# perceptual_fairness_in_image_restoration__0689d929.pdf

Perceptual Fairness in Image Restoration

Guy Ohayon Faculty of Computer Science Technion Israel Institute of Technology ohayonguy@cs.technion.ac.il

Michael Elad Faculty of Computer Science Technion Israel Institute of Technology elad@cs.technion.ac.il

Tomer Michaeli Faculty of Electrical and Computer Engineering Technion Israel Institute of Technology tomer.m@ee.technion.ac.il

Fairness in image restoration tasks is the desire to treat different sub-groups of images equally well. Existing definitions of fairness in image restoration are highly restrictive. They consider a reconstruction to be a correct outcome for a group (e.g., women) only if it falls within the group s set of ground truth images (e.g., natural images of women); otherwise, it is considered entirely incorrect. Consequently, such definitions are prone to controversy, as errors in image restoration can manifest in various ways. In this work we offer an alternative approach towards fairness in image restoration, by considering the Group Perceptual Index (GPI), which we define as the statistical distance between the distribution of the group s ground truth images and the distribution of their reconstructions. We assess the fairness of an algorithm by comparing the GPI of different groups, and say that it achieves perfect Perceptual Fairness (PF) if the GPIs of all groups are identical. We motivate and theoretically study our new notion of fairness, draw its connection to previous ones, and demonstrate its utility on state-of-the-art face image restoration algorithms.

1 Introduction

Tremendous efforts have been dedicated to understanding, formalizing, and mitigating fairness issues in various tasks, including classification [17, 22, 29, 81, 94, 95], regression [2, 7, 8, 12, 43, 65], clustering [4 6, 13, 70, 73], recommendation [25, 26, 46, 52, 92], and generative modeling [15, 24, 44, 66, 74, 75, 96]. Fairness definitions remain largely controversial, yet broadly speaking, they typically advocate for independence (or conditional independence) between sensitive attributes (ethnicity, gender, etc.) and the predictions of an algorithm. In classification tasks, for instance, the input data carries sensitive attributes, which are often required to be statistically independent of the predictions (e.g., deciding whether to grant a loan should not be influenced by the applicant s gender). Similarly, in text-to-image generation, fairness often advocates for statistical independence between the sensitive attributes of the generated images and the text instruction used [24]. For instance, the prompt An image of a firefighter should result in images featuring people of various genders, ethnicities, etc.

While fairness is commonly associated with the desire to eliminate the dependencies between sensitive attributes and the predictions, fairness in image restoration tasks (e.g., denoising, super-resolution) has a fundamentally different meaning. In image restoration, both the input and the output carry sensitive attributes, and the goal is to preserve the attributes of different groups equally well [34]. But what exactly constitutes such a preservation of sensitive attributes? Let us denote by x, y, and ˆx the unobserved source image, its degraded version (e.g., noisy, blurry), and the reconstruction of x

38th Conference on Neural Information Processing Systems (Neur IPS 2024).

ˆ X1 ˆ X2 ˆ X3 ˆ X4

p X p Y p ˆ X

p Y |A( |a)

p ˆ X|A( |a)

GPI of a = 0 GPI of a = 1

GPI of a = 0 GPI of a = 1

GPI of a = 0 > GPI of a = 1

GPI of a = 0 > GPI of a = 1

Figure 1: Illustrative example of the proposed notion of Perceptual Fairness (PF). This figure presents four possible restoration algorithms exhibiting different behaviors and fairness performance. In this example, the sensitive attribute A takes the values 0 or 1 with probabilities P(A = 0) < P(A = 1). The distributions p X and p Y correspond to the ground truth signals (e.g., natural images) and their degraded measurements (e.g., noisy images), respectively. The distribution p X|A( |a) corresponds to the ground truth signals associated with the attribute value a, and p Y |A( |a) is the distribution of their degraded measurements. The distribution of all reconstructions is denoted by p ˆ X, and p ˆ X|A( |a) is the distribution of the reconstructions associated with attribute value a. The Group Perceptual Index (GPI) of the group associated with a is defined as the statistical distance between p ˆ X|A( |a) and p X|A( |a), and good PF is achieved when the GPIs of all groups are (roughly) similar. For example, ˆ X1 achieves good PF since the GPIs of both a = 0 and a = 1 are roughly equal, while ˆ X3 achieves poor PF since the GPI of a = 0 is worse (larger) than that of a = 1. See Section 2 for more details.

from y, respectively. Additionally, let Xa denote the set of images x carrying the sensitive attributes a. Jalal et al. [34] deem the reconstruction of any x Xa as correct only if ˆx Xa. This allows practitioners to evaluate fairness in an intuitive way, by classifying the reconstructed images produced for different groups. For instance, regarding x, y, and ˆx as realizations of random vectors X, Y , and ˆX, respectively, Representation Demographic Parity (RDP) states that P( ˆX Xa|X Xa) should be the same for all a, and Proportional Representation (PR) states that P( ˆX Xa) = P(X Xa) should hold for every a. However, the idea that a reconstructed image ˆx can either be an entirely correct output (ˆx Xa) or an entirely incorrect output (ˆx / Xa) is highly limiting, as errors in image restoration can manifest in many different ways. Indeed, what if one algorithm always produces blank images given inputs from a specific group, and another algorithm produces images that are almost in Xa for such inputs (e.g., each output is only close to some image in Xa)? Should both algorithms be considered equally (and completely) erroneous for that group? Furthermore, quantities of the form P( ˆX Xa| ) completely neglect the distribution of the images within Xa. For example, assuming the groups are women and non-women, an algorithm that always outputs the same image of a woman when the source image is a woman, but produces diverse non-women images when the source is not a woman, still satisfies RDP. Does this algorithm truly treat women fairly?

To address these controversies, we propose to examine how the restoration method affects the distribution of each group of interest (e.g., the distribution of images of women or non-women). Specifically, we define the Group Perceptual Index (GPI) to be the statistical distance (e.g., Wasserstein) between the distribution of the group s ground truth images and the distribution of their reconstructions. We then associate Perceptual Fairness (PF) with the degree to which the GPIs of the different groups are close to one another. In other words, the PF of an algorithm corresponds to the parity among the GPIs of the groups of interest (see Figure 1 for intuition). The rationale behind using such an index is two-fold. First, it solves the aforementioned controversies. For example, an algorithm that always outputs the same image of a woman when the source image is a woman, and diverse non-women images otherwise, would achieve poor GPI for women and good GPI for non-women, thus resulting in poor PF. Second, the GPI reflects the ability of humans to distinguish between samples of a group s ground truth images and samples of the reconstructions obtained from the degraded images of that group [10]. Thus, achieving good PF (i.e., parity in the GPIs) suggests that this ability is the same for all groups.

This paper is structured as follows. In Section 2 we formulate the image restoration task and present the mathematical notations necessary for this paper. This includes a review of prior fairness definitions in image restoration, alongside our proposed definition. We also discuss why PF can be considered as a generalization of RDP. In Section 3 we present our theoretical findings. For instance, we prove that

x y DDRM DDNM+ Pi GDM DPS

DPS Pi GDM DDNM DDRM

Young & Asian Young & non-Asian Old & Asian Old & non-Asian

DPS Pi GDM DDNM DDRM

Figure 2: Examining fairness in face image super-resolution techniques through the lens of RDP [34] or PF (our proposed notion of fairness). Both RDP and PF assess how well an algorithm treats different fairness groups. Specifically, RDP evaluates the parity in the GP of different groups (higher GP is better), and PF evaluates the parity in the GPI of different groups (lower GPI is better). The results show that the groups old&Asian and old&non-Asian attain similar treatment according to RDP (similar GP scores that are roughly zero), while the latter group attains better treatment according to PF. In Section 4 and Appendix G.7, we show why this outcome of PF is the desired one.

achieving perfect GPI for all groups simultaneously is not feasible when the degradation is sufficiently severe. We also establish an interesting (and perhaps counter-intuitive) relationship between the GPI of different groups for algorithms attaining a perfect Perceptual Index (PI) [10], and show that PF and the PI are often at odds with each other. In Section 4 we demonstrate the practical advantages of PF over RDP. In particular, we show that PF detects bias in cases where RDP fails to do so. Lastly, in Section 5 we discuss the limitations of this work and propose ideas for the future.

