### Introduction

The Haar wavelet-based perceptual similarity index (HaarPSI) is a similarity measure for images that aims to correctly assess the perceptual similarity between two images with respect to a human viewer.

In most practical situations, images and videos can neither be compressed nor transmitted without introducing distortions that will eventually be perceived by a human observer. Vice versa, most applications of image and video restoration techniques, such as inpainting or denoising, aim to enhance the quality of experience of human viewers. Correctly predicting the similarity of an image with an undistorted reference image, as subjectively experienced by a human viewer, can thus lead to significant improvements in any transmission, compression, or restoration system.

The HaarPSI has the following advantages over previous full reference quality metrics such as (MS-)SSIM, FSIM, PSNR, GSM, or VIF:

• It achieves higher correlations with human opinion scores on large benchmark databases in almost every case (see experimental results).
• It can be computed very efficiently and significantly faster than most other metrics (see Table II in our paper).

The HaarPSI can also be found on github.

If you use the HaarPSI in your research, please cite the following paper:
• R. Reisenhofer, S. Bosse, G. Kutyniok and T. Wiegand.
A Haar Wavelet-Based Perceptual Similarity Index for Image Quality Assessment. (PDF)
Signal Processing: Image Communication, vol. 61, 33-43, 2018.
doi:10.1016/j.image.2017.11.001

#### Usage

In MATLAB, the HaarPSI can be computed by passing two images of the same size to HaarPSI.m.

HaarPSI.m accepts both RGB and grayscale images as input. Please make sure that the values are given in the $$[0,255]$$ interval!

Please note that by default, both images are being preprocessed by the convolution with a $$2\times2$$ mean filter as well as a subsequent dyadic subsampling step, to simulate the typical distance between an image and its viewer. To omit this preprocessing step, you can pass 0 as a third argument to HaarPSI, as shown in the example below.

%load images
imgDist = imgRef + randn(size(imgRef))*25;
imgDist = min(max(imgDist,0),255);

%compute HaarPSI
haarPsi = HaarPSI(imgRef,imgDist);

%omit preprocessing step
haarPsiWithoutPreprocessing = HaarPSI(imgRef,imgDist,0);


### The Haar Wavelet-Based Perceptual Similarity Index

The HaarPSI expresses the perceptual similarity of two digital images in the interval $$[0,1]$$, that is $$\operatorname{HaarPSI}\colon \ell^2({\mathbb{Z}^2})\times\ell^2({\mathbb{Z}^2}) \rightarrow [0,1],$$ such that the HaarPSI of two identical images will be exactly one and the HaarPSI of two completely different images will be close to zero.

The first step in the computation of the HaarPSI is the construction of similarity maps based on local features obtained for both images.

To assess the similarity of local features $$a, b \in \mathbb{R}$$, a simple similarity measure for scalar values that already appeared in [2] is used: $$\operatorname{S}(a,b,C) = \frac{2ab + C}{a^2 + b^2 + C},$$ with a constant $$C > 0$$.

In the HaarPSI, the features of two grayscale images $$f_{1},f_{2}\in\ell^2(\mathbb{Z}^2)$$ used to construct local similarity maps are based on the coefficients of a discrete wavelet transform. The wavelet chosen for HaarPSI is the so-called Haar wavelet, which was already proposed in 1910 by Alfred Haar [1] and is arguably the simplest and computationally most efficient wavelet there is. The one-dimensional Haar filters are given by $$h^{\text{1D}}_1 = \frac{1}{\sqrt{2}}\cdot[1,1] \text{ and } g^{\text{1D}}_1 = \frac{1}{\sqrt{2}}\cdot[-1,1],$$ where $$h^{\text{1D}}_1$$ denotes the low-pass scaling filter and $$g^{\text{1D}}_1$$ the corresponding high-pass wavelet filter.

