Probabilistic Noise2Void: Unsupervised Content-Aware Denoising

Probabilistic Noise2V oid: Unsup ervised Con ten t-Aw are Denoising Alexander Krull 1 , T om´ a ˇ s Vi ˇ car 2 , Flor ian Jug 1 1 MPI-CBG/PKS (CSBD) 2 Brno Universit y of T ec hnology Abstract. T o day , Con vol utional Neural N etw orks (CNN s) are the lead- ing metho d for image denoising. They are traditionally trained on pairs of images, which are often hard to obtain for p ractical applications. This motiv ates self-superv ised training method s such as Nois e2V oid (N2V) that operate on singl e noisy images. Self-sup erv ised metho ds are, un- fortunately , not comp etitive with mo dels trained on image pairs. Here, w e present Pr ob abilistic Noise2V oid (PN 2V ) , a metho d to train CNNs to predict per-p ixel intensit y distributions. Com b ining these with a suit- able description of th e noise, w e obtain a complete probabilistic m o del for the noisy observ ations and true signal in every pixel. W e ev aluate PN2V on publicly av ailable microscopy datasets, under a b road range of noise regimes, and achiev e comp etitive results with resp ect to sup ervised state-of-the-art metho ds. Keywords: Denoising · CARE · Deep Learning · Microscop y Data 1 In t ro duction Image restoration is the pro ble m of reconstructing a n image from a corrupted version of its e lf. Recent work sho ws how CNNs can b e use d to build powerful conten t-aw are image restoration (CARE) pip elines [11,10,12,13,6,4,1,5]. How- ever, for superv ised CARE mo dels , such as [10], pairs o f clean and no isy images are required. F or many application area s, it is impractical or imp ossible to a cquire clea n ground-truth images [2]. In such cases, Noise2Noise (N2N) tra ining [6] relaxes the problem, only requiring tw o noisy instances of the same data. Unfortunately , even the acquisition o f tw o noisy r e alizations of the same imag e co nten t is often difficult [2]. Self-super vised training metho ds, such as Noise2 V oid (N2V) [4 ], a re a promising alter native, as they oper ate exclusively on single noisy imag es [4,1,5]. This is enabled b y excluding/ masking the cen ter (blind-sp ot) of the netw or k’s receptive fields. Self-super vised training ass umes that the noise is pixel- wis e inde- pendent and that the true intensit y of a pixel can b e predicted from lo ca l image context, excluding b efor e-mentioned blind-spo ts [4]. F or many applications, es- pec ially in the cont ext of micr oscopy imag e s, the firs t assumption is fulfilled, but the second assumption offers ro om fo r improv ements [5]. 2 Alexander Kru ll, T om´ a ˇ s V iˇ car, Florian Jug Hence, self-sup erv is ed mo dels can often not comp ete with super vised tra in- ing [4]. In concurre nt work, by Laine et al. [5 ], this problem was e le g antly ad- dressed by a ssuming a Gaussian noise mo del and predicting Gaussian intensit y distributions p er pix e l. The author s also show ed that the same appro ach an b e applied to other noise distributions, which c an be a pproximated as Gaussia n, or can be descr ibe d a nalytically . Here, we introduce a new training approa ch called Pr ob abilistic Noise2V oid (PN2V) . Similar to [5], PN2V prop o ses a wa y to leverage informa tion of the net- work’s blind-sp ots. Howev er, PN2V is not restricted to Gaussian noise mo dels o r Gaussian in tens ity predictions . Mor e precisely , to compute the pos terior distri- bution of a pixel, we combine ( i ) a ge ne r al noise mo del tha t can b e represe nted as a histog ram (obs erv ation likeliho o d), and ( ii ) a distribution of p ossible true pixel int ensities (prior), represented by a set of pr edicted samples. Having this complete probabilistic mo del fo r each pixel, we are now free to chose whic h statistical estimator to employ . In this work we use MMSE estimates for our fina l pr edictions and show that MMSE-PN2V cons istent ly outp erfor mes other self-sup ervis e d metho ds and, in many cases, leads to results that are com- petitive even with super vised state-of-the-ar t CARE netw or k s (see b e low). 