2 Problem formulation and preliminaries

We adopt the Bayesian perspective of inverse problems, where an image x is regarded as a realization of a random vector X with probability density function p X. Consequently, an input y is a realization of a random vector Y (e.g., a noisy version of X), which is related to X via the conditional probability density function p Y |X. The task of an estimator ˆX (in this paper, an image restoration algorithm) is to estimate X only from Y , such that X Y ˆX is a Markov chain (X and ˆX are statistically independent given Y ). Given an input y, the estimator ˆX generates outputs according to the conditional density p ˆ X|Y ( |y).

2.1 Perceptual index

A common way to evaluate the quality of images produced by an image restoration algorithm is to assess the ability of humans to distinguish between samples of ground truth images and samples of the algorithm s outputs. This is typically done by conducting experiments where human observers vote on whether the generated images are real or fake [18, 20, 28, 32, 33, 72, 101, 102]. Importantly, this ability can be quantified by the Perceptual Index [10], which is the statistical distance between the distribution of the source images and the distribution of the reconstructed ones,

PId := d(p X, p ˆ X), (1)

where d( , ) is some divergence between distributions (Kullback Leibler divergence, total variation distance, Wassersterin distance, etc.).

2.2 Fairness

2.2.1 Previous notions of fairness

Jalal et al. [34] introduced three pioneering notions of fairness for image restoration algorithms: Representation Demographic Parity (RDP), Proportional Representation (PR), and Conditional Proportional Representation (CPR). Formally, given a collection of sets of images {Xai}k i=1, where ai is a vector of sensitive attributes and each Xai represents the group carrying the sensitive attributes ai, these notions are defined by

RDP: P( ˆX Xai|X Xai) = P( ˆX Xaj|X Xaj) for every i, j; (2)

PR: P( ˆX Xai) = P(X Xai) for every i; (3)

CPR: P( ˆX Xai|Y = y) = P(X Xai|Y = y) for every i, y. (4)

While such definitions are intuitive and practically appealing, they have several limitations. First, any reconstruction that falls even slightly off the set Xai is considered an entirely wrong outcome for its corresponding group. In other words, reconstructions with minor errors are treated the same as completely wrong ones. Second, these definitions neglect the distribution of the groups images. Consequently, an algorithm can satisfy RDP, PR, CPR, etc., while treating some groups much worse than others in terms of the statistics of the reconstructed images. For instance, consider dogs and cats as the two fairness groups. Let Xdogs and Xcats be the sets of images of dogs and cats, respectively, and let xdog Xdogs be a particular image of a dog. Furthermore, suppose that the species can be perfectly identified from any degraded measurement, i.e.,

P(X Xdogs|Y = y) = 1 or P(X Xcats|Y = y) = 1 (5)

for every y. Now, suppose that ˆX always produces the image xdog from any degraded dog image, while generating diverse, high-quality cat images from any degraded cat image. Namely, for every y, we have

1 = P( ˆX = xdog|X Xdogs) = P( ˆX Xdogs|X Xdogs) = P( ˆX Xcats|X Xcats), (6)

P( ˆX = xdog|Y = y) = P( ˆX Xdogs|Y = y) = P(X = Xdogs|Y = y), (7)

P( ˆX Xcats|Y = y) = P(X = Xcats|Y = y). (8)

Although this algorithm satisfies RDP (Equation (6)) and CPR (Equations (7) and (8)), which entails PR [34], it is clearly useless for dogs. Should such an algorithm really be deemed as fair, then?

To address such controversies, we propose to represent each group by the distribution of their images, and measure the representation error of a group by the extent to which an algorithm preserves such a distribution. This requires a more general formulation of fairness groups, which is provided next.

2.2.2 Rethinking fairness groups

We denote by A (a random vector) the sensitive attributes of the degraded measurement Y , so that p Y |A( |a) is the distribution of degraded images associated with the attributes A = a (e.g., the distribution of noisy women images). Consequently, the distribution of the ground truth images that possess the sensitive attributes a is given by p X|A( |a), and the distribution of their reconstructions is

given by p ˆ X|A( |a). Moreover, we assume that A Y ˆX forms a Markov chain, implying that knowing A does not affect the reconstructions when Y is given. This assumption is not limiting, since image restoration algorithms are mostly designed to estimate X solely from Y , without taking the sensitive attributes as an additional input. See Figure 1 for an illustrative example of the proposed formulation.

Note that such a formulation is quite general, as it does not make any assumptions regarding the nature of the image distributions, whether they have overlapping supports or not, etc. Our formulation also generalizes the previous notion of fairness groups, which considers only the support of p X|A( |a) for every a. Indeed, one can think of Xa = supp p X|A( |a) as the set of images corresponding to some group, and of {Xa}a supp p A as the collection of all sets. Furthermore, notice that A can also be the degraded measurement itself, i.e. A = Y . In this case, p X|A( |a) = p X|Y ( |a) is the posterior distribution of ground truth images given the measurement a, and p ˆ X|A( |a) = p ˆ X|Y ( |a) is the distribution of the reconstructions of the measurement a. Namely, our mathematical formulation is adaptive to the granularity of fairness groups considered.

2.2.3 Perceptual fairness

We define the fairness of an image restoration algorithm as its ability to equally preserve the distribution p X|A( |a) across all possible values of a. Formally, we measure the extent to which an algorithm ˆX preserves this distribution by the Group Perceptual Index, defined as GPId(a) := d(p X|A( |a), p ˆ X|A( |a)), (9)

where d( , ) is some divergence between distributions. Then, we say that ˆX achieves perfect Perceptual Fairness with respect to d, or perfect PFd in short, if GPId(a1) = GPId(a2) (10) for every a1, a2 supp p A (see Figure 1 to gain intuition). In practice, algorithms may rarely achieve exactly perfect PFd, while the GPId of different groups may still be roughly equal. In such cases, we say that ˆX achieves good PFd. In contrast, if there exists at least one group that attains far worse GPId than some other group, we say that ˆX achieves poor/bad PFd. Importantly, note that achieving good PFd does not necessarily indicate good PId and/or good GPId values.

2.2.4 Group Precision, Group Recall, and connection to RDP

In addition to the PId defined in (1), the performance of image restoration algorithms is often measured via the following complementary measures [45, 71]: (1) Precision, which is the probability that a sample from p ˆ X falls within the support of p X, P( ˆX supp p X), and (2) Recall, which is the probability that a sample from p X falls within the support of p ˆ X, P(X supp p ˆ X). Achieving low precision implies that the reconstructed images may not always appear as valid samples from p X. Thus, precision reflects the perceptual quality of the reconstructed images. Achieving low recall implies that some portions of the support of p X may never be generated as outputs by ˆX. Hence, recall reflects the perceptual variation of the reconstructed images.

Since here we are interested in the perceptual quality and the perceptual variation of a group s reconstructions, let us define the Group Precision and the Group Recall by

GP(a) := P( ˆX Xa|A = a), (11)

GR(a) := P(X ˆ Xa|A = a), (12) where Xa = supp p X|A( |a) and ˆ Xa = supp p ˆ X|A( |a). Hence, when adopting our formulation of fairness groups, satisfying RDP simply means that the GP values of all groups are the same. However, as hinted in previous sections, two groups with similar GP values may still differ significantly in their GR. From the following theorem, we conclude that attaining perfect PFd TV, where d TV(p, q) = 1

2 R |p(x) q(x)|dx is the total variation distance between distributions, guarantees that both the GP and the GR of all groups have a common lower bound. This implies that PFd TV can be considered as a generalization of RDP. Theorem 1. The Group Precision and Group Recall of any restoration method satisfy GP(a) 1 GPId TV(a), (13) GR(a) 1 GPId TV(a), (14) for all a supp p A.

