1. On the Effectiveness of Visible Watermarks Supplemental 1 Image/Watermark Decomp. We derive the solution to the optimization problem in Eq. 11 in the continuous domain, i.e., replacing the sum by an integral. In this case, the optimal solution must satisfy the Euler-Lagrange equations, given by ∂ ∂L ∂L ∂ ∂L − − ∂I k (p) ∂x ∂(I x k (p)) ∂y ∂(I y k (p)) (1) ∂ ∂ ∂L ∂L ∂L − − k k ∂W (p) ∂x ∂(W x (p)) ∂y ∂(W y k (p)) (2) for all pixel locations p, where L is the integrand in Eq. 11. That is, L = L data (I k , W k , α) + λ I L reg (∇I k ) + λ w L reg (∇W k ) + λ α L reg (∇α) + βL f (∇(αW k )) + γL aux (W, W k ), (3) where L x is the corresponding integrand to term E x in the paper (i.e., the expression inside the sum). Keeping only the relevant terms in L, Eq. (1-2) are given by: ∂L data ∂ ∂L reg (∇I k ) ∂ ∂L reg (∇I k ) − λ I − λ I k k ∂I (p) ∂x ∂(I x (p)) ∂y ∂(I y k (p)) (4) ∂(L data + γL aux ) ∂ ∂(βL f + λ w L reg ) ∂ ∂(βL f + λ w L reg ) − − ∂W k (p) ∂x ∂(W x k (p)) ∂y ∂(W y k (p)) (5) Let α = diag(α), ᾱ = diag(1 − α) be diagonal matrices, where α and 1 − α are the diagonals, respectively. We define the following notations: Ψ 0 data = diag(Ψ 0 ((αW k + (1 − α)I k − J k ) 2 )) Ψ 0 w = diag(Ψ 0 ((|α x |W x k + |α y |W y k ) 2 )) Ψ 0 I = diag(Ψ 0 ((|α x |I x k + |α y |I y k ) 2 )) Ψ 0 f = diag(Ψ 0 (k∇(αW k ) − ∇W m )k 2 ) Ψ 0 aux = diag(Ψ 0 ((W k − W ) 2 )) 1

2. Ψ 0 rI = diag(Ψ 0 ((|α x |I x 2 + |α y |I y 2 )) Ψ 0 rw = diag(Ψ 0 ((|α x |W x 2 + |α y |W y 2 ))  With these notations in hand, (1-2) can be explicitly written as     α 2 Ψ 0 data + λ w L w + βA f αᾱΨ 0 data b w W k = b I I k αᾱΨ 0 data ᾱ 2 Ψ 0 data + λ I L I (6) L I = D x T c x Ψ 0 rI D x + D y T c y Ψ 0 rI D y L w = D x T c x Ψ 0 rw D x + D y T c y Ψ 0 rw D y A f = α T (D x T Ψ 0 f D x + D y T Ψ 0 f D y )α + γΨ 0 aux | {z } L f and D x , D y denote the horizontal and vertical derivatives operators. The vectors b w , b I are given by b w = α T Ψ 0 data J k + βL f W m + γΨ 0 aux W b I = ᾱ T Ψ 0 data J k . The weighting matrices c x , c y are given by c x = diag(|α x |) and c y = diag(|α y |). As mentioned in Sec. 3.2, we solve Eq. 6 using iterative reweighed least square, i.e., iterating between computing the non linear terms Ψ 0 based on the current estimate, and updating the solution of I k and W k . II. Matte Update: The EL equation for Eq. 12 is given by ! X Ψ 0 k + λ α L α + β Ã f α = X k A k (J − I k ) + βW T L f W m , (7) k where L f as defined above, W = diag(W ) and Ψ 0 k = diag Ψ 0 (αW + (1 − α)I k − J k ) 2 (W − I k )

L α = D x T Ψ 0 α D x + D y T Ψ 0 α D y à f = W T L f W As before, Ψ 0 α = diag(Ψ 0 (k∇αk 2 )). 2 Blend Factor Estimation We assume a small per-image deviation from a global blend factor c, i.e., the opacity of the k th image watermark is c k · c. We solve for a per-image r c k by minimizing the following objective Ψ((c k cαW + (1 − c k cα)I k − J k ) 2 ) + λ c (c k − 1) 2 , (8) where λ c is the weight of the regularization term (which controls the amount of deviation from the global opacity), and Ψ is a robust function. Minimizing 2

3. Input: A collection of watermarked images {J k }. (optional) In case of random watermark position – bounding box around watermarked region in a single image J i Output: Watermark W , alpha matte α, watermark free collection {I k } 1. 2. 3. 4. Compute initial matted watermark & detect all watermarks (Sec. 3.1) Initialize α using single-image matting Estimate global (average) blend factor c for t = 1 to T do for k = 1 to K do I. Image–Watermark Decomposition: Solve for I k and W k , keeping α and W fixed. II. Opacity Estimation (Optional): Solve for small per-image variation in opacity c k . III. Flow Estimation (Optional): Solve for small per-image watermark perturbation ω k end IV. Watermark Update: Solve for W keeping {I k , W k , c k , ω k }, and α fixed. IIV. Matte Update: Solve for α keeping {I k , W k c k , ω k }, and W fixed. end Algorithm 1: Our automatic multi-image watermark removal algorithm this equation w.r.t. c k , and keeping the rest of the unknowns (W, α, I k , c) fixed, leads to    X X c k = λ c − (Ψ 0 k )(I k − J k )α(W − I k ) λ c + (Ψ 0 k α 2 (W − I k ) 2 ) , (9) where Ψ 0 k = Ψ 0 (c k cαW + (1 − c k cα)I k − J k ) 2 . This estimation is integrated into our multi-image matting and reconstruction algorithm as additional (optional) step (see Alg. 1). 3 The Effect of Number of Images: We tested how the number of images effects on our performance. In particular, we evaluated the impact of two factors: (i) #images used to estimate the initial matted watermark (Sec. 3.1), (ii) #images used in the multi-image matting step (Sec. 3.2). We denote these two factors by N init , and N matting , respectively. The computed PSNR and DSSIM errors for running our algorithm with different values of N init , and N matting , on the CVPR17 dataset, are presented in Fig. 1(a-b). An example of our reconstructions for the minimal (N init = 10, N matting = 5), and maximal (N init = 300, N matting = 70) combinations are shown in Fig. 1(c-d), respectively. As expected, the results improves as more 3

4. 10 50 100 200 300 10 N init 5 5 10 10 30 30 50 50 70 70 N matting (a) PSNR N matting 50 100 200 300 N init (b) DSSIM (c) Our Recon. (N init =10, N matting =5) (d) Our Recon. (N init =300, N matting =70) N init -- #Images used for intial wateramark estimate N matting -- #Images used for multi-image matting Figure 1: The effect of number of images. (a-b) Error matrix measuring PSNR and DSSIM between the ground truth and our results, respectively; the num- ber of images used for initial matted-watermark estimation (N init ) is changing along the columns; the number of images used for the multi-image matting step (N matting ) is changing along the rows. (c-d) An example of our result corre- sponding to locations (1,1) and (5,5) in the error matrix, respectively. images are used. However, with N init = 300 the errors are already visually unnoticeable. Furthermore, this evaluation shows that the accuracy of the initial matted-watermark has much higher impact on the quality of the results than the number of images used for the multi-image matting step. That is, with a good initialization of the watermark in hand, it is enough to have an order of tens images for decomposition step. 4