 5 years' and cerebral imaging experience. Their task was to mark, with a mouse-click, the locations of any perceived lesions, and assign a confidence level on a 0 to 100 integer scale to each mark. Each radiologist performed this free-response task on 100 abdominal CT image-slices and 80 cerebral CT image-slices without time limits. The radiologists were blinded about the number of lesions (0-∞) that might be present on each image-slice and the lesion parameters. Each mark was classified as lesion localization (LL) if it was within an agreed upon distance of a simulated lesion or else it was classified as non-lesion localization (NL).

Note that no lesion-absent image-slices were presented to human observers in the experiment for time-saving purposes, but we can still get a high statistical power by using the wJAFROC1 figure of merit (cf. section III.D).

C. Extended PCJO Configurations

In the Candidates Localization part of the PCJO, the healthy CT image-slices were considered as the reference image for the VDP, while those with simulated lesions were considered as the distorted image. The default parameters in VDP were given in [START_REF] Daly | The visible differences predictor: an algorithm for the assessment of image fidelity[END_REF]. The probability threshold applied on the detection probability map in the Candidates Localization part (T p in [START_REF] Zhang | A perceptually relevant channelized joint observer (PCJO) for the detection-localization of parametric signals[END_REF]) is set to the values varying from 0.1 to 0.9 with a step of 0.1.

For the CJO Detection Decision part, 600 training blocks (300 without signal and 300 with signal) were extracted from healthy abdominal and cerebral CT images, respectively. The size of blocks was set to 65×65. The scale range of the simulated lesions was min max

D. Observer performance data analysis method

Since the data consisted of a-priori unknown numbers of mark-rating pairs on each viewed image-slice, where each slice was a simulated diseased case, the appropriate analysis is to use the area under the equally weighted alternative free-response receiver operating characteristic A wAFROC1 as the performance metric or figure or merit (FOM) [START_REF] Chakraborty | Observer studies involving detection and localization: Modeling, analysis and validation[END_REF][START_REF] Chakraborty | New Developments in Observer Performance Methodology in Medical Imaging[END_REF]. It is defined as follows:

( ) ( )

2 2 2 1 1 2 2 1 ' 1 1 ' 1 2 2 1 max , k L K K wAFROC kl l k l kl k l k A W z z K K ψ = = = = ∑∑ ∑ (12)
In the above expression the index of a diseased case is k or k', NL locations are indexed by l 1 and LL locations by l 2 . The rating of a case is denoted by The weighting assures that each case gets equal importance in the analysis, regardless of the number of lesions in it. In the absence of such weighting, cases with larger numbers of lesions would contribute more to the FOM than cases with fewer lesions, an undesirable characteristic since each case is intended to be an equally representative sample from the population of cases.

Since each free-response experiment (abdominal or cerebral) constitutes a single modality multiple reader study, the method described by Hillis [START_REF] Sl Hillis | A comparison of denominator degrees of freedom methods for multiple observer ROC studies[END_REF] for fixed reader analysis is applicable (see Section 5.4 in cited paper). The necessary co-variances were evaluated using the jackknife method [START_REF] Efron | An Introduction to the Bootstrap[END_REF].

The analysis applies a statistical test, using the F-distribution, of the null hypothesis that all readers have identical performances. The Type I error rate (alpha) of the test was chosen to be 0.05. If the null hypothesis is rejected, i.e., if the p-value of the overall F-test is smaller than 0.05, then one is justified in concluding that at least one pair of readers have different performances. Since the F-test controls the overall probability of a Type I error, no further correction (e.g., Bonferroni) for multiple comparisons is needed. Which reader pairs are different is determined by applying individual t-tests to each reader pairing. A t-test yielding a p-value less than 0.05 means that particular pair of readers have significantly different performances.

The analysis method is implemented in an R package called RJafroc (https://cran.r-project.org/web/packages/RJafroc).

IV. RESULTS AND DISCUSSIONS

A. Results of radiologists & PCJO for abdominal CT image

For the detection-localization of simulated hepatic carcinoma lesions on abdominal CT images, the wAFROC1 FOMs of the four radiologists were calculated and are shown in Table I. From the F-test results, we can see that no significant difference exists between the FOMs of all the four radiologists (p-value = 0.4 > 0.05). For the PCJO, we firstly studied the influence of the probability threshold (T p ) by varying it from 0.1 to 0.9 with a step of 0.1. The numbers of LL and NL marks obtained by different T p settings are shown in Table II. We can see that when T p is between 0.4 and 0.7, the LL and NL numbers of the PCJO seem close to those of radiologists.