Although using d TV( , ) provides a straightforward relationship between PFd TV and RDP, other types of divergences may not necessarily indicate GP and GR so explicitly. The perceptual quality & variation of a group s reconstructions may be defined in many different ways [71], and the GPI implicitly entangles these two desired properties.

The mathematical notations and fairness definitions are summarized in Appendix A. To further develop our understanding of PF, the next section presents several introductory theorems.

3 Theoretical results

Image restoration algorithms can generally be categorized into three groups: (1) Algorithms targeting the best possible average distortion (e.g., good PSNR) [3, 21, 47, 48, 83, 85, 97 100], (2) algorithms that strive to achieve good average distortion but prioritize attaining best PI [1, 19, 27, 47, 61, 80, 83 85, 88, 89, 93, 100, 104], and (3) algorithms attempting to sample from the posterior distribution p X|Y of the given task at hand [16, 40 42, 51, 58, 76, 86, 91]. In Appendix B, we demonstrate on a simple toy example that all these types of algorithms may achieve poor PF, implying that perfect PF is not a property that can be obtained trivially. Namely, even when using common reconstruction algorithms such as the Minimum Mean-Squared-Error (MMSE) estimator or the posterior sampler, one group may attain far worse GPI than another group. It is therefore tempting to ask in which scenarios there exists an algorithm capable of achieving perfect GPI for all groups simultaneously. As stated in the following theorem, this desired property is unattainable when the degradation is sufficiently severe.

Theorem 2. Suppose that a1, a2 supp p A such that

P(X Xa1 Xa2|A = ai) < P(Y Ya1 Ya2|A = ai), (15)

for both i = 1, 2, where Xai = supp p X|A( |ai) and Yai = supp p Y |A( |ai). Then, GPId(a1) and GPId(a2) cannot both be equal to zero.

In words, Theorem 2 states that when the degraded images of different groups are more overlapping than their ground truth images, at least one group must have sub-optimal GPI. Importantly, note that perfect GPI can always be achieved for some group corresponding to A = a individually, by ignoring the input and sampling from p X|A( |a). Hence, Theorem 2 implies that, for sufficiently severe degradations, one may attempt to approach zero GPI for all groups simultaneously, until the GPI of one group hinders that of another one. But what about algorithms that just attain perfect overall PI? Can such algorithms also attain perfect PF? As stated in the following theorem, it turns out that these two desired properties (perfect PI and perfect PF) are often incompatible.

Theorem 3. Suppose that A takes discrete values, ˆX attains perfect PId (p ˆ X = p X), and a, am supp p A such that GPId(a) > 0 and P(A = am) > 0.5. Then, ˆX cannot achieve perfect PFd TV.

In words, when there exists a majority group in the data distribution, Theorem 3 states that an algorithm with perfect PI, whose GPI is not perfect even for only one group, cannot achieve perfect PFd TV. This intriguing outcome results from the following convenient relationship between the GPIs of different groups for algorithms with perfect PI.

Theorem 4. Suppose that A takes discrete values and ˆX attains perfect PId (p ˆ X = p X). Then,

GPId TV(a) 1 P(A = a)

a =a P(A = a )GPId TV(a ) (16)

for every a with P(A = a) > 0.

This theorem is, perhaps, counter-intuitive. Indeed, for algorithms with perfect PI, improving the GPId TV of one group can only improve the GPId TV of other groups, and this is true even if the groups do not overlap1. While this may seem contradictory to Theorem 2, note that such a relationship holds until the algorithm can no longer attain perfect PI. The example in Appendix B demonstrates this theorem.

1Two groups with attributes a1, a2 are overlapping if P(X Xa1 Xa2) > 0, where Xai = supp p X|A( |ai).

4 Experiments

We demonstrate the superiority of PF over RDP in detecting fairness bias in face image superresolution. Our analysis considers various aspects, including different types of degradations, and fairness evaluations across four groups categorized by ethnicity and age. First, we show that RDP incorrectly attributes fairness in a simple scenario where fairness is clearly violated. In contrast, PF successfully detects the bias. Second, we showcase a scenario where PF uncovers potential malicious intent. Specifically, it can detect bias injected into the system via adversarial attacks, a situation again missed by RDP.

4.1 Synthetic data sets

In the following sections we assess the fairness of leading face image restoration methods through the lens of PF and RDP. Such methods are often trained and evaluated on high-quality, aligned face image datasets like Celeb A-HQ [36] and FFHQ [37], which lack ground truth labels for sensitive attributes such as ethnicity. Moreover, these datasets are prone to inherent biases, e.g., they contain very few images for certain demographic groups [31, 35, 69], and it is unclear whether images from different groups have similar levels of image quality and variation (prior work suggests that they might not [11]). To address these limitations, we leverage an image-to-image translation model that takes a text instruction as additional input. This model allows us to generate four synthetic fairness groups with high-quality, aligned face images. Specifically, we translate each image x from the Celeb A-HQ [36] test partition into four different images representing Asian/non-Asian and young/old individuals2. We use a unique text instruction for each translation. For example, the text instruction 120 years old human, Asian, natural image, sharp, DSLR translates x into an image of an old&Asian individual. Finally, we include each resulting image in its corresponding group data only if all translations are successful according to the Fair Face combined age & ethnicity classifier [35]. This involves classifying the ethnicity and age of the translated images and ensuring that old individuals are categorized as 70+ years old, young individuals are categorized as any other age group, Asian individuals are classified as either Southeast or East Asian, and non-Asian individuals are classified as belonging to any other ethnicity group. See Appendix G.1 for more details and for the visualization of the results.

Disclaimer. Importantly, we note that the generated synthetic data sets may impose offensive biases and stereotypes. We use such data sets solely to investigate the fairness of image restoration methods and verify the practical utility of our work. We do not intend to discriminate against any identity group or cultures in any way.

4.2 Perceptual Fairness vs. Representation Demographic Parity

We consider several image super-resolution tasks using the average-pooling down-sampling operator with scale factors s {4, 8, 16, 32}, and statistically independent additive white Gaussian noise of standard deviation σN {0, 0.1, 0.25}. In Appendix I we also conduct experiments on image denoising and deblurring. The algorithms DDNM+ [86], DDRM [42], DPS [16], and Pi GDM [76] are evaluated on all scale factors, and GFPGAN [84], VQFR [27], GPEN [93], Diff BIR [49], Code Former [104], Restore Former++ [89], and Restore Former [88] are evaluated only on the 4 and 8 scale factors (these algorithms produce completely wrong outputs for the other scale factors). To assess the PF of each algorithm, we compute the GPIKID of each group using the Kernel Inception Distance (KID) [9] and the features extracted from the last pooling layer of the Fair Face combined age & ethnicity classifier [35]. In Appendix G.4 we utilize the Fréchet Inception Distance (FID) [30] instead of KID, and in Appendix G.5 we assess other types of group metrics such as PSNR. Additionally, we provide in Appendix G.6 an ablation study of alternative feature extractors. To assess RDP, we use the same Fair Face classifier to compute the GP of each group. As done in [34], we approximate the GP of each group by the classification hit rate, which is the ratio between the number of the group s reconstructions that are classified as belonging to the group and the total number of the group s inputs. Qualitative and quantitative results for s = 32, σN = 0.0 are presented in Figure 2.

2We choose to consider these fairness groups since image restoration algorithms are likely biased towards young and white demographics, given the overrepresentation of such groups in common training datasets (e.g., FFHQ, Celeb A). Namely, groups of Asian and/or old individuals are typically underrepresented in such datasets.

0.0 0.5 1.0

Young Asian

Young Non-Asian

Old Non-Asian

s = 4, σN = 0.0

0.0 0.5 1.0

s = 8, σN = 0.0

0.0 0.5 1.0

s = 16, σN = 0.0

0.0 0.5 1.0

s = 32, σN = 0.0

Young Asian

Young Non-Asian

Old Non-Asian

Restore Former + + Code Former

DiffBIR GFPGAN

GPEN Restore Former

Figure 3: Comparison of the GP and the GPIKID of different fairness groups, using various state-of-the-art face image super-resolution methods. In most experiments, GPIKID suggests a fairness discrepancy between the groups old&non-Asian and old&Asian, while the GP of these groups is roughly equal.