For any scale $$j\in\mathbb{N}$$, we can construct two-dimensional Haar filters by setting \begin{align*} g^{\text{(1)}}_j &= g^{\text{1D}}_j \otimes h^{\text{1D}}_j,\\ g^{\text{(2)}}_j &= h^{\text{1D}}_j \otimes g^{\text{1D}}_j, \end{align*} where $$\otimes$$ denotes the outer product and the one-dimensional filters $$h^{\text{1D}}_j$$ and $$g^{\text{1D}}_j$$ are given for $$j>1$$ by \begin{align*} g^{\text{1D}}_j &= h^{\text{1D}}_{1}*(g^{\text{1D}}_{j-1})_{\uparrow 2}, \\ h^{\text{1D}}_j &= h^{\text{1D}}_{1}*(h^{\text{1D}}_{j-1})_{\uparrow 2}, \end{align*} where $$\uparrow2$$ is the dyadic upsampling operator, and $$*$$ denotes the one-dimensional convolution operator. Note that $$g^{\text{(1)}}_j$$ responds to horizontal structures, while $$g^{\text{(2)}}_j$$ picks up vertical structures.

To correctly predict the perceptual similarity experienced by human viewers, it can be useful to apply an additional non-linear mapping to the local similarities obtained from high-frequency Haar wavelet filter responses. This non-linearity is chosen to be the logistic function, which is widely used as an activation function in neural networks for modeling thresholding in biological neurons and is given for a parameter $$\alpha >0$$ as $$l_\alpha(x) = \frac{1}{1 + e^{-\alpha x}}.$$

For two grayscale images $$f_1,f_2\in\ell^2(\mathbb{Z}^2)$$, we define a local similarity measure based on the first two stages of a two-dimensional discrete Haar wavelet transform, namely $$\operatorname{HS^\text{(k)}_{f_1,f_2}}[x]=l_\alpha\left(\frac{1}{2}\sum_{j= 1}^2\operatorname{S}\left(\vert(g^\text{(k)}_j*f_1)[x]\vert,\vert(g^\text{(k)}_j*f_2)[x]\vert,C\right)\right),$$ where $$C > 0$$ is a constant, $$k\in\{1,2\}$$ selects either horizontal or vertical filters, $$\operatorname{S}$$ denotes the scalar similarity measure, and $$*$$ is the two-dimensional convolution operator.

To define a single similarity index for two images, all values from the previously defined similarity maps are combined by computing their weighted mean. The corresponding weights are based on the third scale of a discrete Haar wavelet transform and given by $$\operatorname{W^\text{(k)}_{f}}[x] = \left\vert(g^{\text{(k)}}_3*f)[x]\right\vert,$$ where $$k\in\{1,2\}$$ again differentiates between horizontal and vertical filters. Note that this weight function can be seen as an analog to phase congruency measure [10] used for a similar purpose in the definition of FSIM [3].

The Haar wavelet-based perceptual similarity index for two grayscale images $$f_1,f_2$$ is finally given by $$\operatorname{HaarPSI_{f_1,f_2}} = l_\alpha^{-1}\left(\frac{\sum\limits_x \sum\limits_{k=1}^2\operatorname{HS^\text{(k)}_{f_1,f_2}}[x] \cdot \operatorname{W^\text{(k)}_{f_1,f_2}}[x]}{\sum\limits_x \sum\limits_{k=1}^2\operatorname{W^\text{(k)}_{f_1,f_2}}[x]}\right)^2,$$ with $$\operatorname{W^\text{(k)}_{f_1,f_2}}[x] = \max(\operatorname{W^\text{(k)}_{f_1}}[x],\operatorname{W^\text{(k)}_{f_2}}[x])$$.