We then further find the relation between the wAFROC1 FOM difference (between the PCJO and radiologists) and the probability threshold (T p ), as shown in Fig. 4. Note that for different thresholds, we tried different combinations of the number of steerable channel K and the number of scale-shiftable channel J to get the smallest wAFROC1 FOM difference. F-test results showed that when T p = 0.5 ~ 0.7, there is no significant difference between the wAFROC1 FOM of the PCJO and that of all the four radiologists. In addition, when the threshold T p = 0.5 the wAFROC1 FOM of the PCJO is closest to those of two experts. In order to make the wAFROC1 FOM of the PCJO closest to that of most radiologists, we retained the value 0.5 for the T p for the abdominal CT images.

With T p = 0.5, we then explored the influence of the number of steerable channels K and the number of scale-shiftable channels J on the PCJO's performance, as shown in Fig. 5. The results illustrate that: (1) the number of scale-shiftable channel J has a much stronger effect on the PCJO's FOM than the number of steerable channels K; (2) even with a small number of channels, (K=3 & J=4, 3×4=12 channels), there is no significant difference between the PCJO's wAFROC1 FOM and that of the four radiologists (p-value > 0.05).

B. Results of radiologists & PCJO for cerebral CT images

Table III summarizes the task performances obtained for the detection-localization of simulated MS lesions on cerebral CT images. Since the p-value of the overall F-test is 0.00 < 0.05 (that means there are significant differences among readers), we also listed the difference for each pair of radiologists. The p-values of each paired t-test demonstrate that there is no significant difference between expert 1 and expert 2 (p = 0.88 > 0.05), all other pairings of readers are significantly different. This finding is consistent with our previous experiment about hypersignals on MR images where there was a significant difference between the experts and the non-experts [START_REF] Zhang | A perceptually relevant channelized joint observer (PCJO) for the detection-localization of parametric signals[END_REF].

In order to get the best probability threshold (T p ), we then varied it from 0.1 to 0.9 with a step of 0.1 and counted the number of LL and NL marks obtained by the model, as shown in table VI. It can be seen that when T p = 0.9, the number of LL and NL is very close to subjective results from the experts. We then plotted the wAFROC1 FOMs of the PCJO vs. different values of T p , as shown in Fig. 6. F-test results showed that when T p = 0.8 and 0.9, there is no significant difference between the PCJO and the two experts. From Fig. 6 we can see that when T p = 0.9 the PCJO reaches the nearest approximation of the experts' task performance. This is also coherent with the setting in [START_REF] Zhang | A perceptually relevant channelized joint observer (PCJO) for the detection-localization of parametric signals[END_REF], where T p was set to 0.9.

With T p = 0.9, we then compared the wAFROC1 FOMs of the PCJO to those of the experts with different combinations of the number of steerable channels K and the number of scale-shiftable channels J, as shown in Fig. 7. Here, we observe Fig. 5 JAFROC1 FOMs of the PCJO under different combinations of the number of steerable channel K and the number of scale-shiftable channel J, and the significance testing results between the PCJO and each of the four radiologists for abdominal CT image again the same influences of K and J as in the above section (section IV.A): K hardly influences the PCJO's FOM while J does. There is no significant difference between the PCJO's performance and that of the experts for any combination of K and J.

