Wavelet and Earth Mover’s Distance Coupling Denoising Techniques

Abstract

The widely used wavelet-thresholding techniques (DWT-H and DWT-S) have a near-optimal behavior that cannot be enhanced by any local denoising filter, but they cannot utilize the similarity of small-size image patches to enhance the denoising performance. Two of the latest improvements (WNLM and NLMW) introduced the Euclidean distance to measure the similarity of image patches, and then used the non-local meaning of similar patches for further denoising. Since the Euclidean distance is not a good similarity measurement, these two improvements are limited. In this study, we introduced the earth mover’s distance (EMD) as the similarity measure of small-scale patches within the wavelet sub-bands of noisy images. Moreover, at higher noise levels, we further incorporated joint bilateral filtering, which can filter both the spatial domain and the intensity domain of images. Denoising simulation experiments on BSDS500 demonstrated that our algorithm outperformed the DWT-H, DWT-S, WNLM, and NLMW algorithms by 4.197 dB, 3.326 dB, 2.097 dB, and 1.162 dB in terms of the average PSNR, and by 0.230, 0.213, 0.132, and 0.085 in terms of the average SSIM.

1. Introduction

The emergence of wavelet techniques has brought fundamental changes in signal and image analysis. The concept of wavelet decomposition appeared first during the 1980s, when Meyer found that the integer translation and dyadic dilation of one single function could form an orthonormal basis (the so-called Meyer wavelet; see [1]). By incorporating a multiresolution idea into computer science, Mallat and Meyer proposed the structure of multiresolution analysis, which is viewed as a rebirth of wavelet theory, as it can provide a general formalism for wavelet constructions [2]. The main highlight of multiresolution analysis is that it can approximate any data at progressively coarser scales, while recording the differences between approximations at consecutive scales, leading to various algorithms on the fast computation of wavelet coefficients (so-called discrete wavelet transforms). The main steps in these fast wavelet algorithms are to calculate the convolution of data and wavelet filters. If the used wavelet filters are not finite-length, then such a convolution cannot be implemented efficiently in the real world. Since Daubechies [3] constructed many finite-length wavelet filters, wavelet techniques have been widely used in diverse domains, e.g., image watermarking [4], image compression [5], image denoising and feature extraction [6], and image retrieval [7].

Gray images are often contaminated by various noises during the process of generation, transportation, and processing, leading to a serious destruction of the visual effect of images. Denoising is an indispensable step, before the images are further subjected to edge detection, feature extraction, and object recognition. The ultimate goal of image denoising is to suppress noise, and then produce clear images without the loss of fine details or edges [8]. Earlier techniques include median filters [9], total variance techniques [10], Wiener filters [11], anisotropic filters [12], and bilateral filters [13]. As these conventional filters are based on Fourier spectral design, and have no capability for local time–frequency extractions, these filters often produce blurring and excessive smoothing in images, that lead to the loss of key edge or texture information. Compared with these filters, due to strong capacity for time–frequency localization, discrete wavelet transform may provide a sparse representation for the smooth regions and texture regions of any image, at the same time. The image information is always conserved in just a few high-magnitude wavelet coefficients, while inherent noise is represented by a large number of coefficients with small magnitudes [14,15]. Johnstone and Silverman [16] showed that for images contaminated by stationary Gaussian noise, level-dependent wavelet thresholding has a near-optimal behavior that cannot be enhanced by any local denoising filter. At the same time, wavelet thresholding provides effective denoising with minimum computational complexity [15]. The whole wavelet-denoising process depends largely on the use of wavelet thresholding, which is performed directly on the wavelet coefficients, then the noise-free wavelet coefficients can be estimated from the noisy ones [17,18,19]. The wavelet thresholds can be divided into hard thresholds and soft thresholds and, generally, a soft threshold has a better denoising performance than a hard threshold [20,21,22,23]. Except for classic image denoising, wavelet-threshold techniques are introduced to denoise laser self-mixing interference signals [24].

Although wavelet thresholding has a near-optimal behavior that cannot be enhanced by any local denoising filter [16], the obvious drawback in conventional wavelet-threshold techniques lies in the fact that they cannot utilize the similarity of small-size image patches to improve the denoising results. This leads to two enhanced wavelet-denoising techniques. The WNLM technique means that, after images are denoised according to wavelet thresholds, the Euclidean distance is used to search existing small-size similar patches within the low-frequency domain of the image, and the value of the center of each patch is replaced by the non-local mean (NLM) of the center of all the similar image patches [25,26,27]. The NLMW technique means that, for each wavelet sub-band of image decomposition, the Euclidean distance is used to search all the similar image patches, the non-local mean of similar patches is used to denoise each wavelet sub-band and, then, the inverse wavelet transform is used to reconstruct the image [28,29]. As these two wavelet improvements (WNLM and NLMW) introduce the Euclidean distance as the similarity measurement for small-size image patches, they demonstrate a better denoising performance. However, it is well known that the Euclidean distance is not a suitable measurement in classification and cluster analyses, so these two improvements are limited.

In this study, we propose a novel approach to improving classic wavelet-denoising techniques and their extensions. As the earth mover’s distance has been witnessed to have many advantages over the Euclidean distance and other distance measures in classification and cluster analyses (e.g., it is very applicable to clustering, naturally reflects nearness, and allows for partial matching), we introduced the earth mover’s distance as the similarity measure for small-scale patches of images, to improve the denoising efficiency of the wavelet-thresholding technique. Instead of the widely used Gaussian filters at higher noise levels, we further incorporated joint bilateral filters, which can filter both the spatial domain and the intensity domain of images simultaneously. Denoising simulation experiments demonstrate that our algorithm achieves a significantly better visual denoising performance than various wavelet techniques, and obtains a higher peak signal-to-noise ratio (PSNR) and structural similarity index measure (SSIM).