The HaarPSI can be extended to color images in the YIQ color space by including the chroma channels I and Q in the local similarity measure. This generalization is given by $$\operatorname{HaarPSIC_{f_1,f_2}} = l_\alpha^{-1}\left(\frac{\sum\limits_x \sum\limits_{k=1}^3\operatorname{HS^\text{(k)}_{f_1,f_2}}[x] \cdot \operatorname{W^\text{(k)}_{f^\text{Y}_1,f^\text{Y}_2}}[x]}{\sum\limits_x \sum\limits_{k=1}^3\operatorname{W^\text{(k)}_{f^\text{Y}_1,f^\text{Y}_2}}[x]}\right)^2,$$ with $$\operatorname{HS^\text{(1)}_{f_1,f_2}}$$ and $$\operatorname{HS^\text{(2)}_{f_1,f_2}}$$ as in the definition of $$\operatorname{HaarPSI_{f_1,f_2}}$$ and a chroma-sensitive local similarity measure $$\operatorname{HS^\text{(3)}_{f_1,f_2}}[x]=l_\alpha\left(\frac{1}{2}\left(\operatorname{S}\left(\vert(m*f^{\text{I}}_1)[x]\vert,\vert(m*f^{\text{I}}_2)[x]\vert,C\right) + \operatorname{S}(\vert(m*f^{\text{Q}}_1)[x]\vert,\vert(m*f^{\text{Q}}_2)[x]\vert,C)\right)\right)$$ with a $$2\times2$$ mean filter $$m$$ and $$\operatorname{W^\text{(3)}_{f^\text{Y}_1,f^\text{Y}_2}}[x] = \frac{1}{2}\left(\operatorname{W^\text{(1)}_{f^\text{Y}_1,f^\text{Y}_2}}[x]+\operatorname{W^\text{(2)}_{f^\text{Y}_1,f^\text{Y}_2}}.[x]\right).$$

The HaarPSI only requires two parameters to be selected, namely $$C$$ and $$\alpha$$. These parameters were optimized to yield a superior overall performance on all benchmark databases on which the HaarPSI was evaluated. The parameters were eventually chosen to be $$C = 30$$, $$\alpha = 4.2$$. It should be noted that for other databases or specific applications, different values might still be favorable (see for example Fig. 4 in our paper).

### Experimental Results

The consistency of HaarPSI with the human perception of image quality was evaluated and compared with most state-of-the-art image similarity measures via four large publicly available benchmark databases of quality-annotated images (LIVE [11], TID 2008 [12], TID 2013 [13] and CSIQ [5]).

The following tables depict the Spearman rank order correlation coefficients (SROCC) with the human opinion scores for all four databases and ten different image similarity measures. The highest correlation in each row is written in boldface.

 SROCC for Grayscale Images PSNR VIF [6] SSIM [2] MS-SSIM [9] GSM [7] MAD [5] SR-SIM [8] FSIM [3] VSI [4] HaarPSI LIVE 0.8756 0.9636 0.9479 0.9513 0.9561 0.9672 0.9619 0.9634 0.9534 0.9690 TID 2008 0.5531 0.7491 0.7749 0.8542 0.8504 0.8340 0.8913 0.8804 0.8830 0.9043 TID 2013 0.6394 0.6769 0.7417 0.7859 0.7946 0.7807 0.8075 0.8022 0.8048 0.8094 CSIQ 0.8058 0.9195 0.8756 0.9133 0.9108 0.9466 0.9319 0.9242 0.9372 0.9546
 SROCC for Color Images PSNR VIF [6] SSIM [2] MS-SSIM [9] GSM [7] MAD [5] SR-SIM [8] FSIM [3] VSI [4] HaarPSI LIVE 0.8756 0.9636 0.9479 0.9513 0.9561 0.9672 0.9619 0.9645 0.9524 0.9683 TID 2008 0.5531 0.7491 0.7749 0.8542 0.8504 0.8340 0.8913 0.8840 0.8979 0.9097 TID 2013 0.6394 0.6769 0.7417 0.7859 0.7946 0.7807 0.8075 0.8510 0.8965 0.8732 CSIQ 0.8058 0.9195 0.8756 0.9133 0.9108 0.9466 0.9319 0.9310 0.9423 0.9604

Detailed tables reporting Spearman and Kendall rank order correlations as well as Pearson correlations before and after non-linear regression can be found here. These tables also include an analysis of statistical significance and distortion specific results for all four databases.

#### References

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### Contact

Feel free to direct any questions or comments regarding HaarPSI to Rafael Reisenhofer (reisenhofer@math.uni-bremen.de).