Quantitative results for all values of s and σN = 0.0 are shown in Figure 3. Complementary details and results are provided in Appendix G.

Figure 3 shows that the group young&non-Asian receives the best overall treatment in terms of both GP and GPIKID. This result is not surprising, since the training data sets of the evaluated algorithms (e.g., FFHQ) are known to be biased towards young and white demographics [50, 63]. However, while most algorithms appear to treat the groups old&Asian and old&non-Asian quite similarly in terms of GP, the GPIKID indicates a clear disadvantage for the former group. Indeed, by examining ethnicity and age separately using the Fair Face classifier, we show in Appendix G.7 that, according to RDP, the group old&non-Asian exhibits better preservation of the ethnicity attribute compared to the group old&Asian, while the age attribute remains equally preserved for both groups. This highlights that RDP is strongly dependent on the granularity of the fairness groups (as suggested in [34]), since slightly altering the groups partitioning may completely obscure the fact that an algorithm treats certain attributes more favorably than others. However, as our results show, this issue is alleviated when adopting GPIKID instead of GP. Namely, the ethnicity bias is still detected by comparing the GPIKID of different groups, even though the fairness groups are partitioned based on age and ethnicity combined.

4.3 Adversarial bias detection

In Section 2.2.1 we discussed the limitations of fairness definitions such as RDP. For instance, an algorithm might satisfy RDP by always generating the same output for degraded images of a particular group, even if it produces perfect results for another. However, such an extreme scenario is not common in practice. Indeed, real-world imaging systems often involve degradations that are not too severe, and well-trained algorithms perform impressively well when applied to different groups (see, e.g., Figure 4b). So what practical advantage does PF have over RDP in such circumstances? Here, we demonstrate that a malicious user can manipulate the facial features (e.g., wrinkles) of a group s reconstructions without violating fairness according to RDP, but violating fairness according to PF. In particular, we consider only the ethnicity sensitive attribute by taking the young&Asian group as Asian, and the young&non-Asian group as non-Asian. Then, we use the Restore Former++ method, which roughly satisfies RDP with respect to these groups (see Figure 4a, where GP is evaluated by

0.00 0.25 0.50 0.75 1.00

GP(non-Asian)

GPIKID(Asian)

GPIKID(non-Asian)

No adv. attacks Adv. attacks on Asian Adv. attacks on non-Asian

(a) Restore Former++ achieves roughly similar GP for both groups (i.e., roughly satisfies RDP). The adversarial attacks on the inputs from the non Asian group remain undetected by GP, while they highly affect the GPI.

y y Adv y y Adv y y Adv y y Adv

(b) Visual results. y and x are the original input and the source image, respectively. y Adv and ˆx Adv are the adversarial input and its corresponding output, respectively. Each y Adv successfully alters the output facial features. Indeed, ˆx Adv clearly contains a face with more wrinkles than x.

Figure 4: Using adversarial attacks to inject bias into the outputs of Restore Former++, in a setting where it (roughly) satisfies RDP. Such attacks are detected by PF but not by RDP.

classifying ethnicity alone), and perform adversarial attacks on the inputs of each group to manipulate the outputs such that they are classified as belonging to the 70+ age category. The fact that the GP of each group is quite large implies that the malicious user can classify ethnicity quite accurately from the degraded images, and then manipulate the inputs only for the group it wishes to harm (we skip such a classification step and simply attack all of the group s inputs). Such attacks are anticipated to succeed due to the perception-robustness tradeoff [59, 60]. Complementary details of this experiment are provided in Appendix H.

In Figure 4, we present both quantitative and qualitative results demonstrating that the attacks on the non-Asian group are not detected by RDP. However, we clearly observe that these attacks are successfully identified by the GPIKID of each group. This again highlights that PF is less sensitive to the choice (partitioning) of fairness groups compared to RDP. Specifically, age must be considered as a sensitive attribute to detect such a bias via RDP. Yet, even then, the malicious user may still inject other types of biases. Conversely, PF does not suffer from this limitation, as any attempt to manipulate the distribution of a group s reconstructions would be reflected in the group s GPI.

5 Discussion

Different demographic groups can utilize an image restoration algorithm, and fairness in this context asserts whether the algorithm treats all groups equally well. In this paper, we introduce the notion of Perceptual Fairness (PF) to assess whether such a desired property is upheld. We delve into the theoretical foundation of PF, demonstrate its practical utility, and discuss its superiority over existing fairness definitions. Still, our work is not without limitations. First, while PF alleviates the strong dependence of RDP on the choice of fairness groups [34] (as demonstrated in Section 4), it still cannot guarantee fairness for any arbitrary group partitioning simultaneously (a property referred to as obliviousness in [34]). Second, our current theorems are preliminary, requiring further research to fully understand the nature of PF. For example, the severity of the tradeoff between the GPI scores of different groups (Theorem 2) and that of the tradeoff between PF and PI (Theorem 3) remain unclear. Third, we do not address the nature of optimal estimators that achieve good or perfect PF. What is their best possible distortion (e.g., MSE) and best possible PI? Fourth, on the practical side, we show in Appendix G.6 that effectively evaluating PF using metrics such as KID necessitates utilizing image

features extracted from a classifier dedicated to handling the considered sensitive attributes (e.g., an age and ethnicity classifier). However, this is not a disadvantage compared to previous fairness notions (RDP, CPR and PR), which also require such a classifier. Lastly, while the proposed GPI may be suitable for evaluating fairness in general-content natural images, we considered only human face images due to their societal implications, namely since fairness issues are particularly critical when dealing with such images. For example, if a general-content image restoration algorithm performs better on images with complex structures than on images of clear skies, this discrepancy is unlikely to be problematic for practitioners, as long as the algorithm attains good performance overall. Moreover, previous works [34] evaluated fairness with respect to non-human subjects (e.g., dogs and cats), but these studies provide limited insights into human-related fairness issues, which often arise due to subtle differences between images (e.g., wrinkles). Expanding our method to other datasets remains an avenue for future work.

6 Societal impact

Designing fair and unbiased image restoration algorithms is critical for various AI applications and downstream tasks that rely on them, such as facial recognition, image classification, and image editing. By proposing practically useful and well-justified fairness definitions, we can detect (and mitigate) bias in these tasks, ultimately leading to fairer societal outcomes. This fosters increased trust and adoption of AI technology, contributing to a more equitable and responsible use of AI in society.

Acknowledgments

This research was partially supported by the Israel Science Foundation (ISF) under Grant 2318/22 and by the Council For Higher Education - Planning & Budgeting Committee.

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Name / Notation Meaning / Formal definition

X Ground truth image (a random vector) Y Degraded measurement (a random vector) ˆX Reconstructed image (a random vector) p X P.d.f of the ground truth images p Y P.d.f of the degraded measurements p ˆ X P.d.f of the reconstructed images Perceptual Index (PId or PI) d(p X, p ˆ X) A Sensitive attribute (a random vector) p X|A( |a) P.d.f of the ground truth images of A = a p Y |A( |a) P.d.f of the degraded measurements of A = a p ˆ X|A( |a) P.d.f of the reconstructed images of A = a Xa supp p X|A( |a) Ya supp p Y |A( |a) ˆ Xa supp p ˆ X|A( |a) Group Perceptual Index (GPId(a), GPId, or GPI) d(p X|A( |a), p ˆ X|A( |a)) Group Precision (GP(a) or GP) P( ˆX Xa|A = a) Group Recall (GR(a) or GR) P(X ˆ Xa|A = a) Representation Demographic Parity (RDP) a1, a2 : GP(a1) = GP(a2) Proportional Representation (PR) a : P(X Xa) = P( ˆX Xa) Conditional Proportional Representation (CPR) a, y : P(X Xa|Y = y) = P( ˆX Xa|Y = y) Perceptual Fairness (PFd or PF) a1, a2 : GPId(a1) = GPId(a2)

Table 1: Summary of mathematical notations and fairness definitions used in this paper.

