Conditional Diffusion Process for Inverse Halftoning
Anonymous Author(s)
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Abstract
Inverse halftoning is a technique used to recover realistic images from ancientprints ( e.g. , photographs, newspapers, books).
The rise of deep learning has ledto the gradual incorporation of neural network designs into inverse halftoningmethods.
Most of existing inverse halftoning methods adopt the U-net architecture,which uses an encoder to encode halftone prints, followed by a decoder for imagereconstruction.
However, the mainstream U-net architecture has poor generalizationability in practical applications.
Specifically, when there is a large gap between thehalftone patterns of the training and test data, the reconstructed continuous-toneimages have obvious artifacts.
This is an important issue in practical applications,since the algorithms for generating halftones are ever-evolving.
Even for the samealgorithm, different parameter choices will result in different halftone ditheringpatterns.
In this paper, we propose the first generative halftoning method inthe literature, which regards the black pixels in halftones as physically movingparticles, and makes the randomly distributed particles move under some certainguidance through the reverse diffusion process, so as to obtain the desired halftonepatterns.
In particular, we propose a Conditional Diffusion model for image
Halftoning (CDH), which consists of a halftone dithering process and an inversehalftoning process.
By changing the initial state of the diffusion model, ourmethod can generate visually plausible halftones with different dithering patternsunder the condition of image gray level and Laplacian prior.
To avoid introducingredundant patterns and undesired artifacts, we propose a meta-halftone guidednetwork to incorporate blue noise guidance in the diffusion process.
In this way,halftone images subject to more diverse distributions are fed into the inversehalftoning model, which helps the model to learn a more robust mapping fromthe halftone distribution to continuous-tone distribution, thereby improving thegeneralization ability to unseen samples.
Quantitative and qualitative experimentalresults demonstrate that the proposed method achieves state-of-the-art results.
Introduction
Halftoning refers to the task of simulating the brightness change of a continuous-tone image bychanging the size or frequency of halftone dots (such as ink dots).
In the last century, halftoningtechnology has been widely used in the printing industry to store precious image data through oldnewspapers, books, photographs, etc .
At the same time, inverse halftoning technology emerged torecover stored continuous-tone image from vintage materials.
The goal of the inverse halftoningtechnique is to minimize the loss of information in the restoration process, so that the restored printhas the highest possible visual quality.
Traditional inverse halftoning methods are mainly based on image filtering ( e.g. , wavelet domainfiltering (Xiong et al., 1999), edge-preserving filtering (Kite et al., 2000), SUSAN filtering (Siddiquiand Bouman, 2007)), bilateral filtering (Sun et al., 2014) and statistical learning methods ( e.g. , least mean square filtering (Chen and Hang, 1997), maximum a posteriori (Stevenson, 1997), look-uptable (Mese and Vaidyanathan, 2001; Chung and Wu, 2005), dictionary learning (Zhang et al., 2018)).
With the revival of deep learning technology, inverse halftoning methods based on deep neuralnetworks have made great progress and attracted more and more attention (Hou and Qiu, 2017; Xiaoet al., 2017; Kim and Park, 2018).
The most representative method is the U-net (Ronneberger et al.
,2015) based architecture, which has an encoder to learn the hidden encoding of the halftone printing,followed by a decoder to reconstruct the image (Xia and Wong, 2018; Gao et al., 2019).
However, the U-net architecture has poor generalization ability in practical applications.
Specifically, when there is a large gap between the halftone patterns of the training and test data, thereconstructed continuous-tone images often have obvious artifacts.
This is an important issue inpractical applications, since the algorithms used for generating halftones evolve with time.
Even forthe same algorithm, different parameter choices will result in different halftone dithering patterns.
Taking frequency modulation (FM) halftoning as an example, we select 9 classical error diffusionprocesses with different dither patterns, namely Floyd-Steinberg Dithering, Jarvis-Judice-Ninke
Dithering, Stucki Dithering, Atkinson Dithering, Burkes Dithering, Sierra Dithering, and several oftheir variants (Lau and Arce, 2018).
We train the U-net model on the images generated by 5 of thealgorithms, and test on the halftone images obtained with the rest 4 algorithms.