2. Background Techniques

Wavelets are well suited to catching and isolating the discontinuity across edges inside images, meaning that wavelets are a mainstream tool in image analysis and processing. Johnstone and Silverman [16] showed that, for images contaminated by stationary Gaussian noise, level-dependent thresholding on wavelet coefficients has a near-optimal behavior that cannot be enhanced by any local denoising filter. Moreover, hard wavelet thresholds always demonstrate a better denoising performance than soft wavelet thresholds. The main steps in wavelet-denoising techniques are:

➢

Starting from the wavelet coefficients of an image Y:

$${\omega }_{jk}=\mathrm{DWT}(\mathrm{Y}).$$

(1)

we can estimate the level of noise ${\hat{\sigma }}_j$ at each wavelet level $j=m,\cdots ,J-1$ using the median of absolute deviation of wavelet coefficients:

$${\hat{\sigma }}_j=\frac{\mathrm{med}\left(\left|{\omega }_{jk}\right|\right)}{0.6745}$$

(2)

➢

The wavelet threshold T_j at each wavelet level j is set to

$${\mathrm{T}}_j={\hat{\sigma }}_j\sqrt{2\ln 2^j}$$

(3)

➢

The wavelet coefficients after soft thresholds are

$${\hat{\mathrm{d}}}_{jk}^{\mathrm{S}}=\{\begin{array}{cc}\mathrm{sgn}(\left(\left|{\omega }_{jk}\right|-{\mathrm{T}}_j\right)& \mathrm{if} \left|{\mathsf{\omega }}_{jk}\right|>{\mathrm{T}}_j\\ 0& \mathrm{otherwise}\end{array}$$

(4)

or the wavelet coefficients after hard thresholds are

$${\hat{\mathrm{d}}}_{jk}^{\mathrm{H}}=\{\begin{array}{cc}{\omega }_{jk}& \mathrm{if} \left|{\mathsf{\omega }}_{jk}\right|>{\mathrm{T}}_j\\ 0& \mathrm{otherwise}\end{array}$$

(5)

➢

The inverse discrete wavelet transform of thresholding wavelet coefficients is just the denoised image.

Classic wavelet-thresholding techniques in denoising cannot utilize the similarity of different image patches to improve the denoising results. This leads to the incorporation of Euclidean-distance-similarity measures and non-local-mean filters into wavelet-denoising techniques. The WNLM algorithm uses the Euclidean distance to search small-size similar patches in low-frequency wavelet sub-bands after thresholding and, then, perform non-local means on these similar patches [25,26,27]. The NLMW algorithm is used to perform non-local-mean filtering with the Euclidean distance as a weight to denoise each wavelet sub-band without thresholding [28,29,30]. Subsequently, wavelet techniques have been extensively adopted in diverse domains, such as image watermarking [4], compression [5], denoising, transmission, and feature extraction [6].

The Earth Mover’s Distance (EMD) [31] is the minimum cost to transform one histogram into another. It has the intuitive interpretation of the minimum amount of work required by earth movers to move piles of earth into holes, as follows:

Let $P=\left\{\left(p_1,w_{p_1}\right),\dots ,\left(p_m,w_{p_m}\right)\right\}$ represent $m$ earth piles with location $p_k$ and weight ${\omega }_{p_k}$ and $Q=\left\{\left(q_1,w_{q_1}\right),\dots ,\left(q_n,w_{q_n}\right)\right\}$ represent $n$ holes with location $q_k$ and size ${\omega }_{q_k}$. Let $D=\left[d_{ij}\right]$ be the distance matrix among earth piles and holes. We want to find a flow $F=\left[f_{ij}\right]$ of earth movers from earth piles to holes, with minimal cost:

$$WORK\left(P,Q,F\right)={\displaystyle \sum }_{i=1}^m{\displaystyle \sum }_{j=1}^n{d}_{ij}{f}_{ij}$$

(6)

subject to the following constraints:

$$f_{ij}\ge 0, {\displaystyle \sum }_{j=1}^n{f}_{ij}\le w_{p_i}, {\displaystyle \sum }_{i=1}^n{f}_{ij}\le w_{q_j},$$

$${\displaystyle \sum }_{i=1}^m{\displaystyle \sum }_{j=1}^n{f}_{ij}=\min \left({\displaystyle \sum }_{i=1}^m{w}_{p_i},{\displaystyle \sum }_{j=1}^n{w}_{q_j} \right)$$

(7)

the Earth mover’s distance (EMD) between earth piles and holes is

$${\mathrm{dist}}_{\mathrm{EMD}}\left(P,Q\right)=\underset{f_{ij}}{\min }\frac{{{\displaystyle \sum }}_{i=1}^m{{\displaystyle \sum }}_{j=1}^n{d}_{ij}{f}_{ij}}{{{\displaystyle \sum }}_{i=1}^m{{\displaystyle \sum }}_{j=1}^n{f}_{ij}}$$

(8)

The EMD is closer to the way in which humans perceive distance, and is robust to outlier noise and quantization effects. Pele and Werman (2008 & 2010) improved the classic EMD as

$${\mathrm{dist}}_{\widehat{\mathrm{EMD}}}\left(P,Q\right)=\underset{f_{ij}}{\min }{\displaystyle \sum }_{i=1}^m{\displaystyle \sum }_{j=1}^n{d}_{ij}{f}_{ij}+\left|{\displaystyle \sum }_i{P}_i-{\displaystyle \sum }_j{Q}_j\right|\times \alpha \underset{i,j}{\min }{d}_{ij}$$

(9)

which is more suited to dealing with cases in which the net weight of earth is not equal to the net size of holes. The EMD has many advantages over the Euclidean distance and other distance measures, including the fact that it naturally reflects nearness, and allows for partial matching. The earth mover’s distance (EMD) has seen a widespread utilization across a range of data analysis domains, encompassing, for example, cardiovascular disease determination [32], computational aesthetics [33], and visible thermal person re-identification [34].

The joint bilateral filter [35,36] is the combination of a Gaussian filter and a range filter. The range filter in the joint bilateral filter can detect high-frequency oscillations, and then preserve image edges, meaning that the joint bilateral filter has significant advantages over classic Gaussian filters.