2 Bac kground Image F ormation and the Denoisi ng T a sk: An image x = ( x 1 , . . . , x n ) is the co rrupted version of a cle a n ima ge (sig na l) s = ( s 1 , . . . , s n ). O ur goa l is to recov er the original signal from x , th us implementing a function f ( x ) = ˆ s ≈ s . In this pap er , we as sume tha t ea ch observed pixel v a lue x i is indep endently drawn from the conditional distribution p ( x i | s i ) such that p ( x | s ) = n Y i =1 p ( x i | s i ) . (1) W e will r efer to p ( x i | s i ) a s observatio n likeliho o d . It is descr ibe d by an arbitrar y noise mode l. T raditional T raini ng and Noise2N oise: The function f ( x ) c an b e imple- men ted by a F ully Convolutional Net work (F CN) [7] (see e.g. [11,10,12,6]), a t yp e of CNN that takes an image as input and pr o duces an entire (in this case denoised) image as output. Howev er, in this setup ev ery predicted output pixel ˆ s i depe nds only o n a limited receptive field x RF( i ) , i.e. a patch o f input pixels surrounding it. F CN based image deno is ing in fact implemen ts f ( x ) by pro duc- ing indep endent predictions ˆ s i = g ( x RF( i ) ; θ ) ≈ s i for ea ch pixel i , dep ending only on x RF( i ) instead of on the entire ima ge. The pr ediction is par ametrized b y the w eights θ of the netw o rk. In traditional training, θ are learned from pairs of noisy x j and cor r esp onding clean tr aining images s j , which provide training examples ( x j RF( i ) , s j i ) c o nsisting of noisy input patc hes x j RF( i ) and their corresp onding clean tar get v a lues s j i . Probabilistic Noise2V oid: Unsup ervised Con tent-Aw are D enoising 3 The parameters θ are traditionally tuned to minimize an empirical risk function such as the a verage squared distance argmin θ n X i =1 m X j =1 ( ˆ s j i − s j i ) 2 (2) ov er all training images j and pixels i . In Noise2Noise [6], Leh tinen et al. show that clea n data is in fac t not necess ary for training and that the same training s cheme can b e used with noisy data alone. Noise2Noise us es pairs of corresp onding no isy tr aining ima ges x j and x ′ j , whic h are based on the same signal s j , but ar e c o rrupted indep endently b y no ise (see Eq. 1). Such pair s can for example b e acquired by imag ing a static sample twice. Noise2Noise us es training examples ( x j RF( i ) , x ′ j i ), with the input patch x j RF( i ) cropp ed from the fir s t ima ge x j and the noisy tar get x ′ j i extracted from the patch center in the second one x ′ . It is of c o urse impo ssible for the netw or k to pre dict the noisy pixel v alue x ′ j i from the independently cor rupted input x j RF( i ) . Howev er, as s uming the nois e is zer o centered, i.e. E h x ′ j i i = s j i , the b est achiev able prediction is the clean signal s j i and the net work will lear n to denoise the images it is pre s ent ed with. Noise2V oid T raining: In No ise2V oid, K rull et al. [4 ] show that tra ining is still poss ible when not even noisy training pa ir s are av ailable. The y use single images to extract input and target for their