The experimentalresults are shown in Figure 1.
Artifacts can be clearly observed in the restored halftones.
To address this problem, in this paper, we propose a Conditional Diffusion model for image Halftoning(CDH), as shown in Figure 2, which consists of a halftone dithering process and an inverse halftoningprocess.
We regard the black pixels in halftones as physically moving particles, and make therandomly distributed particles move under some certain guidance through the reverse diffusionprocess, so as to obtain the desired halftone distribution.
Specifically, for the halftone ditheringprocess, we train a conditional diffusion model to generate halftones with different dithering patternsunder the condition of image gray level and Laplacian prior.
By changing the initial state of thediffusion model, it can simulate different point processes to generate diverse halftone images.
Toavoid introducing redundant patterns and undesired artifacts during halftone generation, we proposea meta-halftone guided network to incorporate the blue noise guidance into the diffusion process.
For inverse halftoning, we train an inverse halftoning diffusion model to learn the mapping functionfrom the halftone distribution to the continuous-tone distribution.
In this way, halftones subject tomore diverse distributions are input to the inverse halftoning model, which helps the model to learn amore robust mapping to perform inverse halftoning and improve the generalization ability to unseensamples.
Our contributions are summarized as follows:
• This is the first work to propose a generative halftoning method, which regards the blackpixels in halftones as physically moving particles, and makes the randomly distributedparticles move under some certain guidance through the reverse diffusion process, so as toobtain the desired halftone dithering patterns.• To avoid introducing redundant patterns and undesired artifacts during halftone generation,we propose a meta-halftone guided network to incorporate the blue noise guidance into thehalftone diffusion process.•
To obtain better generalization ability, we use the x 0 state of halftone dithering diffusion asthe condition for inverse halftoning diffusion, so that the inverse halftoning diffusion modelbenefits from a wider range of dithering patterns and learns a more robust mapping.
We conduct experiments on the dataset consisting of
halftoning algorithms
and quantitative and
qualitative experiments demonstrate that the proposed method achieves state-of-the-art results.
Werelease the code of this paper anonymously to facilitate further research 1 .
Related Work
Inverse Halftoning.
Traditional inverse halftoning methods are mainly based on image filtering ( e.g. ,edge-preserving filtering (Kite et al., 2000), wavelet domain filtering (Xiong et al., 1999), bilateralfiltering (Sun et al., 2014), SUSAN filtering (Siddiqui and Bouman, 2007)) and statistical learningmethods ( e.g. , look-up table (Chung and Wu, 2005; Mese and Vaidyanathan, 2001), least mean squarefiltering (Chen and Hang, 1997), maximum a posteriori (Stevenson, 1997), dictionary learning (Zhanget al., 2018)).
For example, Sun et al.
(Sun et al., 2014) proposed to use an anisotropic Gaussian filterand an edge-preserving filter for inverse halftoning.
With the revival of deep learning technology,some researchers have tried to use deep neural networks to accomplish inverse halftoning, e.g. ,
U-net based models (Hou and Qiu, 2017; Xiao et al., 2017; Gao et al., 2019), residual learningbased models (Xia and Wong, 2018) and contextual learning based models (Kim and Park, 2018).
For example, Xia et al.
(Xia and Wong, 2018) proposed a progressively residual based U-net thatsynthesizes the global tone and subtle details to generate inverse halftones.
However, these encoderdecoder based architectures suffer from unacceptable artifacts in the generated images when facedwith domain generalization problems.
Diffusion Models.
Diffusion models are a class of deep generative models developed from nonequilibrium thermodynamics (Sohl-Dickstein et al., 2015).
They define a Markovian process for thediffusion steps, incrementally adding random noises to the original data, and then learn the reversediffusion process to resample the data from noises (Sohl-Dickstein et al., 2015; Ho et al., 2020;
Luo and Hu, 2021).
Diffusion models are closely related to score-based generative models, whichgenerate samples by Langevin dynamics based on estimated gradients of the data distribution (Songand Ermon, 2019, 2020; Song et al., 2020b, 2021).
Some improved techniques are proposed to helpthe diffusion model converge to a lower negative log-likelihood or to speed up sampling (Nichol and
Dhariwal, 2021; Dhariwal and Nichol, 2021; Song et al., 2020a).