A Summary of mathematical notations and fairness definitions

We summarize in Table 1 the mathematical notations and fairness definitions used in this paper.

B Toy signal restoration example

The following toy signal restoration example demonstrates that common estimators (e.g., the stochastic estimator which samples from the posterior distribution p X|Y ) do not trivially achieve perfect PF.

0.0 0.5 1.0

GPId TV (1)

GPId TV (0)

ˆ XPosterior

0.0 0.5 1.0

ˆ XPosterior

p ˆ XMSE|A( |a)

p ˆ XPosterior|A( |a)

p ˆ XMSE+PI|A( |a)

Figure 5: Illustration of Example 1. Left: Conditional probability density functions p X|A( |a), p ˆ XMSE|A( |a), p ˆ XPosterior|A( |a), and p ˆ XMSE+PI|A( |a), where a = 1 (left plot) or a = 0 (right plot). Right: The GPId TV and GPIW1 of each group (associated with a = 1 or a = 0). The dotted lines PFd TV or PFW1 correspond to the points where perfect PFd TV or perfect PFW1 is achieved, respectively. It is clear that all three estimators achieve sub-optimal PFd TV and sub-optimal PFW1. See Appendix B for more details.

Example 1. Suppose that X, N N(0, 1) are statistically independent random variables, and let Y = X + N. In this case, it is known that ˆXMSE = 1

2Y is the estimator that attains the lowest possible Mean-Squared-Error (MSE), ˆXPosterior = 1

2Y + W where W N(0, 1

2) is statistically independent of X and Y , is the estimator that samples from the posterior distribution p X|Y , and ˆXMSE+PI = 1

is the estimator that attains the lowest possible MSE among all estimators that satisfy p ˆ X = p X (perfect PId) [10, 23]. Now, consider the sensitive attribute A = 1X 1. All of these commonly used estimators produce much better (lower) GPId TV and GPIW1 for the group associated with A = 0, which, in this case, is a majority satisfying P(A = 0) 0.8413 (see Figure 5).

B.1 Conditional density plots

The density p X|A(x|a) is obtained using the closed form solution of a truncated normal distribution,

p X|A(x|1) = ϕ(x) Φ( ) Φ(1), (17)

p X|A(x|0) = ϕ(x) Φ(1) Φ( ), (18)

where ϕ(x) is a normal density and Φ(x) is its cumulative distribution,

and p X|A(x|1) = 0 and p X|A(x|0) = 0 for every x 1 and x 1, respectively. The densities p ˆ XMSE|A( |a), p ˆ XMSE+PQ|A( |a) and p ˆ XPosterior|A( |a) are obtained by feeding these algorithms with the degraded measurements corresponding to X 1 (for a = 1) and to X < 1 (for a = 0), separately. This is achieved by generating samples x p X and y p Y |X( |x), and then partitioning these samples into two sets of measurements based on the value of x. We then perform Kernel Density Estimation (KDE) [64] on the reconstructions of each group to obtain their density, using the function seaborn.kdeplot [90] with the arguments bw_adjust=2, common_norm=False, gridsize=200. The number of samples used to compute the KDE is set to 200,000 for both a = 1 and a = 0.

B.2 Computation of the total variation distance d TV and of the Wasserstein distance W1

The value of GPId TV(a) for a given algorithm ˆX is defined by the total variation distance

GPId TV(a) = d TV(p X|A( |a), p ˆ X|A( |a)) = 1

Z p X|A(x|a) p ˆ X|A(x|a) dx. (21)

To compute this integral, we use the function scipy.integrate.quad [82] with parameters (a=-1000, b=1000, limit=500, points=[1.0]). At each point x, the integrand p X|A(x|a) p ˆ X|A(x|a) (22)

is evaluated using the closed form solution of p X|A( |a) and the pre-computed KDE density of each p ˆ X|A( |a).

The value of GPIW1(a) for a given algorithm ˆX is the Wasserstein 1-distance between p X|A( |a) and p ˆ X|A( |a). To approximate this distance, we utilize the function scipy.stats.wasserstein_distance with the previously obtained 200,000 samples from p X|A( |a) and 200,000 samples from p ˆ X|A( |a).

C Proof of Theorem 1

Theorem 1. The Group Precision and Group Recall of any restoration method satisfy

GP(a) 1 GPId TV(a), (13) GR(a) 1 GPId TV(a), (14)

for all a supp p A.

Proof. For every a, x, it holds that

p ˆ X|A(x|a) min n p X|A(x|a), p ˆ X|A(x|a) o . (23)

Moreover, the value of min n p X|A(x|a), p ˆ X|A(x|a) o is zero for every x / supp p X|A( |a), so Z

supp p X|A( |a) min n p X|A(x|a), p ˆ X|A(x|a) o dx = Z min n p X|A(x|a), p ˆ X|A(x|a) o dx. (24)

Thus, GP(a) = P( ˆX Xa|A = a) (25)

= P( ˆX supp p X|A( |a)|A = a) (26)

supp p X|A( |a) p ˆ X|A(x|a)dx (27)

supp p X|A( |a) min n p X|A(x|a), p ˆ X|A(x|a) o dx (28)

= Z min n p X|A(x|a), p ˆ X|A(x|a) o dx (29)

p ˆ X|A(x|a) + p X|A(x|a) p ˆ X|A(x|a) p X|A(x|a) dx (30)

Z p ˆ X|A(x|a) + p X|A(x|a) dx 1

Z p ˆ X|A(x|a) p X|A(x|a) dx (31)

= 1 d TV(p X|A( |a), p ˆ X|A( |a)) (32)

= 1 GPId TV(a). (33) By replacing the roles of p ˆ X|A(x|a) and p X|A(x|a), the result GR(a) 1 GPId TV(a) can be derived with identical steps using the same mathematical arguments.

D Proof of Theorem 2

Theorem 2. Suppose that a1, a2 supp p A such that P(X Xa1 Xa2|A = ai) < P(Y Ya1 Ya2|A = ai), (15) for both i = 1, 2, where Xai = supp p X|A( |ai) and Yai = supp p Y |A( |ai). Then, GPId(a1) and GPId(a2) cannot both be equal to zero.

Proof. Suppose by contradiction that p ˆ X|A( |ai) = p X|A( |ai) for both i = 1, 2. Thus,

1 = P(X Xai|A = ai) (34)

= P( ˆX Xai|A = ai) (35)

Xai p ˆ X|A(x|ai)dx (36)

Xai p ˆ X,Y |A(x|ai)dxdy (37)

Xai p ˆ X|A,Y (x|ai)p Y |A(y|ai)dxdy (38)

Xai p ˆ X|Y (x|y)p Y |A(y|ai)dxdy (39)

Yai p Y |A(y|ai)

Xai p ˆ X|Y (x|y)dx

Yai p Y |A(y|ai)P( ˆX Xai|Y = y)dy, (41)

where Equation (39) holds from the assumption that A and ˆX are statistically independent given Y , and from the fact that p Y |A(y|ai) = 0 for every y / Yai. We will show that P( ˆX Xai|Y = y) = 1 for almost every y Yai. Indeed, if this does not hold, then for some Ti Yai with P(Y Ti|A = ai) > 0 we have P( ˆX Xai|Y = y) < 1 for every y Ti. Thus,

Yai p Y |A(y|ai)P( ˆX Xai|Y = y)dy (42)

Yai\Ti p Y |A(y|ai)P( ˆX Xai|Y = y)dy + Z

Ti p Y |A(y|ai)P( ˆX Xai|Y = y)dy (43)

Yai\Ti p Y |A(y|ai)P( ˆX Xai|Y = y)dy + Z

Ti p Y |A(y|ai)dy (44)