C. Discussion

When comparing the results on the two types of CT images, we observed a behavioral difference between the experts and the radiologists (a significant difference was pointed out between the two groups for hypersignals, but not for hyposignals). This may indicate that the detection-localization task is more difficult on hypersignals than on hyposignals, since human observers have to read more images to familiarize themselves with this type of abnormalities and to have more experiences of diagnoses on this type of images (the differences between experts and average radiologists) to reach a good task performance. It is known that as the difficulty increases, the HVS's detection probability threshold rises. In order to approach human performance, the PCJO involves an HVS model (the VDP), on the output of which we applied a detection probability threshold T p . Thus, the different difficulty levels of the two tasks can consequently explain why the T p should be higher in the cerebral experiment (0.5~0.7) than that in the abdominal experiment (0.8~0.9). The results in this study are highly consistent with the findings reported in our previous study [START_REF] Zhang | A perceptually relevant channelized joint observer (PCJO) for the detection-localization of parametric signals[END_REF], even though the experiments are conducted by radiologists from two different countries (China and France, respectively) and on different modalities (CT and MRI, respectively). These cannot be merely coincident. We thus some conclusions on the PCJO: 1. Though the PCJO was originally proposed based on the MR image backgrounds, it can be generalized to other modalities, where the CGB could model the background more or less correctly. 2. With the number of steerable channels K=3 and the number of scale-shiftable channels J=4, the PCJO can approach well (with no statistically significant difference) the radiologist's and the expert's detection-localization performance in the hyposignal and hypersignal task, respectively. The small number of channels then renders the PCJO more efficiently (the PCJO executes faster with less channels). 3. The reference value of the T p could be 0.5 for hyposignals and 0.9 for hypersignals to mimic experts. The reference value may need to be adjusted according to the circumstances of a particular application, considering that the uncertainties in the clinical studies are often greater than those in the lab. In addition, it is worth to notice that human readers (both experts and general radiologists) had high false negative rates in our 2D image experiments, cf. the low detection rates in Tables I andIV. Indeed, by observing the experiments and talking to the radiologists after the experiments, we noted that they could not really tell the difference between the lesions and the cerebral cortex that also appears as signals, without referring to the information in the adjacent slices, in some difficult cases. Thus when the simulated lesions were close to the cerebral cortex, which may also happen in the real clinical cases, the radiologists may miss them. Actually, in our previous studies on MR images, we inquired into the changes in radiologists' performances from single-slice (2D) experiment to multi-slice (3D) experiment [START_REF] Zhang | A multi-slice model observer for medical image quality assessment[END_REF][START_REF] Zhang | Numerical observers for the objective quality assessment of medical images[END_REF]. The comparison results [START_REF] Zhang | Numerical observers for the objective quality assessment of medical images[END_REF] showed that both experts and radiologists improved their detection rate with the help of volumetric information (adjacent slices): an expert's detection rate could be increased from 0.72 to 0.90; general radiologists' detection rates could be increased from 0.65 to 0.77 on average. In this study, we extracted 2D non-contiguous slices from 3D CT images, since we wanted to concentrate firstly on the extension of the PCJO for 2D images. But we believe that the human readers' performances can also be improved when we show them volumetric CT images. This is also one of our future works, to compare human readers' performances on reading multi-slice images with that of an extension of our previously proposed multi-slice PCJO (msPCJO).

V. CONCLUSIONS AND PERSPECTIVES

In this paper, we extended the PCJO for hyposignals detection-localization task in order to propose a highly clinical relevant MO. Then, we applied the new PCJO on both cerebral and abdominal CT images. Experimental results showed that it can predict well the experts' performances in the joint detection-localization task for both hypersignals and hyposignals. Combined with its results on MR images in the previous study, the PCJO also showed its generality on another modality -CT. Although the PCJO model is parametric, the highly coherent results of the two studies allow us setting the best empirical values of all the three parameters (K=3, J=4, T p = 0.5 for hyposignals or 0.9 for hypersignals) for its application on MR and CT image quality assessments. All these features make the PCJO a promising method for evaluating image processing algorithms from the medical experts' point of view in the future.

One limitation of the study is the small number of experts and the limited patient data. The test CT images were selected from only 10 healthy subjects. Even though the slices are non-contiguous, it is still likely that a lot of them are correlated. This may have an impact on the statistical analysis. Since finding available experts in one hospital is very difficult and collecting real patient data is very costly, they remain two difficulties in the medial domain. We are open for collaborations, hoping to have more experts and real patient data to further validate the conclusions of this study. Another limitation of the study is that we did not verify how PCJO performs in the benchmark of different image processing algorithms. If the PCJO can output the same ranking as radiologists do, this can further prove its usefulness in the medical image quality assessment. This is actually our short-term objective: to use the PCJO for ranking different medical compression ratios and comparing the low-dose CT image qualities to the high-dose ones [START_REF] Chen | Improving low-dose abdominal CT images by weighted intensity averaging over large-scale neighborhoods[END_REF][START_REF] Chen | Artifact suppressed dictionary learning for low-dose CT image processing[END_REF].

  Extended PCJO for the detection-localization of hypersignals and hyposignals in CT images M (a) (b) Fig. 1 A cerebral CT image of a multiple sclerosis (MS) patient (a) and an abdominal CT image of a hepatic carcinoma patient (b). Note the arrow in each image points to one abnormality (lesion).

  cerebral MR image of a hematoma patient (a) and an abdominal MR image of a pancreatic cancer patient (b). Note the arrow in each image points to one abnormality (lesion).