For example, Song et al.
(Songet al., 2020a) generalizes denoising diffusion probabilistic models via a class of non-Markovianprocesses, which can correspond to deterministic generative processes and gives rise to implicitmodels that produce high quality samples much faster.
However, these diffusion models cannot bedirectly applied in halftone generation, since they do not take into account the blue noise propertiesof halftone prints, making it difficult to generate visually pleasing dithering patterns.
Method
3.1 Halftone Dithering Diffusion Conditioned on Gray-scale and
Laplacian
Prior
Traditional halftone dithering methods are mainly based on techniques of amplitude modulation (Blatner et al., 1993;
Campbell et al., 1966) and frequency modulation (Eschbach, 1997; Floyd, 1976;
Me¸se and Vaidyanathan, 1999).
With the development of deep learning, some modern approaches
[Figure] [Figure]
try to generate halftone dithering images using convolutional networks Kim and Park (2018)
; Xiaet al. (2021).
However, these methods are limited to specific dithering patterns, and cannot achieveflexible generative halftones.
To address this problem, considering that the halftone prints are onlycomposed of black and white pixels, we regard the black pixels in halftones as particles, and makethese particles move from the random Gaussian distribution to the halftone distribution under thereverse diffusion process, resulting in a generative halftoning model.
Given the distribution q s 1 ( x s 1 0 ) of halftone prints, the diffusion process q s 1 is a Markovian noisingprocess (Ho et al., 2020) that gradually adds noise to x s 1 0 to obtain x s 1 1: T , where s 1 denotes thehalftone dithering diffusion stage.
Specifically, at each step t , the diffusion step adds the random
Gaussian noise with a β t -controlled variance:
[Equation]
[Equation]
where β t ∈ (0 , 1) , t = 1 , ..., T .
With the reparameterization trick (Kingma and Welling, 2013), wecan sample x s 1 t from any time step t in a closed form: x s 1 t =
√ ¯ α t x
s 1 0 + √ 1 − ¯ α t ϵ , ϵ ∼ N ( 0 , I ) ,that is
[Equation]
where α t = 1 − β t , ¯ α t = (cid:81) ti =
1 α i .
In this way, we can directly derive x
s 1 t by q s 1 ( x s 1 t | x s 1 0 ) withoutrepeatedly applying the Markovian process q s 1 and calculating q s 1 ( x s 1 t | x s 1 t − 1 ) .
In the halftone dithering scenario, we would like to obtain the halftone sample x s 1 0 via the reversediffusion process q s 1 ( x s 1 t − 1 | x s 1 t ) through a random Gaussian distribution, i.e. , the state x s 1 T ∼
N ( 0 , I ) .
However, considering that the particles cannot directly form the halftone distributionsthrough the reverse diffusion process, we use the image gray level r and the Laplacian prior l asconditions to guide the reverse diffusion process.
Formally, the reverse halftoning diffusion processis defined on the distribution p s 1 θ 1 ( x s 1 0: T | r , l ) , which is a Markov chain with start state p s 1 ( x s 1 T ) =
N ( x s 1 T ; 0 , I ) :
[Equation] [Equation]
Different from previous methods like using class labels (Dhariwal and Nichol, 2021) or shapelatents (Luo and Hu, 2021) as conditions for the diffusion model, we use the pixel-wise image priors
[Figure]
[Equation]
[Figure]
[Equation]
[Figure]
[Equation]
Feature
[Figure]
Affine
[Figure]
as conditions.
The conditions are performed in a simple way, i.e. , we concatenate the state vector x s 1 tat diffusion step t with r and l along the channel dimension to provide pixel-wise guidance:
[Equation]
(6) where ⊕ denotes the concatenate operation.
Due to the properties of halftones, local ditheringpatterns are more important than global semantic information for generating high-quality halftones.
We also observe in the experiments that excessive introduction of global information can be harmfulto dithering results.
We would like to learn the model p s 1 θ 1 to approximate the conditional probabilities
q s 1 ( x s 1 t − 1 | x s 1 t ) .