netw orks. If this was done naively , the netw or k would simply learn the ident ity tra nsformation, directly outputting the v alue at the center of each pixel’s rece ptiv e field. Kr ull et al. a ddr ess the issue by effectively removing the central pixel from the netw or ks receptive field. T o achiev e this, they mask the pixel during training, replacing it with a ra ndom v alue from the vicinity . Th us, a Noise2V o id trained netw o r k can b e seen a s a function ˆ s i = ˜ g ( ˜ x RF( i ) ; θ ) ≈ s i , making a prediction for a single pixel based on the mo dified patch ˜ x RF( i ) that excludes the c ent ral pixel. Such a netw o rk can no longer describ e the identit y , and c a n b e tra ined from single no isy images. How ever, this a bilit y co mes at price. The a ccuracy of the predictions is re- duced, as the netw ork has to exclude the central pixel of its receptive field, th us having less information av aila ble. T o allow efficient training of a CNN with No is e2V oid, Krull et al. simultane- ously mask multiple pixels in lar ger training patches and joint ly calculate their gradients. 3 Metho d Maxim um Lik eli ho o d T raining: In P N2V, we build on the idea of masking pixels [4] to obta in a pr ediction from the mo dified rec e ptiv e field ˜ x RF( i ) . Howev er, instead of directly pr edicting an estimate fo r ea ch pixel v a lue , PN2V trains a CNN to describ e a pro bability distribution p ( s i | ˜ x RF( i ) ; θ ) . (3) 4 Alexander Kru ll, T om´ a ˇ s V iˇ car, Florian Jug 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 s i 0 . 0 0 0 . 0 1 0 . 0 2 0 . 0 3 p r o b a b i l i t y d e n si t y input image average prior MMSE estimate p o st e r i o r p r i o r ( CNN p r e d i c t i o n ) l i k e l i h o o d ( n o i s e m o d e l ) GT MMSE avg. prior input Fig. 1. Image denoising with PN2V. The final MMSE estimate (orange dashed line) for the true signal s i of a pixel (p osition marked in the image insets on the right) corresponds to th e cen ter of mass of the p osterior distribut ion (orange cu rve). Given an observe d n oisy input v alue x i (dashed green line), the p osterior is prop ortional to the p rodu ct of the prior (blue curve) and the observ ation lik eliho o d (green curve). PN2V describes the prior b y a set of samples predicted by our CNN. The likelihoo d is provided b y an arbitrary noise mo del. Blac k dashed line is the true signal of the pixel (GT). Prior and p osterior are visualized using a kernel densit y estimator. W e will r efer to p ( s i | ˜ x RF( i ) ; θ ) a s prior , as it des crib es our knowledge o f the pixel’s signal consider ing only its surro undings, but not the obser v ation at the pixel itself x i , since it has b een excluded fr om ˜ x RF( i ) . W e choos e a s ample based representation for this prior, whic h will be discus sed b elow. Remem be ring that the o bserved pixels v alues a re drawn indepe ndently (E q . 1), we can com bine Eq. 3 with our noise mo del, and obtain the join t distr ibution p ( x i , s i | ˜ x RF( i ) ; θ ) = p ( s i | ˜ x RF( i ) ; θ ) p ( x i | s i ) . (4) By in teg rating ov er a ll po ssible clean s ignals, we can derive p ( x i | ˜ x RF( i ) ; θ ) = Z ∞ −∞ p ( s i | ˜ x RF( i ) ; θ ) p ( x i | s i ) d s i , (5) the probability of obs erving the pixel v alue x i , g iven we know its surr ound- ings ˜ x RF( i ) . W e can now view CNN training as an unsup ervised lear ning task. F ollowing the maximum likelihoo d approa ch, w e tune θ to minimize argmin θ n X i =1 − ln  Z ∞ −∞ p ( s i | ˜ x RF( i ) ; θ ) p ( x i | s i ) d s i  . (6) Note