Yai\Ti p Y |A(y|ai)dy + Z

Ti p Y |A(y|ai)dy (45)

Yai p Y |A(y|ai)dy (46)

which is not possible. So, P( ˆX Xai|Y = y) = 1 for almost every y Yai. Now, from basic rules of probability theory, we have

P(X Xa1 Xa2|A = a1) =P(X Xa1|A = a1) + P(X Xa2|A = a1) P(X Xa1 Xa2|A = a1), (48)

where the first and last terms on the right hand side cancel out (from the definition of Xa1, they are both equal to 1). Thus, we have

P(X Xa1 Xa2|A = a1) = P(X Xa2|A = a1), (49)

and finally,

P(X Xa1 Xa2|A = a1) = P(X Xa2|A = a1) (50)

= P( ˆX Xa2|A = a1) (51)

Ya1 p Y |A(y|a1)P( ˆX Xa2|Y = y)dy (52)

Ya1 Ya2 p Y |A(y|a1)P( ˆX Xa2|Y = y)dy (53)

Ya1 Ya2 p Y |A(y|a1)dy (54)

= P(Y Ya1 Ya2|A = a1), (55)

where Equation (51) follows from the contradictory assumption that p ˆ X|A( |ai) = p X|A( |ai), Equation (52) follows from the same steps that led to Equation (41), and Equation (54) follows from our previous finding that P( ˆX Xai|Y = y) = 1 for every y Yai (we have y Ya1 Ya2 in the integrand, so y Ya2). However, it is given that P(X Xa1 Xa2|A = a1) < P(Y Ya1 Ya2|A = a1), so we have established a contradiction.

E Proof of Theorem 3

Theorem 3. Suppose that A takes discrete values, ˆX attains perfect PId (p ˆ X = p X), and a, am supp p A such that GPId(a) > 0 and P(A = am) > 0.5. Then, ˆX cannot achieve perfect PFd TV.

Proof. Suppose that GPId TV(am) = 0. From the assumptions, there exists a = am such that GPId(a) > 0, so GPId TV(a) > 0. This means that PFd TV is not perfect.

Otherwise, suppose that GPId TV(am) > 0. Thus, from Theorem 4 we have

GPId TV(am) 1 P(A = am)

P(A = am) max a =am GPId TV(a ) (56)

< max a =am GPId TV(a) (57)

= GPId TV(a ), (58)

where Equation (57) holds since 1 P(A=am)

P(A=am) < 1, and Equation (58) holds by defining

a = arg max a =a GPId TV(a ). (59)

Thus, we have found two groups am and a such that GPId TV(am) < GPId TV(a ), so PFd TV cannot be perfect.

F Proof of Theorem 4

Theorem 4. Suppose that A takes discrete values and ˆX attains perfect PId (p ˆ X = p X). Then,

GPId TV(a) 1 P(A = a)

a =a P(A = a )GPId TV(a ) (16)

for every a with P(A = a) > 0.

Proof. For every a, let us denote Pa = P(A = a). Suppose that ˆX attains perfect perceptual index, so p ˆ X = p X. From the marginalization of probability density functions, it holds that

a Pap X|A(x|a), (60)

p ˆ X(x) = X

a Pap ˆ X|A(x|a), (61)

and since p ˆ X = p X we have X

a Pap X|A(x|a) = X

a Pap ˆ X|A(x|a). (62)

Let a be some group with Pa > 0. By rearranging Equation (62) we get

Pa(p X|A(x|a) p ˆ X|A(x|a)) = X

a =a Pa (p ˆ X|A(x|a ) p X|A(x|a )). (63)

Taking the absolute value on both sides, we have

Pa p X|A(x|a) p ˆ X|A(x|a) =

a =a Pa (p ˆ X|A(x|a ) p X|A(x|a ))

a =a Pa p ˆ X|A(x|a ) p X|A(x|a )) , (65)

where Equation (65) follows from the triangle inequality. Thus, it holds that

d TV(p X|A( |a), p ˆ X|A( |a)) = 1

Z p X|A(x|a) p ˆ X|A(x|a) dx

a =a Pa p ˆ X|A(x|a ) p X|A(x|a ) dx (66)

Z p ˆ X|A(x|a ) p X|A(x|a ) dx (67)

a =a Pa d TV(p X|A( |a ), p ˆ X|A( |a )). (68)

This concludes the proof.

G Face image super-resolution - complementary details and results

G.1 Synthetic data sets

All the Celeb A-HQ images we use are of size 512 512. The image-to-image translation model we utilize, stabilityai/stable-diffusion-xl-refiner-1.0, is sourced from Hugging Face [77] and boasts over 1,200,000 downloads (at the time writing this paper). This model integrates SDXL [67] with SDEdit [53]. For all groups, we adjust the hyperparameters strength and guidance_scale from their default settings, with strength set to 0.4. When translating a Celeb AHQ image x into a group image using its specified text instruction (see Table 2), we choose the smallest value from [8.5, 9.5, 10.5, 11.5, 12.5] as the guidance_scale hyperparameter, such that the resulting image is classified as belonging to the group. Otherwise, if none of these guidance_scale values work for some group (i.e., their class is incorrect), we discard all the translations of x from all groups. To clarify, this means that the translated images for different groups may use different guidance_scale values, as long as all translations are correctly classified. The text instructions we use for each group are provided in Table 2. For all groups, we use the same negative_prompt text instruction ugly, deformed, fake, caricature . Each of the resulting groups contains 1,356 images of size 512 512. In Figures 17 to 20 we present 130 image samples from each group.

Group Image-to-image translation text instruction

Old&Asian 120 years old human, Asian, natural image, sharp, DSLR Young&Asian 20 years old human, Asian, natural image, sharp, DSLR Old&non-Asian 120 years old human, natural image, sharp, DSLR Young&non-Asian 20 years old human, natural image, sharp, DSLR

Table 2: Text instructions for the image-to-image translation model to generate images of each fairness group. See Section 4 and Appendix G.1 for more details.

G.2 Visual results

Visual results of all algorithms (the reconstructions of each fairness group) for s {4, 8, 16, 32} and σN {0, 0.1} are provided in Figures 21 to 28.

G.3 Additional levels of additive noise

Figure 3 presents quantitative results with all scaling factors, and without adding white Gaussian noise (σN = 0). Here, in Figures 6 and 7 we report the results with σN {0.1, 0.25}. We observe similar trends and conclusions as in Figure 3 (please refer to Section 4.2 for more details).

G.4 Comparing GPIFID instead of GPIKID

We report in Figures 8 to 10 the GPIFID of each group, where FID is the Fréchet Inception Distance [30]. These results show trends similar to those observed in Figure 3. Namely, using the statistical distance FID instead of KID does not alter the trends and conclusions of the results.

G.5 Additional group metrics

We report, compare and analyze additional group performance metrics.

GPNN and GRNN We approximate the GP and GR of each group using [45], a method which evaluates the precision and recall between two distributions in their feature space. We denote the results by GPNN and GRNN, respectively. Note that this approach to approximate GP differs from our previous experiments, where we use the classification hit rate (Figures 3, 6 and 7). Similarly to the experiments where we compute GPIKID (Section 4.2) and GPIFID (Appendix G.4), GPNN and GRNN are computed by extracting image features using the last average pooling layer of the Fair Face combined age & ethnicity classifier [35].

GPSNR and GLPIPS For each group we compute the Peak Signal-to-Noise Ratio (PSNR) and the Learned Perceptual Image Patch Similarly (LPIPS) [103]3, where these metrics are evaluated by feeding the restoration algorithm only with the group s inputs and with respect to the group s ground truth images. Formally, we define the Group PSNR (GPSNR) and the Group LPIPS (GLPIPS) as

GPSNR(a) = E[PSNR(X, ˆX)|A = a], (69)

GLPIPS(a) = E[LPIPS(X, ˆX)|A = a], (70)

where the expectation is taken over the joint distribution of a group s ground truth images and their reconstructions, p X, ˆ X|A( , |a).