The joint bilateral filtering of any noisy image is processed as

$${\hat{g}}_{i,j}=\frac{{{\displaystyle \sum }}_{s=-M}^M{{\displaystyle \sum }}_{t=-M}^M{W}_G\left(s,t\right)W_R\left(s,t\right)g_{i+s,j+t}}{{{\displaystyle \sum }}_{s=-M}^M{{\displaystyle \sum }}_{t=-M}^M{W}_G\left(s,t\right)W_R\left(s,t\right)}$$

(10)

where the Gaussian filter $W_G$ is

$$W_G\left(s,t\right)=e^{-\left(s^2+t^2\right)/2{\sigma }_d^2}$$

(11)

and the range filter $W_R$ is

$$W_R\left(s,t\right)=e^{-{\left(g_{i,j}+g_{i+s,j+t}\right)}^2/2{\sigma }_r^2}$$

(12)

where ${\sigma }_d$ and ${\sigma }_r$ are the geometric spread and photometric spread, respectively. For the noise level $\sigma$, the optimal choice of ${\sigma }_d$ and ${\sigma }_r$ are $2\sigma$ and $3+\sqrt{\sigma }$, respectively [35,37].

3. Proposed Methods

In this study, we propose a novel approach to improving classic wavelet-denoising techniques. We introduced the earth mover’s distance as the similarity measure of small-scale patches of images to improve the noise efficiency of the wavelet-thresholding technique. This combination of the wavelet and earth mover’s distance makes full use not only of the localization feature of the wavelet-denoising technique, but also the interior similarity and redundancy embedded in the whole image.

The main steps of the full version of our method are as below:

Step 1. The noisy image is decomposed via a stationary wavelet transform. Under a high noise level ($\sigma \ge 50$), all the wavelet coefficients are pre-processed via joint bilateral filtering (JBF).

Step 2. Wavelet soft thresholds are applied to the wavelet coefficients in all the wavelet sub-bands, to roughly denoise local noise.

Step 3. In order to utilize the similarity of small image patches to denoise the image further, the earth mover’s distance (EMD) is used to measure the similarity among different image patches inside the search window $S_q$. After that, a non-local mean with the earth mover’s distance as a weight is used to denoise the image:

$${\hat{X}}_p=\frac{{{\displaystyle \sum }}_{q\in S_p}E{w}_{pq}{X}_q}{{{\displaystyle \sum }}_{q\in S_p}E{w}_{pq}}$$

(13)

where the weight $E{w}_{pq}$ is an exponential decay function whose decaying rate is controlled by the earth mover’s distance between two similar image patches with centers $p$ and $q$:

$$E{w}_{pq}=exp\left(-\frac{{\mathrm{dist}}_{\widehat{\mathrm{EMD}}}\left(p,q\right)/{\sigma }^2}{h^2}\right)$$

(14)

It is clear that the more similar two image patches are, the greater the weight assigned to the non-local mean.

Step 4. The inverse stationary wavelet transform is used to obtain the denoised image.

Our denoising algorithm embeds the EMD similarity measurement into the wavelet algorithm. As a result, our algorithm is called the wavelet–EMD coupling algorithm. The simple version of our algorithm is just to remove Step 2 from the above full version of our algorithm.

4. Denoising Experiments

We conducted denoising experiments using the Berkeley Segmentation Dataset and Benchmark500 (BSDS500), as shown in Figure 1 (128 × 128). We selected 24 test images through the following process: we eliminated images with occlusion, noise, or irregular alterations from the image datasets; then we selected images which encompassed diverse scenes, lighting scenarios, viewpoints, and complexities, and covered various types, including landscape, portrait, objects, and textures. This diversity allows for an examination of the generalization ability of our denoising models. Moreover, the selected images have complex textures, patterns, or subtle details which further support rigorous testing of the resilience of denoising algorithms. We added different Gaussian noises to these images, ranging from low noise levels $\sigma =10, 20$, to medium noise levels $\sigma$ = 30, 40, and strong noise levels $\sigma =50, 70$. By denoising these images, we compared the full version (and simple version) of our denoising algorithms with classic wavelet denoising with hard/soft thresholds (DWT-H, DWT-S), and two improvements (WNLM, NLMW). The denoising performance was evaluated according to the metrics of the peak signal-to-noise ratio (PSNR), and the structure similarity index measure (SSIM), as well as visual comparisons among the denoised images.

4.1. Low Noise Level

Both the full version and simple versions of our algorithm consistently demonstrated a superior performance in terms of the PSNR and SSIM, compared to the traditional DWT-H/DWT-S/WNLM/NLMW algorithms across very different types of images (

Table 1. The PSNR (dB) and SSIM results of six wavelet-denoising algorithms at low noise levels.