that in order to improv e re adability , we fro m here on omit the index j , and refrain from explicitly referring to the tra ining ima ge. Sample Based Prior: T o a llow an efficient optimization of Eq. 6 we cho ose a sample bas ed representation of our prior p ( s i | ˜ x RF( i ) ; θ ). F or ev er y pixel i , our netw ork directly predicts K = 800 output v a lues s k i , which we interpret as indep endent sa mples, drawn from p ( s i | ˜ x RF( i ) ; θ ). W e can now approximate Eq. 6 a s argmin θ n X i =1 − ln 1 K K X k =1 p ( x i | s k i ) ! . (7) Probabilistic Noise2V oid: Unsup ervised Con tent-Aw are D enoising 5 During tra ining we use Eq. 7 as loss function. Note that the summation ov er k can be efficiently perfo rmed o n the GPU. Since every sample s k i is effectively a function of the para meters θ , we can calculate the deriv ative with resp ect to any net work parameter θ l as ∂ ∂ θ l n X i =1 − ln 1 K K X k =1 p ( x i | s k i ) ! = n X i =1 −   P K k =1 ∂ ∂ s k i p ( x i | s k i ) ∂ s k i ∂ θ l P K k =1 p ( x i | s k i )   . (8) Minimal Mean Squar ed Error (MMSE) Inference: Assuming o ur netw o rk is sufficiently trained, we are now interested in pro cess ing images and finding sensible estimates for every pixel’s signal s i . B a sed on our probabilistic mode l, we derive the MMSE estimate, which is defined as s MMSE i = arg min ˆ s i E p ( s i | x RF( i ) ) h ( ˆ s i − s i ) 2 i = E p ( s i | x RF( i ) ) [ s i ] , (9) where p ( s i | x RF( i ) ) is the p osterior distribution of the signa l g iven the complete surrounding patch. The p os terior is prop ortio nal to the joint distribution given in Eq. 4. W e can thus approximate s MMSE i by weighing our predicted s amples with the corresp onding observ ation lik eliho o d and calculating their average s MMSE i ≈ P K k =1 p ( x i | s k i ) s k i P K k =1 p ( x i | s k i ) . (10) Figure 1 illustrates the pr o cess a nd s hows the inv olved distributions for a c o n- crete pixel in a r eal example. 4 Exp erimen t s The results of o ur e xpe riments can b e found in T a ble 1. In Fig ure 2 we provide qualitative results on realistic test images. Datasets: W e ev aluate PN2V on datasets provided by Zhang et al. in [13]. Since PN2V is not yet implemented for multi-c hannel ima ges, we use all av ailable single-channel datasets. These da ta sets a re r ecorded with different samples and under different ima g- ing conditions. Each of them consists of a total of 20 fields o f view (F OVs). One F OV is reserved for testing. The other 19 ar e use d for tra ining a nd v alida tion. F or each FO V, the data is compos e d of 50 ra w microsco py images, ea ch containing different nois e rea lizations of the same static sample. F or every FOV , Zhang et al. additionally s imulate four reduced noise regimes (NRs) by averaging different subsets o f 2, 4 , 8 , and 16 r aw images [13]. W e will r efer to the r aw images as NR1 and to the regimes c r eated through av era g ing 2, 4, 8, and 1 6 imag es as NR2, NR3, NR4, a nd NR5, re sp e ctively . W e find that in one of the datas ets ( Two -Photon Mic e ) the av er age pixel in- tensity fluctuates heavily over the co urse of the 50 image s, even thoug h it should 6 Alexander Kru ll, T om´ a ˇ s V iˇ car, Florian Jug be approximately constant for each FO V. Considering that a single g round truth image (the av erag e) is used for the ev alua tio n on a ll 50 images, this lea ds to fluc- tuations and distortions in the c a lculated PSNR v a lue s , whic h are a ls o reflected in the compara tively high standard errors (SEMs) for all metho