With Bayes theorem, it has
[Equation]
and q s 1 ( x s 1 t − 1 | x s 1 t , x s 1 0 ) can be represented as a Gaussian distribution:
[Equation]
[Equation] [Equation]
The simplified objective function (Ho et al., 2020) is used to train the diffusion model:
[Equation]
3.2 Meta-halftone Guided Network
Blue noise is critical for generating high-quality halftones.
It avoids noticeable low-frequency visualartifacts in the resulting halftones by forcing random pixel dithering (Lau and Arce, 2018).
Sometraditional methods have been proposed to derive blue noise dither patterns, such as simulatedannealing based methods (Sullivan et al., 1991), void-and-cluster techniques (Ulichney, 1993), powerspectrum manipulation algorithms (Yao and Parker, 1994), and dither pattern ordering methods (Lauet al., 1999).
However, most of these approaches are based on statistical methods or manuallydesigned blue-noise dithering matrices, which are difficult to directly apply to the learning-basedneural frameworks.
Some researchers have tried to inject blue noise properties into the L1 norm forneural learning objectives (Xia et al., 2021), however, since the calculation of the L1 norm requiresknown halftone generation results, it cannot be used in generative models.
So far, how to incorporatethe blue-noise dithering properties into halftone generative models remains an unexplored problem.
To address this issue, we propose a meta-halftone guided network (as shown in Figure 3), whichintroduces blue noise guidance into halftone dithering diffusion process and avoids dithering resultscontaining artifacts or redundant patterns.
Formally, for step t of the halftone dithering diffusionprocess, we consider the state vector x s 1 t and take it as input to the meta-halftone guided network.
We first feed x s 1 t into a feature extraction network E to obtain the extracted hidden feature f ( x s 1 t ) ,i.e. , f ( x s 1 t ) =
E ( x s 1 t ) .
We use pre-trained VGG (Simonyan and Zisserman, 2015) as the network
E in the experiment, other networks such as InceptionNet (Szegedy et al., 2016) or ResNet (Heet al., 2016) are also available.
Next, we define the meta-halftone set M , which consists of a set ofmeta-halftone vectors m i (1 ≤ i ≤ k ) :
[Equation]
where k represents the number of meta halftones.
We obtain m i by the halftone dithering diffusionstates of a set of low-frequency images.
Specifically, we construct k images I 1 , I 2 , I 3 ..., I k withrelatively more low-frequency components ( e.g. , images with constant gray levels), and obtainthe halftones H 1 , H 2 , H 3 , ..., H k accordingly using conventional halftone algorithms ( e.g. , FloydSteinberg Algorithm (Lau and Arce, 2018)).
We first train the model for a certain step using the U-netarchitecture alone, which is common in (Ho et al., 2020; Dhariwal and Nichol, 2021), without addingthe meta-halftone guided network.
We have
[Equation]
12) where the superscript h i denotes the constructed halftone data from H i ∈
{ H 1 , H 2 , ..., H k } , andr h i and l h i can be derived from I i ∈ { I 1 , I 2 , ..., I k } .
The superscript s 1 of x h i t − 1 , x h i t are omittedhere for brevity.
We can get ϵ s 1 θ 1 ( √ ¯ α t x h
i 0 + √ 1 − ¯ α t ϵ , t ) by parameterization of µ s 1 θ 1 :
[Equation]
We use m i in Equation 11 to represent the model prediction and construct the meta-halftone set Maccordingly:
[Equation]
Next, the model learns the affine relationship between the hidden feature f ( x s 1 t ) and the meta-halftoneset M through the affine learning layer, and then obtains the affine factors g ( x s 1 t ← i ) :
[Equation]
15) where w g ← i and b g ← i represent learnable weights and biases, respectively.
With the calculatedg ( x s 1 t ← i ) , we perform the depth-wise aggregation of the meta-halftone set M :
[Equation]
Besides depth information, spatial information is also important for the guidance of meta-halftoning.
We perform the spatial-wise aggregation of (cid:102) m and x s 1 t , and denote the result as (cid:102) m ′ :
[Equation]
where ∗ denotes the spatial convolution and d represents the convolution kernel.
