A Note on Convex Relaxations for Non-Metric Smoothness Priors Christopher Zach, Microsoft Research Cambridge∗ August 13, 2012 In  the following convex relaxation for labeling problems with non-metric but isotropic smoothness costs is proposed: ( ) X X X def i T i i i R(u) = max (ps ) ∇us + qs us , (1) (ps ,qs )∈Cs

s

i

i

where def  Cs = (ps , qs ) : kpis − pjs k2 + qsi ≤ θij ∀i, j .

(2)

Here θij ≥ 0 are the costs for a transition from label i to j. us ∈ ∆ is a (relaxed) one-hot encoding (or node marginal) of chosing label i at pixel s (uis = 1, ujs = 0 for j 6= i). Since the objective above decouples into independent problems over pixels s, we focus on a single pixel and drop the subscript indicating s in the def def following. We introduce rij = pi − pj and sij = q i , then one term in the objective above can be written as   X  X X X X max (pi )T ∇ui + q i ui − ı{krij k2 + sij ≤ θij } − ı{pi − pj − rij = 0} − ı{sij − q i = 0} .  (p,q)  i

i

i,j

i,j

i,j

(3) We will derive the primal of this expression using the following variant of Fenchel duality, min f (Ax) = max −f ∗ (y). x

(4)

y:AT y=0

The primal variables correspond to Lagrange multipliers y ij for pi − pj − rij = 0 and z ij for sij − q i = 0. We have X X X −f ∗ (p, q, r, s) = (pi )T ∇ui + q i ui − ı{krij k2 + sij ≤ θij }, (5) i

i

i,j

where each summand is now independent and f can be computed term-wise. Since the convex conjugate of y 7→ aT y is x 7→ ı{x = a}, the first two terms translate to constraints (Ay y + Az z)pi = −∇ui and (Ay y + Az z)qi = −ui in the primal. Here we decomposed A = [Ay |Az ], and e.g. (Ay y)pi refers to the rows def

in Ay y corresponding to the dual variable pi . The conjugate of h∗θ (z, w) = ı{kzk2 + w ≤ θ} requires a little derivation: hθ (x, y) =

max z,w:kzk2 +w≤θ

xT z + yw =

xT z + yw

max z,w,ε≥0:kzk2 +w+ε=θ

= max xT z + y(θ − kzk2 − ε) = θy + max xT z − y(kzk2 + ε) z,ε≥0 z,ε≥0  T = θy + max x z − ykzk2 + max −εy Ansatz: z = cx, c ≥ 0 z ε≥0  = θy + max ckxk22 − ckxk2 y + ı{y ≥ 0} = θy + kxk2 max {c(kxk2 − y)} + ı{y ≥ 0} c≥0

c≥0

= θy + kxk2 ı{y ≥ kxk2 } + ı{y ≥ 0} = θy + ı{y ≥ kxk2 }. ∗I

am grateful to Christian H¨ ane for discussion and proof-reading.

1

(6)

We used the particular ansatz, since xT z − ykzk2 will be maximized w.r.t. z if z is pointing into the same direction as x. In order to finally derive the primal energy, we need to specify the rows of Ay y + Az z. First note that the constraints in the dual form two separate partitions, hence (Ay y)qi = 0, (Ay y)sij = 0 and (Az z)pi = 0, (Az z)rij = 0. Since pi appears in constraints y ij (for j 6= i) with +1 coefficient, and in y ji for j 6= i with -1 coefficient (note that pi cancels in constraint y ii ), we have X X (Ay y)pi = y ij − y ji . (7) j:j6=i

j:j6=i

Since rij appears exactly in the constraint corresponding to the multiplier y ij (with -1), P we have (Ay y)rij = −y ij . q i appears with a -1 coefficient in all constraints z ij for j, and (Az z)qi = − j z ij . Thus, we can already state the two linear constraints appearing in the primal problem, X X X ∇ui = y ji − y ij and ui = z ij . j:j6=i

j

j:j6=i

The dual capacity constraints krij k2 + sij ≤ θij translate to X  X ij ij  θij z ij + ı{z ij ≥ k−y ij k2 } = θ z + ı{z ij ≥ ky ij k2 } . i,j

(8)

i,j

Overall we can state R(u) in its primal form XX  R(u) = min θij z ij + ı{z ij ≥ ky ij k2 } x,y

s.t. ∇uis =

s

X j:j6=i

(9)

i,j

ysji −

X

ysij ,

zsij ≥ kysij k2 ,

uis =

X

zsij .

j

j:j6=i

The first set of constraints are the differential marginalization constraints as derived in . The second set of constraints forces zsij to be no smaller than kysij k2 . If the final set of constraints is dropped, any minimizer will satisfy zsij = kysij k2 and we have Etight in  (which is the primal form of the saddlepoint energy proposed in ). The last set of constraints tell us, that the zsij are pairwise pseudo-marginals for a transition from label i to j regardless of the orientation of the discontinuity. This also means, that zsij is bounded from above by 1 (since uis ≤ 1). Hence, the contribution of a jump from i to j at pixel s is at most θij regardless of the direction. In particular, diagonal boundaries on regular pixel grids will be underestimated in their cost. Our understanding therefore is, that R(u) is not a stronger relaxation than Etight addressing non-metric smoothness (since even for metric pairwise potentials the minimizer will be in general not the same as Etight ), but a different relaxation in the first place. This is not a contradiction with Section 2.4 in , since there the qsi are explicitly forced to be 0, which again leads directly to the primal energy Etight . Etight-marginals as proposed in  is a proper way to strengthen Etight such that minimizers for metric smoothness costs coincide in both settings.

References  A. Chambolle, D. Cremers, and T. Pock. A convex approach for computing minimal partitions. Technical report, Ecole Polytechnique, 2008.  E. Strekalovskiy, C. Nieuwenhuis, and D. Cremers. Nonmetric priors for continuous multilabel optimization. In Proc. ECCV, 2012. to appear.  C. Zach, C. H¨ ane, and M. Pollefeys. What is optimized in convex relaxations for multi-label problems: Connecting discrete and continuously-inspired MAP inference. Technical report, MSR Cambridge, 2012.  C. Zach, C. H¨ ane, and M. Pollefeys. What is optimized in tight convex relaxations for multi-label problems? In Proc. CVPR, 2012.

2

## A Note on Convex Relaxations for Non-Metric ...

13 Aug 2012 - i,j Ä±{rij. 2 + sij â¤ Î¸ij} â â i,j Ä±{pi â pj â rij = 0} â. â i,j Ä±{sij â qi = 0}... . (3). We will derive the primal of this expression using the following variant of Fenchel duality, min x f(Ax) = max y:AT y=0. âfâ(y). (4). The primal variables correspond to Lagrange multipliers yij for pi â pj â rij = 0 and zij for sij â qi = 0.

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