σ = 10
Methods	DWT-H		DWT-S		WNLM		NLMW		Simple Version		Full Version
Image	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM
1	27.579	0.763	28.113	0.785	28.650	0.799	29.354	0.840	31.028	0.877	31.107	0.882
2	27.815	0.765	28.410	0.786	28.536	0.819	29.721	0.822	30.950	0.882	31.027	0.884
3	27.027	0.775	27.442	0.788	27.637	0.832	29.138	0.826	30.370	0.890	30.501	0.892
4	26.938	0.711	27.580	0.745	27.216	0.736	28.936	0.819	29.420	0.845	29.579	0.855
5	27.493	0.763	27.880	0.775	28.002	0.841	29.118	0.788	30.316	0.863	30.383	0.869
6	26.471	0.811	27.023	0.827	26.728	0.831	28.813	0.874	29.36	0.901	29.596	0.906
7	29.009	0.767	29.389	0.775	30.768	0.876	29.476	0.750	32.074	0.861	32.274	0.870
8	26.854	0.753	27.554	0.780	27.617	0.774	29.115	0.848	30.178	0.873	30.316	0.882
9	28.044	0.702	28.526	0.711	29.054	0.825	29.279	0.697	31.079	0.813	31.211	0.826
10	28.411	0.741	28.641	0.747	29.439	0.856	29.393	0.725	31.494	0.840	31.641	0.850
11	26.519	0.856	26.914	0.865	26.706	0.874	28.913	0.904	29.455	0.927	29.613	0.930
12	29.819	0.650	30.126	0.663	31.884	0.809	29.541	0.623	32.132	0.766	32.452	0.783
13	29.217	0.712	29.555	0.724	31.531	0.836	29.588	0.720	32.614	0.844	32.938	0.852
14	27.765	0.668	28.325	0.698	28.718	0.755	29.418	0.757	31.235	0.846	31.350	0.851
15	29.164	0.642	29.518	0.661	30.973	0.738	29.449	0.689	31.907	0.789	32.135	0.793
16	27.209	0.765	27.790	0.786	27.855	0.803	29.079	0.832	30.309	0.879	30.397	0.882
17	27.656	0.823	28.015	0.828	28.033	0.899	29.056	0.817	30.356	0.892	30.438	0.899
18	27.072	0.777	27.540	0.783	27.725	0.827	29.115	0.817	30.196	0.879	30.316	0.881
19	28.211	0.704	28.743	0.722	29.318	0.810	29.302	0.724	31.104	0.832	31.220	0.838
20	24.968	0.836	25.524	0.852	23.787	0.798	26.009	0.881	26.187	0.884	26.503	0.894
21	27.948	0.693	28.571	0.707	29.020	0.821	29.382	0.693	31.246	0.812	31.373	0.827
22	27.904	0.732	28.204	0.741	28.420	0.815	29.164	0.745	30.670	0.832	30.743	0.839
23	28.539	0.663	28.975	0.680	29.723	0.774	29.413	0.698	31.649	0.810	31.848	0.818
24	27.994	0.733	28.556	0.754	29.091	0.811	29.325	0.780	31.137	0.859	31.210	0.862
σ = 20
Methods	DWT-H		DWT-S		WNLM		NLMW		Simple Version		Full Version
Image	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM
1	23.935	0.604	24.465	0.634	25.913	0.690	26.712	0.740	28.035	0.791	28.121	0.796
2	24.117	0.585	24.734	0.611	26.040	0.682	27.068	0.732	27.642	0.791	27.876	0.797
3	23.251	0.600	23.888	0.625	25.080	0.686	26.364	0.746	26.47	0.798	26.868	0.807
4	23.563	0.553	24.177	0.594	25.102	0.632	26.138	0.696	26.225	0.720	26.525	0.730
5	23.604	0.580	24.129	0.595	25.45	0.671	26.432	0.709	26.764	0.790	27.057	0.795
6	22.930	0.665	23.746	0.697	24.614	0.736	25.746	0.786	26.058	0.816	26.326	0.824
7	24.433	0.538	24.701	0.542	26.714	0.656	27.270	0.680	29.346	0.830	29.481	0.826
8	23.475	0.602	24.206	0.642	25.378	0.678	26.325	0.741	27.127	0.769	27.273	0.778
9	23.948	0.467	24.351	0.475	26.001	0.576	26.836	0.616	27.775	0.750	28.006	0.751
10	24.022	0.509	24.379	0.513	26.140	0.626	27.092	0.657	27.972	0.796	28.354	0.797
11	23.083	0.739	23.802	0.760	24.634	0.795	25.933	0.840	25.950	0.859	26.298	0.866
12	24.821	0.375	24.915	0.377	27.243	0.510	27.594	0.534	29.993	0.726	30.101	0.720
13	24.492	0.473	24.771	0.483	27.041	0.603	27.501	0.635	30.158	0.801	30.224	0.797
14	23.965	0.478	24.495	0.509	25.907	0.580	26.974	0.656	27.695	0.720	27.96	0.727
15	24.513	0.402	24.779	0.425	26.955	0.526	27.439	0.574	29.866	0.704	29.915	0.704
16	23.511	0.592	24.115	0.621	25.359	0.678	26.367	0.737	27.101	0.784	27.307	0.790
17	23.674	0.659	24.170	0.668	25.383	0.740	26.453	0.764	27.057	0.863	27.32	0.862
18	23.334	0.595	23.976	0.613	25.192	0.679	26.294	0.733	26.417	0.782	26.836	0.793
19	24.035	0.475	24.459	0.492	26.154	0.593	26.907	0.634	28.224	0.766	28.39	0.766
20	21.845	0.730	22.887	0.767	22.552	0.744	22.509	0.731	24.349	0.821	23.091	0.763
21	23.863	0.446	24.403	0.463	26.070	0.566	26.921	0.613	27.809	0.735	28.067	0.740
22	23.778	0.528	24.214	0.538	25.591	0.623	26.692	0.660	27.027	0.760	27.436	0.765
23	24.235	0.429	24.574	0.447	26.366	0.548	27.163	0.599	28.576	0.729	28.787	0.731
24	24.101	0.541	24.550	0.563	26.096	0.641	26.864	0.687	28.065	0.772	28.192	0.774

The bold blue indicates the best results, and the underlining indicates the second-best results.

). In terms of PSNR results (Table 1), the simple version of our algorithm achieved significant improvements, with 1.218–5.666 dB, 0.662–5.388 dB, 0.248–3.117 dB, and 0.017–3.025 dB over the DWT-H/DWT-S/WNLM/NLMW algorithm, respectively. The incorporation of wavelet thresholding into the full version of our algorithm further enhanced the denoising performance, leading to an additional improvement of approximately 0.163 dB over the simple version. Regarding the SSIM results (Table 1), the simple version of our algorithm also consistently outperformed the other four wavelet methods, with an average advantage of 0.171, 0.153, 0.086, and 0.083. Our full-version algorithm further enhanced the average SSIM result by approximately 0.004, compared to the simple version.