The results for all noise levels σN {0.0, 0.1, 0.25} are provided in Figures 8 to 10. First, note that both the GPSNR and the GLPIPS metrics are unreliable indicators of bias. For example, the metrics GP, GPNN GPIKID, and GPIFID all indicate that the group young&non-Asian receives better treatment than the group young&Asian (e.g., the GP of the former group is clearly higher than that of the latter group across all noise levels and scaling factors). However, both groups exhibit roughly similar GPSNR and GLPIPS scores. This highlights why assessing the fairness of image restoration algorithms solely based on GPSNR, GLPIPS or similar metrics (MSE, SSIM [87], etc.) might not be sufficient. This result regarding GPSNR is not surprising, as it is well known that such a metric often does not correlate with perceived image quality [10]. Regarding GLPIPS, it might be more effective to use image features extracted by a classifier trained to identify the sensitive attributes in question. We leave exploring this option for future work. Second, the GPNN values in Figures 8 to 10 are almost identical to the GP scores reported in Figures 3, 6 and 7. This suggests that approximating the true GP either through the classification hit rate (as in Figures 3, 6 and 7) or via [45] (as done in this section), are consistent. Third, the GRNN scores suggest potential unfairness in the perceptual variation across different groups. For example, when s = 16, σN = 0, we observe that all algorithms consistently produce higher GRNN scores for the young&non-Asian group compared to the young&Asian group.

G.6 Feature extractors ablation

We employ the dinov2-vit-g-14 [62], clip-vit-l-14 [68], and inception-v3-compat [78] feature extractors via torch-fidelity [57] to compute the GPIKID for each fairness group (previously, we used the image features extracted from the Fair Face classifier s final average pooling layer). The results are presented in Figures 11 to 13.

The outcomes from both the dinov2-vit-g-14 and clip-vit-l-14 feature extractors generally align with those of the Fair Face image classifier, though the biases exposed by these extractors are less pronounced. Put differently, computing GPIKID with either of these general-purpose feature extractors leads to a smaller disparity in the GPIKID of the different fairness groups. Moreover, the inception-v3-compat image feature extractor yields inconsistent results, suggesting that the old&Asian group receives more favorable treatment compared to the old&non-Asian group (contrary to the biases indicated by the other feature extractors). The following section strengthens our argument that this behavior of inception-v3-compat is undesirable. Overall, relying on such general-purpose image feature extractors seems unsatisfactory for the purpose of uncovering nuanced biases in face image restoration methods.

G.7 Considering age and ethnicity as separate sensitive attributes

In Section 4.2 we reveal a significant discrepancy between PF and RDP regarding whether the groups old&Asian and old&non-Asian are treated equally. Specifically, both groups achieve similar GP, while the GPIKID of the latter group (old&non-Asian) is notably better (lower) than that of the former group (old&Asian). In other words, GPIKID indicates that the old&non-Asian group enjoys a better preservation of ethnicity.

Let us support our claim in Section 4.2 that this outcome of PF is the desired one, by showing that RDP may obscure the fact that some sensitive attributes are treated better than others. Indeed, as shown in Figure 14, the ethnicity of the old&non-Asian group is better preserved than that of the old&Asian group, while Figure 15 confirms that the age of these two groups is equally preserved.

3Future work may investigate the utility of no-reference perceptual quality measures (e.g., [54, 55, 79]) to assess fairness in image restoration.

While RDP fails to uncover this ethnicity bias when the fairness groups are determined based on both age and ethnicity, PF clearly reveals it.

G.8 Final details

All algorithms are evaluated using the official codes and checkpoints provided by their authors. We use the torch-fidelity package [57] (Git Hub commit a61422f) to compute the KID [9], FID [30], precision and recall [45]. The GPSNR and the GLPIPS are computed using the piq package [38, 39] (version 0.8.0 in pip).

Finally, note that some of the evaluated algorithms generate output images of size 256 256 (e.g., DDNM), while others produce images of size 512 512 (e.g., Restore Former). Consequently, for fair quantitative evaluations, we resize the outputs of the latter algorithms, along with the ground truth images, to 256 256. To clarify, the super-resolution scaling factors are calculated based on the 256 256 image size. For instance, when s = 4, the resolution of the input images is 64 64.

H Adversarial attacks - complementary details

The degradation we apply consists of three consecutive steps: (1) Average pooling down-sampling with a scale factor of s = 4, (2) additive white Gaussian noise with a standard deviation of σN = 0.1, and then (3) JPEG compression with a quality factor of 50. We attack each degraded image using a tweaked version of the I-FGSM basic attack [14] with α = 6/255 and T = 200. In particular, instead of using the L2 loss in I-FGSM like in [14], we forward each attacked output through a classifier that predicts the age category of the output face image [56], and then maximize the log-probability of the oldest age group category. In other words, we forward each degraded image through Restore Former++ and then feed the result to the age classifier. We then use the I-FGSM update rule to maximize the soft-max probability of the oldest age category (this adversarial attack technique was employed in [60]).

I Additional experiments on image denoising and deblurring

We conduct additional experiments on image denoising and deblurring to further demonstrate the utility of the proposed notion of perceptual fairness. Specifically, for image denoising we use additive white Gaussian noise with standard deviation σN = 0.5, and for image deblurring we use a Gaussian blur kernel of size k = 5 and σ = 10, and add to the blurred image a white Gaussian noise of standard deviation σN {0.1, 0.25, 0.5}. Since these degradations are not handled well by the GAN-based methods, we only compare DPS, DDNM+, DDRM, and Pi GDM.

Quantitative results are reported in Figure 16, and visual comparisons are provided in Figures 29 to 32. As in the super-resolution experiments, PF is able to expose bias (which is also visually clear) when RDP fails to do so, and not vice versa.

J Computational resources

All our experiments are conducted on a NVIDIA RTX A6000 GPU.

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Figure 6: Experiments similar to Figure 3, but when the standard deviation of the additive white Gaussian noise is σN = 0.1.

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Figure 7: Experiments similar to Figure 3, but when the standard deviation of the additive white Gaussian noise is σN = 0.25.

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Figure 8: Evaluation of additional group metrics where the additive noise level is σN = 0.0 and the superresolution scaling factor is s {4, 8, 16, 32}. Please refer to Appendix G for more details.

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Figure 9: Evaluation of additional group metrics where the additive noise level is σN = 0.1 and the superresolution scaling factor is s {4, 8, 16, 32}. Please refer to Appendix G for more details.

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Figure 10: Evaluation of additional group metrics where the additive noise level is σN = 0.25 and the superresolution scaling factor is s {4, 8, 16, 32}. Please refer to Appendix G for more details.

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Figure 11: Using the dinov2-vit-g-14 feature extractor [62] via torch-fidelity [57] to compute the GPIKID of each group. This general-purpose feature extractor network is somewhat able to detect bias between the old&Asian and old&non-Asian (as detected before by extracting features from the Fair Face image classifier). However, the bias is significantly less pronounced in this case.

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Figure 12: Using the clip-vit-l-14 feature extractor [68] via torch-fidelity [57] to compute the GPIKID of each group. Even this general purpose feature extractor network is somewhat able to detect some bias between the old&Asian and old&non-Asian (as detected before by extracting features from the Fair Face image classifier). However, the bias is significantly less pronounced in this case.

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Figure 13: Using the inception-v3-compat feature extractor [78] via torch-fidelity [57] to compute the GPIKID of each group. These results of inception-v3-compat hint that the old&Asian group in some cases receive better treatment than the old&non-Asian group, while all the other feature extractors suggest the opposite bias. This outcome inception-v3-compat is also inconsistent with the experiments in Appendix G.7, which demonstrate that the old&non-Asian group is the one receiving the better treatment.