Fig. 3

 3 Fig.3 Examples of experimental image-slices, the left two image-slices are original image-slices from healthy subjects and the right ones are those with simulated hyposignals (top image) and hypersignals (bottom image).

  π; the shape was defined by 2 b = and the amplitude range was [a min , a max ] = [20,35] for hyposignals and [a min , a max ] = [40,55] for hypersignals.

  number of lesions on case k is denoted by L k . The function ( ) x ψ is defined as unity if x > 0, one half if x = 0 and zero if x<0. The max() function is the maximum over the indicated subscript. The quantity 2 kl W is the weight of lesion l 2 ondiseased-case k. The weights, which and sum to unity on each case, denote the clinical importance of each lesion. In the current study the lesions were weighted equally, so

FIG. 6

 6 FIG. 6 The wAFROC1 FOM of the PCJO under different threshold Tp for cerebral CT images vs. those of two experts

Fig. 7 JAFROC1

 7 Fig. 7 JAFROC1 FOMs of the PCJO under different combinations of the number of steerable channel K and the number of scale-shiftable channel J, and the significance testing results between the PCJO and each of the four radiologists for cerebral CT image

TABLE I THE

 I TASK PERFORMANCES OF ALL RADIOLOGISTS FOR ABDOMINAL CT IMAGES

	Overall F-test results
	F statistic		0.98
	ndf (numerator degrees of freedom)	3
	ddf (denominator degrees of freedom)	297
	p-value		0.40
	FOM for individual radiologists
	wAFROC1 FOM	Detection rate
	E1	0.80	0.73
	E2	0.80	0.79
	R1	0.78	0.75
	R2	0.77	0.53

TABLE II NUMBER

 II OF LL MARKS AND NL MARKS UNDER DIFFERENT THRESHOLD TP OF THE PCJO FOR ABDOMINAL CT IMAGES

	Tp	LL	NL
	0.1	180	24
	0.2	167	24
	0.3	161	27
	0.4	146	21
	0.5	128	27
	0.6	117	17
	0.7	107	14
	0.8	80	5
	0.9	63	1
	FIG. 4 The

WAFROC1 FOM of the PCJO under different threshold Tp for abdominal CT images vs. those of four radiologists

TABLE III THE

 III TASK PERFORMANCES OF ALL RADIOLOGISTS FOR CEREBRAL CT IMAGES

		Overall F-test results
		F statistic		13.08
	ndf (numerator degrees of freedom)	3
	ddf (denominator degrees of freedom)	237
		p-value		0.00
		FOM for individual radiologists
		wAFROC1 FOM	Detection rate
	E1	0.81		0.64
	E2	0.81		0.66
	R1	0.75		0.78
	R2	0.68		0.54
		Paired Differences
		Difference	p-value	Confidence Interval
	E1-E2	0.00	0.88	[-0.03, 0.04]
	E1-R1	0.06	0.00	[0.03, 0.10]
	E1-R2	0.14	0.00	[0.07, 0.20]
	E2-R1	0.06	0.00	[0.03, 0.09]
	E2-R2	0.13	0.00	[0.07, 0.20]
	R1-R2	0.07	0.02	[0.01, 0.13]

TABLE IV NUMBER

 IV OF LL MARKS AND NL MARKS UNDER DIFFERENT THRESHOLD TP OF THE PCJO FOR CEREBRAL CT IMAGES

	Tp	LL	NL
	0.1	216	11
	0.2	216	6
	0.3	216	10
	0.4	216	12
	0.5	215	9
	0.6	212	8
	0.7	197	7
	0.8	182	6
	0.9	144	5

ACKNOWLEDGEMENTS.

The authors would like to thank Professor Jean-Louis Coatrieux (IEEE Life Fellow) for reviewing and giving valuable comments on this paper, and Professor Dev Chakraborty for providing and explaining to us the wAFROC1 tool.

This work was supported in part by the State's Key Project of Research and Development Plan 2017YFC0109202 and 2017YFA0104302, the National Natural Science Foundation under Grant 81370040, Grant 81530060, in part by the Fundamental Research Funds for the Central Universities, and in part by the Qing Lan Project in Jiangsu Province.