Figure 2 demonstrates the denoising visual effects of different algorithms for flower images at a low noise level ($\sigma =20$). Notably, compared to the four traditional wavelet algorithms, our algorithm demonstrated more convincing improvements in the visual quality of the flower image, especially in reconstructing the edges of the petals and the blurry background. DWT-H and DWT-S over-smooth the image, causing the loss of most details in the flower. NLMW and WNLM manage to retain some structures and details, but noticeable noise remains around the edges of the flower. Compared with these, our algorithms preserve a higher level of detailed information; the reconstruction of the flower exhibits sharper edges, and the overall background appears cleaner. This highlights the enhanced denoising capabilities of our algorithm, especially in terms of producing sharper and more visually pleasing results.

4.2. Middle Noise Level

For middle noise levels ($\sigma$ = 30, 40), the PSNR results (

Table 2. The PSNR (dB) and SSIM results of six wavelet-denoising algorithms at middle noise levels.

σ = 30
Methods	DWT-H		DWT-S		WNLM		NLMW		Simple Version		Full Version
Image	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM
1	20.284	0.441	21.745	0.476	24.064	0.587	25.413	0.675	26.225	0.718	26.258	0.719
2	20.474	0.415	21.882	0.454	24.166	0.585	25.660	0.674	26.054	0.718	26.069	0.724
3	19.737	0.453	20.912	0.463	22.703	0.565	24.679	0.680	24.698	0.711	24.641	0.714
4	20.279	0.416	21.462	0.431	23.284	0.515	24.750	0.614	24.767	0.636	24.784	0.636
5	19.937	0.428	21.221	0.454	23.312	0.576	24.921	0.654	25.165	0.705	25.183	0.713
6	19.725	0.523	20.868	0.542	22.499	0.621	24.150	0.718	24.470	0.744	24.412	0.743
7	20.231	0.348	21.895	0.401	24.859	0.575	26.126	0.641	27.314	0.740	27.425	0.756
8	20.089	0.455	21.438	0.477	23.414	0.555	24.952	0.664	25.493	0.691	25.478	0.686
9	19.934	0.303	21.505	0.329	23.989	0.475	25.481	0.560	25.991	0.638	26.052	0.654
10	20.051	0.342	21.435	0.373	23.850	0.537	25.684	0.615	25.935	0.695	25.963	0.711
11	19.884	0.604	20.830	0.621	22.324	0.694	24.287	0.788	24.310	0.801	24.216	0.799
12	20.252	0.199	22.158	0.244	25.710	0.440	26.644	0.491	28.009	0.610	28.263	0.637
13	20.297	0.294	21.983	0.338	25.124	0.514	26.395	0.590	27.852	0.699	27.977	0.715
14	20.337	0.340	21.716	0.350	23.985	0.465	25.591	0.584	25.975	0.621	26.002	0.628
15	20.167	0.241	22.031	0.279	25.380	0.444	26.537	0.520	27.919	0.608	28.135	0.623
16	19.861	0.423	21.351	0.457	23.488	0.567	25.037	0.672	25.483	0.705	25.523	0.708
17	20.128	0.517	21.282	0.552	23.266	0.670	24.975	0.730	25.336	0.792	25.364	0.801
18	19.813	0.438	21.062	0.449	22.895	0.553	24.706	0.665	24.751	0.691	24.738	0.696
19	20.004	0.308	21.664	0.349	24.411	0.509	25.719	0.583	26.533	0.669	26.607	0.683
20	19.193	0.631	19.732	0.603	20.384	0.606	21.398	0.663	22.330	0.726	21.301	0.655
21	20.000	0.288	21.623	0.308	24.220	0.459	25.568	0.550	26.149	0.621	26.220	0.638
22	19.856	0.373	21.253	0.399	23.363	0.528	25.222	0.609	25.286	0.668	25.307	0.680
23	20.391	0.279	21.836	0.301	24.594	0.460	25.968	0.543	26.738	0.625	26.836	0.640
24	20.396	0.381	21.853	0.416	24.320	0.542	25.603	0.625	26.362	0.684	26.434	0.693
σ = 40
Methods	DWT-H		DWT-S		WNLM		NLMW		Simple Version		Full Version
Image	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM
1	20.662	0.434	21.608	0.453	23.064	0.537	24.245	0.610	24.202	0.617	25.307	0.666
2	20.408	0.425	21.176	0.438	22.473	0.558	24.149	0.600	24.184	0.700	25.603	0.712
3	19.879	0.438	20.637	0.436	21.754	0.508	23.063	0.599	23.313	0.658	23.755	0.686
4	20.659	0.407	21.371	0.416	22.504	0.480	23.364	0.538	24.030	0.570	24.226	0.588
5	20.129	0.420	20.987	0.434	22.299	0.516	23.357	0.582	23.910	0.688	24.650	0.736
6	19.964	0.517	20.719	0.522	21.734	0.582	22.575	0.636	22.667	0.645	23.372	0.693
7	20.740	0.362	22.042	0.406	23.966	0.531	24.483	0.564	26.369	0.758	27.070	0.786
8	20.257	0.436	21.215	0.450	22.472	0.509	23.473	0.584	23.691	0.590	24.545	0.635
9	20.225	0.301	21.494	0.320	23.106	0.423	24.040	0.485	25.426	0.678	25.581	0.711
10	20.385	0.344	21.248	0.361	22.731	0.474	24.022	0.618	24.819	0.722	25.446	0.760
11	20.025	0.599	20.463	0.596	21.345	0.653	22.548	0.716	22.299	0.721	23.131	0.757
12	20.620	0.205	22.230	0.244	24.550	0.375	25.288	0.423	28.216	0.696	28.683	0.737
13	20.743	0.305	22.152	0.343	24.233	0.470	24.843	0.511	26.916	0.701	27.875	0.743
14	20.629	0.338	21.482	0.338	22.978	0.445	24.191	0.508	25.344	0.640	25.798	0.640
15	20.344	0.238	22.015	0.273	24.201	0.384	25.162	0.449	27.778	0.629	28.391	0.661
16	20.220	0.419	21.315	0.441	22.726	0.523	23.595	0.592	24.410	0.640	24.732	0.659
17	20.595	0.534	21.213	0.554	22.411	0.651	23.285	0.663	23.797	0.797	24.655	0.819
18	20.063	0.430	20.924	0.427	22.122	0.502	23.164	0.586	23.691	0.652	24.049	0.674
19	20.209	0.304	21.586	0.339	23.381	0.445	24.264	0.507	25.677	0.664	26.262	0.707
20	18.981	0.580	19.108	0.537	19.485	0.540	20.274	0.578	20.941	0.655	20.593	0.612
21	20.247	0.275	21.619	0.291	23.360	0.389	24.225	0.477	25.712	0.654	25.971	0.699
22	19.923	0.362	20.933	0.377	22.262	0.464	23.675	0.540	24.163	0.672	24.805	0.714
23	21.336	0.303	22.247	0.315	24.122	0.446	24.595	0.472	26.433	0.670	26.664	0.688
24	21.098	0.395	21.999	0.415	23.607	0.521	24.049	0.544	25.095	0.659	25.812	0.683

The bold blue indicates the best results, and the underlining indicates the second-best results.