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Ethnicity is the only considered sensitive attribute

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Figure 14: Evaluating the GP of each group, where ethnicity is the only considered sensitive attribute. Here, the groups old&Asian and young&Asian are each considered as Asian, and the groups old&non-Asian and young&non-Asian are each considered as non-Asian. For clarity, we still specify in each bar plot the corresponding age of each group, but the classifier operates solely on ethnicity (i.e., the GP is approximated with respect to ethnicity alone). As we claim in Section 4.2, the ethnicity of the old&non-Asian group is clearly preserved better than that of the old&Asian group.

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Age is the only considered sensitive attribute

Restore Former + + Code Former

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Figure 15: Evaluating the GP of each group, where age is the only considered sensitive attribute. Here, the groups old&Asian and old&non-Asian are each considered as old, and the groups young&Asian and young&non Asian are each considered as young. For clarity, we still specify in each bar plot the corresponding ethnicity of each group, but the classifier operates solely on age (i.e., the GP is approximated with respect to age alone). As we claim in Section 4.2, the age of both the old&non-Asian and old&Asian groups is (roughly) equally preserved.

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DPS Pi GDM DDNM DDRM

Figure 16: Experiments similar to Figure 3, but on the image denoising and deblurring tasks described in Appendix I. We observe similar trends in these tasks as well. Namely, as in the super-resolution tasks, PF exposes a clear bias when RDP does not (but not vice versa).

Figure 17: Examples of generated images for the old&Asian user group. These samples were generated by passing images from the Celeb A-HQ test partition [36] through the SDXL image-to-image model. The text instruction used was 120 years old human, Asian, natural image, sharp, DSLR . The Fair Face ethnicity and age classifier [35] categorizes all of these images as belonging to either the Southeast Asian or East Asian ethnicities, and to the 70+ age group.

Figure 18: Examples of generated images for the young&Asian user group. These samples were generated by passing images from the Celeb A-HQ test partition [36] through the SDXL image-to-image model. The text instruction used was 20 years old human, Asian, natural image, sharp, DSLR . The Fair Face ethnicity and age classifier [35] categorizes all of these images as belonging to either the Southeast Asian or East Asian ethnicities, and to any age group younger than 70 years old.

Figure 19: Examples of generated images for the old&non-Asian user group. These samples were generated by passing images from the Celeb A-HQ test partition [36] through the SDXL image-to-image model. The text instruction used was 120 years old human, natural image, sharp, DSLR . The Fair Face ethnicity and age classifier [35] categorizes all of these images as belonging to ethnicities other than Southeast Asian or East Asian, and to the 70+ age group.

Figure 20: Examples of generated images for the young&non-Asian user group. These samples were generated by passing images from the Celeb A-HQ test partition [36] through the SDXL image-to-image model. The text instruction used was 20 years old human, natural image, sharp, DSLR . The Fair Face ethnicity and age classifier [35] categorizes all of these images as belonging to ethnicities other than Southeast Asian or East Asian, and to any age group younger than 70 years old.

Young & non-Asian

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Visual results for s = 4, σN = 0.0

Figure 21: Face image super-resolution for each fairness group, where s = 4, σN = 0. (0) DDRM, (1) VQFR, (2) Code Former, (3) DDNM+, (4) Restore Former + +, (5) GPEN, (6) DPS, (7) GFPGAN, (8) Pi GDM, (9) Restore Former, (10) Diff BIR. Zoom in for best view.

Young & non-Asian

Old & non-Asian

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Visual results for s = 4, σN = 0.1

Figure 22: Face image super-resolution for each fairness group, where s = 4, σN = 0.1. (0) DDRM, (1) VQFR, (2) Code Former, (3) DDNM+, (4) Restore Former + +, (5) GPEN, (6) DPS, (7) GFPGAN, (8) Pi GDM, (9) Restore Former, (10) Diff BIR. Zoom in for best view.

Young & non-Asian

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Visual results for s = 8, σN = 0.0

Figure 23: Face image super-resolution for each fairness group, where s = 8, σN = 0. (0) DDRM, (1) VQFR, (2) Code Former, (3) DDNM+, (4) Restore Former + +, (5) GPEN, (6) DPS, (7) GFPGAN, (8) Pi GDM, (9) Restore Former, (10) Diff BIR. Zoom in for best view.

Young & non-Asian

Old & non-Asian

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Visual results for s = 8, σN = 0.1

Figure 24: Face image super-resolution for each fairness group, where s = 8, σN = 0.1. (0) DDRM, (1) VQFR, (2) Code Former, (3) DDNM+, (4) Restore Former + +, (5) GPEN, (6) DPS, (7) GFPGAN, (8) Pi GDM, (9) Restore Former, (10) Diff BIR. Zoom in for best view.

Young & non-Asian

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Visual results for s = 16, σN = 0.0

Figure 25: Face image super-resolution for each fairness group, where s = 16, σN = 0. (0) DDRM, (1) DDNM+, (2) DPS, (3) Pi GDM. Zoom in for best view.

Young & non-Asian

Old & non-Asian

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Visual results for s = 16, σN = 0.1

Figure 26: Face image super-resolution for each fairness group, where s = 16, σN = 0.1. (0) DDRM, (1) DDNM+, (2) DPS, (3) Pi GDM. Zoom in for best view.

Young & non-Asian

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Visual results for s = 32, σN = 0.0

Figure 27: Face image super-resolution for each fairness group, where s = 32, σN = 0. (0) DDRM, (1) DDNM+, (2) DPS, (3) Pi GDM. Zoom in for best view.

Young & non-Asian

Old & non-Asian

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Visual results for s = 32, σN = 0.1

Figure 28: Face image super-resolution for each fairness group, where s = 32, σN = 0.1. (0) DDRM, (1) DDNM+, (2) DPS, (3) Pi GDM. Zoom in for best view.

Young & non-Asian

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Visual results for denoising, σN = 0.5

Figure 29: Face image denoising for each fairness group, where σN = 0.5. (0) DDRM, (1) DDNM+, (2) DPS, (3) Pi GDM. Zoom in for best view.

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Visual results for deblurring, σN = 0.1

Figure 30: Face image deblurring for each fairness group, where σN = 0.1. (0) DDRM, (1) DDNM+, (2) DPS, (3) Pi GDM. Zoom in for best view.

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Visual results for deblurring, σN = 0.25

Figure 31: Face image deblurring for each fairness group, where σN = 0.25. (0) DDRM, (1) DDNM+, (2) DPS, (3) Pi GDM. Zoom in for best view.

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Visual results for deblurring, σN = 0.5

Figure 32: Face image deblurring for each fairness group, where σN = 0.5. (0) DDRM, (1) DDNM+, (2) DPS, (3) Pi GDM. Zoom in for best view.

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Guidelines:

The answer NA means that the authors have not reviewed the Neur IPS Code of Ethics. If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).

10. Broader Impacts Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [Yes]

Justification: We dedicate Section 6 to discuss the societal impacts of our paper.

Guidelines:

The answer NA means that there is no societal impact of the work performed. If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).

11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [NA]

Justification: We do not release data or models. The paper poses no such risks.

Guidelines:

The answer NA means that the paper poses no such risks. Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.

12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [Yes]

Justification: We cite the use of publicly available datasets and conform to their license.

Guidelines:

The answer NA means that the paper does not use existing assets. The authors should cite the original paper that produced the code package or dataset. The authors should state which version of the asset is used and, if possible, include a URL. The name of the license (e.g., CC-BY 4.0) should be included for each asset.

For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. If this information is not available online, the authors are encouraged to reach out to the asset s creators.

13. New Assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

Answer: [NA]

Justification: The paper does not release new assets.

Guidelines:

The answer NA means that the paper does not release new assets. Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc. The paper should discuss whether and how consent was obtained from people whose asset is used. At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.

14. Crowdsourcing and Research with Human Subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

Answer: [NA]

Justification: The paper does not involve crowdsourcing nor research with human subjects.

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper. According to the Neur IPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.

15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects

Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?

Answer: [NA]

Justification: The paper does not involve crowdsourcing nor research with human subjects.

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper. We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the Neur IPS Code of Ethics and the guidelines for their institution. For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.