The output of themeta-halftone guided network o is determined by the meta-halftone guidance (cid:102) m ′ and x s 1 t :
[Equation] [Equation]
and o is subsequently fed into the traditional U-net architecture as in previous work (Dhariwal and
Nichol, 2021; Nichol and Dhariwal, 2021).
It is worth noting that the proposed meta-halftone guidednetwork has the advantage of being very lightweight, containing only 81 K training parameters.
3.3 Inverse Halftoning Diffusion Conditioned on x s 1 0
The inverse halftoning diffusion process is substantially similar to the halftone dithering diffusionin Section 3.1, except that the diffusion conditions are different.
The goal of the inverse halftoningprocess is to learn a mapping from the halftone distribution to the continuous-tone distribution.
Forthe inverse halftoning diffusion process, we deal with the process
q s 2 ( x s 2 1: T ′ | x s 2 0 ) that gradually addsnoise to the data q s 2 ( x s 2 0 ) from a continuous-tone distribution, and the process q s 2 ( x s 2 t − 1 | x s 2 t , x s 2 0 )of gradually denoising from Gaussian noise to obtain desired samples.
Unlike Section 3.1, we use a model p s 2 θ 2 ( x s 2 t − 1 | x s 2 t , h ) to estimate q s 2 ( x s 2 t − 1 | x s 2 t ) , which is conditionedon the halftone distribution h ( i.e. , sampling x s 1 0 state from halftone dithering diffusion model):
[Equation]
Note that the sampling speed of the halftone dithering diffusion model is relatively fast, usually only∼ 10s per sample (see Section 4.1 for details).
Pixel-wise guidance is adopted to the diffusion statex s 2 t and halftone conditions h :
[Equation]
The rest parts (such as the training objective) are similar to the halftone dithering diffusion process,and we omit them here to save space.
Experiments
4.1 Experimental Setup
Datasets.
In order to better evaluate the generalization ability of different methods, we construct adataset with relatively strong domain shifts.
The domain shifts mainly arise from two aspects: thehalftone algorithm and the image semantic.
We construct the training set and validation set based onthe UTKFace dataset (Zhang et al., 2017) and the test set based on the VOC2012 dataset (Everinghamet al., 2010).
We collect 9 different halftone dithering algorithms, namely Floyd-Steinberg Dithering,
Jarvis-Judice-Ninke Dithering, Stucki Dithering, Atkinson Dithering, Burkes Dithering, Sierra
Dithering, and several of their variants (Lau and Arce, 2018).
We use 5 of them in the training setand validation set, and the remaining 4 algorithms are used as the test set.
There are 7 , 857 images inthe training set, 400 images in the validation set, and 400 images in the test set.
Baselines.
We choose the baselines that are commonly used in inverse halftoning tasks, which includethe U-net based methods, generative adversarial network based methods, and super-resolution basedmethods: PRL (Xia and Wong, 2018) proposes an inverse halftoning network with progressivelyresidual learning, which synthesizes the global tone and subtle details from halftone images in aprogressive manner.
ESRGAN (Wang et al., 2018) is used as baselines in the previous inversehalftoning task, which introduces the residual-in-residual dense block to generate realistic texturesand avoid unpleasant artifacts.
GLEAN (Chan et al., 2021) is applied to the super-resolution task,using pre-trained generative adversarial networks to solve ill-posed problems in image restoration.
We first smooth the halftone images with Gaussian kernels of different sizes, and then use GLEANfor image restoration.
RealESRGAN (Wang et al., 2021) is extended to a practical restorationapplication, which introduces a high-order degradation modeling process to better simulate complexreal-world degradations and employs a U-net discriminator with spectral normalization to increasediscriminator capability.
Evaluation Metrics.
Following previous methods (Xia and Wong, 2018), we use two metrics toevaluate different methods: Peak Signal to Noise Ratio (PSNR) and Structural Similarity (SSIM).
PSNR expresses the ratio between the power of the peak signal and the power of noise, and SSIMevaluates the image quality in terms of luminance, contrast and structure.
Implementation Details.
The image size for halftone dithering and inverse halftoning is 256 × 256 .
The number of channels for the diffusion model is 128 , and we use 3 residual blocks.
The linear noiseschedule is adopted throughout diffusion.