) demonstrated significant advantages in our simple version over four wavelet algorithms (DWT-H/DWT-S/WNLM/NLMW), with the maximum PSNR differences reaching up to 7.757 dB, 5.985 dB, 3.665 dB, and 2.927 dB, respectively. Moreover, our full version exhibited a superior performance over the simple version in almost all the denoised images. The SSIM results (Table 2) also showed substantial advantages in our algorithms over four traditional wavelet algorithms (DWT-H/DWT-S/WNLM/NLMW). The maximum SSIM differences reached up to 0.491, 0.452, 0.321, and 0.273, respectively. The considerable improvements in PSNR and SSIM values further highlight the effectiveness and potential of our algorithms in handling noise reduction tasks.

Figure 3 demonstrates an example of denoising at a middle noise level ($\sigma$ = 30). The DWT-H/DWT-S algorithms yield significantly artifacts, and NLMW and WNLM manage to reduce them in some sense; compared with these algorithms, the simple version of our algorithm exhibited a better performance, by effectively separating the koala’s silhouette from the black background, and producing a clearer image. The full version of our algorithm output the clearest details of the koala silhouette, resulting in a more visually pleasing and aesthetically appealing image. The higher PSNR and SSIM values further support the superiority of our algorithm.

4.3. High Noise Level

At the high noise level, the PSNR/SSIM values of the full and simple versions of our algorithm also outperformed the four known wavelet algorithms (

Table 3. The PSNR (dB) and SSIM results of six wavelet-denoising algorithms at high noise levels.

σ = 50
Methods	DWT-H		DWT-S		WNLM		NLMW		Simple Version		Full Version
Image	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM
1	20.199	0.421	21.312	0.437	22.328	0.505	23.173	0.545	23.964	0.603	24.395	0.621
2	19.799	0.412	20.659	0.425	21.540	0.524	23.142	0.607	23.437	0.663	24.742	0.668
3	19.48	0.427	20.310	0.411	21.110	0.473	21.587	0.541	22.533	0.608	22.913	0.630
4	20.173	0.395	20.881	0.395	21.643	0.448	22.396	0.494	23.136	0.533	23.492	0.553
5	19.683	0.409	20.653	0.417	21.562	0.482	21.918	0.585	23.081	0.631	23.775	0.677
6	19.560	0.501	20.364	0.498	21.080	0.546	20.997	0.521	22.353	0.631	22.462	0.629
7	20.426	0.367	21.959	0.410	23.325	0.513	24.588	0.663	25.468	0.711	25.868	0.718
8	19.829	0.422	20.855	0.427	21.721	0.473	22.484	0.507	23.389	0.577	23.752	0.575
9	19.937	0.298	21.365	0.312	22.559	0.400	23.814	0.547	24.623	0.618	24.800	0.638
10	20.002	0.340	20.914	0.350	21.937	0.442	22.562	0.600	23.819	0.657	24.372	0.688
11	19.616	0.588	19.916	0.566	20.530	0.613	20.199	0.598	22.045	0.704	22.172	0.706
12	20.261	0.205	22.281	0.251	23.997	0.359	26.538	0.583	27.150	0.625	27.605	0.667
13	20.349	0.306	22.070	0.344	23.604	0.449	25.164	0.599	25.993	0.653	26.634	0.679
14	20.153	0.330	21.098	0.322	22.167	0.412	23.695	0.527	24.269	0.587	24.985	0.588
15	19.921	0.236	22.012	0.272	23.628	0.362	26.098	0.539	26.809	0.574	27.293	0.607
16	19.934	0.412	21.139	0.424	22.180	0.493	23.105	0.553	23.782	0.603	23.984	0.616
17	20.219	0.534	20.762	0.544	21.567	0.624	21.306	0.658	22.950	0.757	23.626	0.760
18	19.747	0.420	20.651	0.405	21.533	0.472	22.154	0.545	22.938	0.599	23.302	0.623
19	19.802	0.300	21.475	0.334	22.764	0.418	23.879	0.556	24.863	0.608	25.364	0.647
20	18.591	0.548	18.457	0.474	18.740	0.480	18.446	0.435	19.820	0.562	19.943	0.550
21	19.873	0.267	21.522	0.280	22.810	0.359	24.449	0.551	24.902	0.582	25.240	0.635
22	19.509	0.352	20.601	0.360	21.543	0.432	22.262	0.560	23.244	0.608	23.910	0.656
23	21.050	0.314	22.044	0.315	23.368	0.426	24.688	0.549	25.614	0.625	25.750	0.628
24	20.722	0.391	21.699	0.406	22.818	0.495	23.548	0.562	24.300	0.618	24.911	0.633
σ = 70
Methods	DWT-H		DWT-S		WNLM		NLMW		Simple Version		Full Version
Image	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM
1	19.550	0.384	20.292	0.407	20.975	0.464	22.100	0.480	22.161	0.509	22.591	0.516
2	18.918	0.380	19.493	0.401	20.097	0.485	21.187	0.530	22.312	0.578	22.707	0.548
3	19.095	0.395	19.692	0.387	20.310	0.442	20.742	0.467	21.119	0.510	21.257	0.507
4	19.269	0.355	19.739	0.362	20.282	0.407	21.515	0.431	21.388	0.458	21.869	0.458
5	19.313	0.380	19.973	0.393	20.639	0.449	21.005	0.499	21.534	0.524	21.712	0.530
6	19.072	0.461	19.628	0.465	20.159	0.505	20.251	0.464	20.737	0.540	20.924	0.537
7	20.079	0.353	21.170	0.398	22.154	0.490	23.081	0.549	23.647	0.554	23.460	0.615
8	19.214	0.385	19.906	0.393	20.540	0.434	21.493	0.448	21.728	0.486	22.104	0.493
9	19.641	0.273	20.764	0.301	21.697	0.383	22.601	0.434	23.105	0.460	22.997	0.506
10	19.505	0.318	20.167	0.334	20.924	0.415	21.569	0.488	21.958	0.540	22.142	0.505
11	18.911	0.541	19.108	0.531	19.546	0.572	19.448	0.544	20.410	0.619	20.541	0.620
12	20.124	0.196	21.668	0.247	22.961	0.344	24.574	0.443	25.201	0.464	24.977	0.500
13	19.999	0.290	21.272	0.332	22.372	0.423	23.568	0.483	24.053	0.512	24.081	0.552
14	19.460	0.300	20.118	0.303	20.864	0.375	22.543	0.442	22.296	0.458	22.964	0.483
15	19.876	0.222	21.525	0.265	22.788	0.344	24.294	0.430	24.978	0.451	24.698	0.468
16	19.661	0.382	20.516	0.399	21.308	0.459	22.087	0.486	22.487	0.513	22.406	0.524
17	19.485	0.510	19.783	0.520	20.323	0.586	20.362	0.568	21.116	0.607	21.510	0.664
18	19.346	0.383	20.032	0.385	20.690	0.443	21.273	0.474	21.584	0.504	21.680	0.496
19	19.625	0.284	20.899	0.320	21.881	0.395	22.675	0.449	23.290	0.482	23.180	0.502
20	17.836	0.473	17.795	0.423	18.031	0.427	17.977	0.409	18.697	0.511	18.975	0.485
21	19.700	0.248	20.927	0.267	21.903	0.337	23.181	0.440	23.442	0.462	23.425	0.457
22	19.231	0.325	20.000	0.341	20.716	0.403	21.301	0.463	21.698	0.496	21.925	0.497
23	20.309	0.300	20.919	0.307	21.797	0.401	23.344	0.440	23.664	0.528	23.697	0.460
24	19.847	0.355	20.495	0.380	21.236	0.454	22.379	0.474	22.449	0.527	22.875	0.501