ds, see T a ble 1 . T o account for this incons istency in the data, we additio nally use a v aria nt of the PSNR calculation that is inv aria nt to arbitrary shifts and linear transforma tions in the ground truth sig nal. These v alues are ma r ked by an a s terisk (*). Details can also be found in the supplementary material 1 . Acquiring Noi se Mo dels: In our exp eriments, w e use a histogram based metho d to measure and de s crib e the noise distribution p ( x i | s i ). W e start with corres p o nding pairs of clean s j and no isy x j images. Her e, we use the av ailable training data from [13] for this purp ose. Ho wev er, in gener al these image s c ould show an a rbitrary test pattern that co vers the desir ed range o f v alues a nd do no t hav e to res emb le the sa mple we a re interested in. W e construct a 2D histog ram (256 × 2 56 bins), with the y- and x-a xis cor resp onding to the clean s j i and noisy pixel v a lues x j i , r esp ectively . By nor malizing every row, we obtain a a probabil- it y distribution for every signal. Considering Eq . 7 , we r equire our mo del to b e differentiable with r esp ect to the s i . T o ensur e this different iability , w e linearly int erp olate along the y - axis of the nor malized histogram, obtaining a mo del for p ( x i | s i ) that is contin uous in s i . Ev al uated Denoising M o dels/ Metho ds: T o put the deno ising r esults of PN2V in to p ersp ective, we compare to v ar io us sta te-of-the-art baselines, includ- ing the s tr ongest published n umbers on the data sets. U-Net (PN2V): W e use a sta ndard U-Net [9]. Our netw or k has a U-Net depth of 3, 1 input channel, and K = 80 0 output c hannels, which ar e int erpreted as samples. W e use a initia l feature channel num b er of 64 in the fist U-Net lay er. W e train our netw ork separately for each NR in each data set. W e use the same masking technique as [4]. F urther details and training parameter s can b e found in the supplemen tary material 1 . U-Net (N2V): W e use the same netw or k a rchitecture as for U-Net (PN2V) but mo dify the outputlay er to pro duce only a s ingle prediction instead of K = 800. The netw ork is trained using the Nois e 2V oid scheme as descr ib ed in [4 ]. All training parameters are identical to U-Net (P N2 V). U-Net (tr ad.): W e use the exa ct same architecture as U-Net (N2V), but tr ain the netw or k using the a v ailable gr ound-truth data and the standard MSE loss (see Eq. 2). All training para meters a re iden tical to U-Net (PN2V) and U-Net (N2V). VST+BM3D: Num ber s ar e taken fro m [13]. The authors fit a Poisson Gaussian noise mo de l to the da ta and then a pply a combination of v aria nce-stabilizing transformatio n (VST) [8] and BM3D filtering [3]. DnCNN: Num ber s a re taken from [13]. DnCNN [12] is an established CNN based denoising architecture that is trained in a s uper vised fashion. N2N: Num ber s ar e taken from [1 3]. The authors tra in a net work according to the N2N sch eme, us ing a n a rchitecture similar to the one pr esented in [6]. 