For the halftone dithering diffusion model, the learningrate is set to 0 . 0001 , and k is set to 20 .
We use 800 diffusion steps in the training phase.
During the
[Table]
[Figure]
[Table]
sampling phase, we use strided sampling schedule (Nichol and Dhariwal, 2021) to reduce the reversediffusion process to 80 steps.
For the inverse halftoning diffusion model, we set the learning rate to0 . 0001 and use 4000 steps for the diffusion process.
We run the experiments on A40 GPU clusters.
The time required to train the halftone dithering diffusion model is about 4 hours, and the time totrain the inverse halftoning diffusion model is about 10 hours.
AdamW (Loshchilov and Hutter, 2018)is used to optimize the model.
4.2 Experimental Results
We evaluate the performance of the proposed method CDH and baseline methods in terms of
PSNR and SSIM, and the experimental results are shown in Table 1.
We evaluate the effect ofdifferent Gaussian kernel sizes on the baseline performance, where ESRGAN lkernel , GLEAN lkernel ,
RealESRGAN lkernel denote the use of Gaussian kernels of size 7 , and ESRGAN, GLEAN, RealESRGAN represent the use of Gaussian kernels of size 3 , respectively.
The experimental results showthat our method achieves the best performance on both metrics.
Previous methods such as PRL (Xiaand Wong, 2018), due to the lack of considering the generalization ability of the model on differenthalftoning algorithms, show sub-optimal performance on test data that has severe domain shifts.
Different Gaussian kernel sizes affect the performance of image restoration baselines, such as (Wanget al., 2018; Chan et al., 2021; Wang et al., 2021).
A larger Gaussian kernel erases the dot ditheringpattern in the halftone to a greater extent, resulting in restored images with less halftone artifacts,but also loss of high frequency information.
Our method performs better than the image restorationbaselines with different Gaussian kernel sizes.
To verify the effectiveness of the proposed meta-halftone guided network, we conduct ablationstudies to observe the effect on the generated halftone dithering qualities with and without themeta-halftone guided network.
Experimental results are demonstrated in Figure 4.
From the resultsshown in Figure 4, it can be observed that when the meta-halftone guided network is removed, thegenerated halftones have more obvious dithering artifacts, resulting in poorer visual quality. With the
[Figure]
meta-halftone guided network, the visual quality of the generated halftones is improved and moresimilar to the groundtruth.
The experimental results demonstrate the effectiveness of the proposedmeta-halftone guided network.
4.3 Qualitative Results
To further verify the effectiveness of the proposed method, we also conduct qualitative experimentsto compare the performance of our methods and baselines, which are illustrated in Figure 5.
Fromthe qualitative experimental results, it can be obversed that the continuous-tone images generatedby the baseline method still have some redundant artifacts, which are dot dithering patterns that arenot removed during the inverse halftoning process.
In contrast, our model removes the artifacts wellwithout introducing redundant patterns, validating the effectiveness of the proposed method.
Conclusion
In this work, we propose a generative halftoning method, which regards the black pixels in halftonesas physically moving particles, and makes the randomly distributed particles move under some certainguidance through the reverse diffusion process, so as to obtain the desired halftone dithering patterns.
A meta-halftone guided network is introduced to avoid redundant patterns and undesired artifactsin generated halftones.
By adopting more diverse halftones from halftone dithering diffusion, wefurther improve the generalization ability of inverse halftoning model.
We conduct experimentson the dataset consisting of 9 halftoning algorithms, and quantitative and qualitative experimentsdemonstrate the effectiveness of the proposed method.
Broader Impact
A potential negative side effect of this work is that the sample generation model may be used togenerate fake images in the halftoning or inverse halftoning process for some certain purpose.
Inaddition, most of the images we used in the dataset are collected from the Internet and may containsome bias.
The biases are preserved in the learning of the model and may be reflected in the generatedimage samples.
Limitation Analysis
A limitation of our model is that the diffusion process is relatively slow and thus requires moreinference time than traditional inverse halftoning models, which is limited when the model is appliedto mobile devices.
One way to address this limitation is to design parallel inverse halftoning methods,such as restoring low-frequency and high-frequency components of the image in parallel, therebyincreasing the sampling speed.
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