The bold blue indicates the best results, and the underlining indicates the second-best results.

). At the noise level $\sigma =50$, the PSNR/SSIM values of the simple version of our algorithm were consistently, on average, 2.415 dB/0.172 higher than the other four algorithms, and the full version of our algorithm, utilizing wavelet thresholding, achieved an additional improvement of approximately 0.449 dB/0.019. Similarly, at the noise level $\sigma =70$, the PSNR/SSIM values of the simple version of our algorithm were, on average, 1.652 dB/0.1045 higher than four known wavelet algorithms, and the full version of our algorithm achieved an additional 0.151 dB/0.006 improvement. Especially when $\sigma =70$, all the competing wavelet-denoising algorithms only achieved a low SSIM value, while our algorithm showed a better resilience to a high noise level, by maintaining an SSIM value of 0.5, or even 0.6.

Figure 4 shows a building image after denoising via six wavelet algorithms at $\sigma =50$, with a partial zoom in the red box in the lower left corner, to allow for a clearer and more intuitive sense of the denoising effect. It can be seen from the zoomed area that DWT-H and DWT-S brought many ringtone artifacts to the background of the denoised images. As a result, it is difficult to recognize the demarcation and details of the sky and houses. NLMW and WNLM made the image too smooth, leaving some noise at the edge of the chimney. Our algorithm produced clear edges, and restored the basic contours of the dark clouds in the sky.

4.4. Average Denoising Performance

Table 4. The average PSNR (dB)/SSIM for the denoised images via six wavelet-denoising methods.

Noise Levels	DWT-H	DWT-S	WNLM	NLMW	Simple Version	Full Version
σ = 10	27.735/0.742	28.205/0.757	28.601/0.815	29.129/0.777	30.687/0.854	30.841/0.861
σ = 20	23.772/0.549	24.287/0.569	25.707/0.644	26.566/0.687	27.572/0.778	27.742/0.779
σ = 30	20.055/0.393	21.447/0.419	23.734/0.543	25.228/0.629	25.799/0.688	25.800/0.694
σ = 40	20.348/0.390	21.324/0.405	22.787/0.496	23.747/0.557	24.712/0.670	25.291/0.698
σ = 50	19.951/0.383	21.042/0.390	22.086/0.466	23.008/0.559	23.938/0.623	24.387/0.641
σ = 70	19.461/0.353	20.245/0.369	21.008/0.434	21.856/0.472	22.295/0.513	22.446/0.518

The bold blue indicates the best results, and the underlining indicates the second-best results.

presents the average PSNR and SSIM values for the denoising of the 24 test images at different noise levels. The full version and simple version of our algorithm achieved the best and the second-best denoising results, significantly better than the classic wavelet algorithms (DWT-H, DWT-S) and their extensions (WNLM, NLMW). The full/simple versions of our algorithm outperformed the DWT-H, DWT-S, WNLM, and NLMW algorithms by 4.197/3.946 dB, 3.326/3.075 dB, 2.097/1.846 dB, and 1.162/0.911 dB in terms of average PSNR, and by 0.230/0.219, 0.213/0.202, 0.132/0.121, and 0.085/0.074 in terms of average SSIM. The full and simple versions of our algorithm achieved the smallest mean square error (MSE), with an average reduction of 306.791 compared to other algorithms (Figure 5)

4.5. Denoising Experiments on Kodak24 Dataset

To further demonstrate the feasibility of our improved wavelet-denoising algorithms, we conducted 16 image-denoising experiments images with the higher-resolution Kodak24 (500 × 500) dataset (Figure 6).