1 The Supp lemen t wil l b e made av ailable so on. Probabilistic Noise2V oid: Unsup ervised Con tent-Aw are D enoising 7 input image i nput crop U-Net (trad.) U-Net (N2V) U-Net (PN2V) ground truth T wo-Photon Mice Confocal Zebrafish Confocal Mice Fig. 2. Qualitative results for three images (rows) from the datasets w e used in th is manuscri pt. Left to right: raw image (NR 1), zoomed inset, predictions by U-Net (trad.), U-Net (N2V), U- Net (PN2V), and ground truth d ata. Confocal Mice NR1 NR2 NR3 NR4 NR5 Mean Input 29.38 ± 0.01 32.44 ± 0.01 35.59 ± 0.01 38.90 ± 0.01 42.64 ± 0.03 35.79 U-Net ( P N2V ) 38.24 ± 0.02 39.72 ± 0.03 41.34 ± 0.03 43.02 ± 0.04 45.11 ± 0.05 41.49 U-Net ( N 2V) 37.56 ± 0.02 38.78 ± 0.02 39.94 ± 0.02 41.01 ± 0.02 41.95 ± 0.02 39.85 VST+BM3D 37.95 39.47 41.09 42.73 44.52 41.15 U-Net ( t rad.) 38.38 ± 0.02 39.90 ± 0.03 41.37 ± 0.03 43.06 ± 0.04 45.16 ± 0.05 41.58 DnCNN 38.15 39.78 41.41 43.11 45.20 41.53 N2N 38.19 39.77 41.28 42.83 44.56 41.33 Confocal Zebrafish Input 22.81 ± 0.02 25.89 ± 0.02 29.05 ± 0.03 32.39 ± 0.03 36.21 ± 0.04 29.27 U-Net ( P N2V ) 32.45 ± 0.02 33.96 ± 0.03 35.48 ± 0.05 37.07 ± 0.06 39.08 ± 0.07 35.61 U-Net ( N 2V) 32.10 ± 0.02 33.34 ± 0.03 34.43 ± 0.04 35.39 ± 0.04 36.21 ± 0.03 34.30 VST+BM3D 32.00 33.75 35.30 36.78 38.32 35.23 U-Net ( t rad.) 32.93 ± 0.03 34.35 ± 0.04 35.67 ± 0.05 37.11+0.06 39.09 ± 0.07 35.83 DnCNN 32.44 34.16 35.75 37.28 39.07 35.74 N2N 32.93 34.37 35.71 37.06 38.65 35.74 Two-Ph oton Mice Input 24.94 ± 0.07 27.83 ± 0.1 30.69 ± 0.15 33.67 ± 0.19 37.72 ± 0.14 30.97 U-Net ( P N2V ) 33.67 ± 0.33 34.58 ± 0.39 35.42 ± 0.42 36.58 ± 0.37 39.78 ± 0.24 36.00 U-Net ( N 2V) 33.42 ± 0.31 34.31 ± 0.36 35.09 ± 0.38 36.08 ± 0.33 37.80 ± 0.14 35.34 VST+BM3D 33.81 34.78 35.77 36.97 39.39 36.14 U-Net ( t rad.) 34.35 ± 0.19 35.32+0.23 36.14 ± 0.27 37.48 ± 0.27 40.28 ± 0.2 36.72 DnCNN 33.67 34.95 36.10 37.43 40.30 36.49 N2N 34.33 35.32 36.25 37.46 39.89 36.65 ∗ U- N et ( PN2V ) 34.84 ± 0.06 36.02 ± 0.07 37.08 ± 0.08 38.28 ± 0.09 40.89 ± 0.07 37.42 ∗ U- N et (N2V) 34.60 ± 0.09 35.77 ± 0.1 36.71 ± 0.1 37.64 ± 0.09 38.49 ± 0.05 36.64 ∗ U- N et (trad.) 35.05 ± 0.05 36.22 ± 0.06 37.28 ± 0.07 38.78 ± 0.1 41.34 ± 0.07 37.73 T able 1. Results of PN2V and b aseline meth o ds on three d atasets from [13 ]. Compar- isons are p erformed on five n oise regimes (N R1-NR5). N umbers rep ort PSNR (dB) ± 2 SEM, av eraged ov er all 50 images in each NR. W e group all sup erv ised/non-sup ervised metho d s and mark the highest val ues in b old. Rows marked by asterisk ( ∗ ) use a scale- and shift-in v ariant PSNR calculation to address inconsisten t acqu isitions in the Two- Photon mic e dataset (see main text ). Comp. times: All CNN based met h od s required b elo w 1s p er image ( NVIDI A TI T AN Xp ); VST+BM3D required on a vg. 6.22s . 8 Alexander Kru ll, T om´ a ˇ s V iˇ car, Florian Jug 5 Discussion W e have intro duced PN2V, a fully probabilis tic approa ch extending self-s uper vised CARE training. PN2V makes use of an arbitrary noise mo del which can be de- termined by analyzing any set of av ailable images that are sub ject to the s ame t yp e of noise. This is a decisive adv antage compared to state-of-the-ar t sup er- vised methods a nd allows PN2V to b e used for many practica l applicatio ns . The muc h improved per formance o f P N2V lies consistently b eyond self- sup e rvised training and can often comp ete with state- of-the-art sup erv is ed meth- o ds. W e see a plethor a of unique applications for PN2V, for ex ample in challeng- ing low-ligh t conditions, where noise t ypically is the limiting facto r for down- stream analysis. Ac knowledgemen ts W e thank Uwe Schmidt, Mar tin W eigert, and T o bias P ietzsch for the helpful discussions. W e thank T obias Pie tzsch for pr o of reading. The computations were per formed on an HPC Cluster at the C e nter for Information Service s and High Performance Computing (ZIH) at TU Dres den. References 1. Batson, J., Roye r, L.: Noise2self: Blind denoising by self-sup ervision. arXiv preprint arXiv:1901.11 365 (2019) 2. 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