We present the final PSNR/SSIM results for the denoising of images from the Kodak24 image dataset in

Table 5. The average PSNR (dB)/SSIM for the denoised images on Kodak24 via six wavelet-denoising methods.

Noise Levels	DWT-H	DWT-S	WNLM	NLMW	Simple Version	Full Version
σ = 10	27.51/0.754	27.98/0.772	28.40/0.811	29.14/0.803	30.50/0.863	30.54/0.886
σ = 20	23.64/0.570	24.19/0.594	25.53/0.659	26.49/0.711	27.44/0.765	27.62/0.786
σ = 30	19.97/0.411	21.36/0.438	23.59/0.552	25.12/0.645	25.56/0.677	25.68/0.698
σ = 40	19.88/0.393	20.90/0.405	21.91/0.478	22.95/0.552	23.30/0.559	24.28/0.632
σ = 50	22.02/0.515	20.90/0.405	21.91/0.478	22.95/0.552	23.30/0.560	24.05/0.654
σ = 70	19.42/0.370	20.14/0.380	20.88/0.441	21.32/0.448	21.77/0.466	22.31/0.505

The bold blue indicates the best results, and the underlining indicates the second-best results.

. It is clear that the full version and simple version of our algorithm achieved a better PSNR/SSIM than the other competing algorithms. The full/simple versions of our algorithm outperformed the DWT-H, DWT-S, WNLM, and NLMW by approximately 3.67/3.34 dB, 3.01/2.68 dB, 1.94/1.60 dB, and 1.08/0.82 dB in terms of average PSNR, and by approximately 0.191/0.158, 0.182/0.149, 0.110/0.077, and 0.068/0.038 in terms of average SSIM (Table 5).

4.6. Other Denoising Experiments

We have demonstrated the effectiveness of our improvement in successfully restoring images corrupted by Gaussian noise. However, the practice of image formation often involves a combination of Gaussian and Poisson noises. Hence, we conducted denoising experiments on the images in Figure 1, and considered a mixture of Gaussian noise ($\sigma =20$) and different Poisson noises (λ = 0.4, 5, 10). Across all levels of noise, the full/simple versions of our algorithm consistently exhibited a superior performance compared to the DWT-H, DWT-S, WNLM, and NLMW algorithms (

Table 6. The average PSNR (dB)/SSIM for the denoised images via six wavelet-denoising methods.

Noise Levels	DWT-H	DWT-S	WNLM	NLMW	Simple Version	Full Version
λ = 0.4	23.94/0.527	24.43/0.548	26.00/0.631	26.87/0.678	28.17/0.757	28.43/0.779
λ = 5	23.52/0.524	23.97/0.545	25.39/0.627	26.13/0.674	27.19/0.755	27.37/0.777
λ = 10	22.52/0.519	22.88/0.540	23.95/0.621	24.46/0.667	25.13/0.748	25.24/0.770

The bold blue indicates the best results, and the underlining indicates the second-best results.

), manifesting advantages of approximately 3.88/3.68 dB, 3.44/3.24 dB, 2.02/1.83 dB, and 1.28/1.08 dB in terms of average PSNR, respectively, and by 0.252/0.229, 0.231/0.209, 0.149/0.127, and 0.102/0.080 in terms of average SSIM, respectively.

5. Conclusions

Compared with classic wavelet algorithms (DWT-H, DWT-S) and their extensions (WNLM, NLMW), our denoising algorithms coupling the wavelet and earth mover’s distance not only yields higher PSNR and SSIM values, but also produces more visually appealing images. Denoising simulations demonstrated that the robustness of our algorithms to various noise intensities highlights its versatility and adaptability to real-world scenarios. The denoising process using our wavelet algorithm does not introduce unwanted artifacts or distortions, resulting in clearer and more aesthetically pleasing results.

The main mechanisms behind our superior denoising performance comprise:

➢

Our algorithm makes full use of not only the wavelet-denoising technique at a local scale, but also the interior similarity and redundancy embedded in the whole image, leading to our superior denoising performance over classic wavelet algorithms (DWT-H, DWT-S). Moreover, the use of joint bilateral filtering as a processing step, which detects high-frequency oscillations inside images, and then preserves image edges, further enhanced the denoising performance.

➢

We used the earth mover’s distance as the similarity measure of small-scale patches of images. The earth mover’s distance (EMD) naturally extends the concept of distance between individual elements to the concept of distance between sets of elements. As the EMD tolerates the distortion of some moving features, it is well recognized as a much more robust clustering measure than the Euclidean distance, leading to our superior denoising performance over WNLM and NLMW, which use the Euclidean distance to measure the similarity.

Denoising experiments on 40 images demonstrated that our algorithm achieved the best and the second-best denoising results, significantly better than classic wavelet algorithms (DWT-H, DWT-S) and their extensions (WNLM, NLMW). While our improvement consistently demonstrates a remarkable performance in contrast to conventional DWT-H, DWT-S, WNLM, and NLMW on various gray images, residual artifacts and subtle discontinuities are possibly discernible along certain boundaries of minor blocks in a few denoised images. Although there is an increasing trend in the application of deep learning in denoising, the deep learning technique must use a huge number of noisy images; the numbers of parameters used in deep learning techniques are very large; and the computation cost is high. However, wavelet-denoising techniques have no such drawbacks. In the future, we intend to couple EMD distance and wavelet techniques into deep-learning-driven denoising models, to reduce the computational cost, and enhance the interpretability. Recently, simple nonlocal similarity measurements are incorporated into classic compressive sensing, and demonstrated the advantage of such an improvement on the denoising of three images (Barbara, Cameraman, and Lena) [38]. Inspired by this approach, we plan to use the EMD distance to further enhance the denoising performance of compressive sensing in the future. Meanwhile, our future efforts will expand our algorithm’s scope, to cover color image denoising, image segmentation, focal point detection, and image deblurring.