Single Image Enhancement in Sandstorm Weather via Tensor Least Square

Guanlei Xu, Xiaotong Wang, and Xiaogang Xu

Abstract—In this paper, we present a tensor least square based model for sand/sandstorm removal in images. The main contributions of this paper are as follows. First, an important intrinsic natural feature of outdoor scenes free of sand/sandstorm is found that the outlines in RGB channels are somewise similar,which discloses the physical validation using the tensor instead of the matrix. Second, a tensor least square optimization model is presented for the decomposition of edge-preserving base layers and details. This model not only decomposes the color image(taken as an inseparable indivisibility) in X, Y directions, but also in Z direction, which meets the statistical feature of natural scenes and can physically disclose the intrinsic color information. The model’s advantages are twofold: one is the decomposition of edgepreserving base layers and details that can be employed for contrast enhancement without artificial halos, and the other one is the color driving ability that makes the enhanced images as close to natural images as possible via the inherent color structure.Thirdly, the tensor least square optimization model based image enhancement scheme is discussed for the sandstorm weather images. Finally, the experiments and comparisons with the stateof-the-art methods on real degraded images under sandstorm weather are shown to verify our method’s efficiency.

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I. Introduction

RECENTLY, there has been an increased interest in the image processing [1]–[19] and vision communities[20]–[32] on issues related to imaging under bad weather such as fog, haze, mist, smoke and sandstorm such as [1], [2], [4],[6], [7]–[13], [20], [21], [23], [33]–[39]. We know that most outdoor vision applications such as surveillance and autonomous navigation require robust detection of image features. Under such bad weather conditions, however, the contrast and color of images are drastically altered or degraded. Hence, it is imperative to remove weather effects from images in order to make vision systems more reliable.

There have been lots of methods [1]–[15], [19]–[21], [23]related to the image enhancement under bad weather. Among all these methods, the haze (including fog, mist) removal is highly desired in photography and computer vision applications [33]–[39]. Haze removal is a challenging problem because the haze is mainly dependent on the unknown depth information as shown in [2], [9]. The problem is underconstrained if the input is only a single haze image. Therefore,many methods have been proposed by using multiple images or additional information. For example, polarization based methods [14], [15] remove the haze effect through two or more images taken with different degrees of polarization. In[8], [10], [12], more constraints are obtained from multiple images of the same scene under different weather conditions.Depth based methods [5], [11] require the rough depth information either from the user inputs or from known 3D models.

Nowadays, single image haze removal [2], [16] has made significant progresses. The success of these methods lies in employing a stronger prior or assumption. Tan [7] observes that the haze-free image must have higher contrast compared with the input haze image and he removes the haze by maximizing the local contrast of the restored image. The results are visually compelling but may not be physically valid and may result in unpleasing artificial halos. Fattal [2]estimates the albedo of the scene and then infers the medium transmission, under the assumption that the transmission and surface shading are locally uncorrelated. Fattal’s approach is physically sound and can produce impressive results.However, this approach cannot well handle heavy haze images and may fail in the case that the assumption is broken.Heet al. [1] proposed a novel prior (dark channel prior) for single image haze removal. The dark channel prior is based on the statistics of haze-free outdoor images. Note that in most of the local regions which do not cover the sky, it is very often that some pixels (called as “dark pixels”) have very low intensity in at least one color (RGB) channel. In the haze image, the intensity of the dark pixels in that channel is mainly contributed by the airlight. Therefore, these dark pixels can directly provide accurate estimation of the haze’s transmission. In their work, combining a haze model and the matting interpolation method, they can recover a hi-quality haze-free image and produce a good depth map. Lately, Heet al. [14] proposed the fast filter called as guided image filtering (GIF) to speed up the haze removal. In addition, GIF can be employed for image decomposition and contrast enhancement.

Image contrast enhancement (such as multi-scale decomposition based enhancement methods [10], [14]–[19]) is another approach for haze removal [10], [15]. Farbmanet al.[15] advocated an alternative edge-preserving operator, based on the weighted least squares (WLS) framework. This framework was originally used to reduce ringing while deblurring images in the presence of noise [16], and it is closely related to biased anisotropic diffusion [17]. Farbmanet al. [15] showed that the WLS-based operator is robust and versatile, and may be used in many applications. The WLSbased operator is particularly well-suited for progressive coarsening of images, and for extraction of detail at various spatial scales. Thus, it can construct a new kind of edgepreserving multi-scale image decomposition and contrast enhancement for haze removal. Similarly, Subret al. [18]constructed a new kind of edge-preserving multi-scale image decomposition based on extrema and can be used for contrast enhancement such as haze removal. However, Image contrast enhancement based methods only can deal with thin haze removal. Meanwhile, very unfortunately, all the methods as shown in above fail to deal with the images under the sandstorm weather. In [23], a fusion based method is proposed. Unfortunately, how to determine the important parameters for color correction is hard in [23], in addition the contrast enhancement for heavy sand/sandstorm is not ideal(see Fig. 1(e)), and the most important thing is that it cannot perform haze removal so its applications are very limited.

In this paper, we advocate an alternative edge-preserving color image decomposition model based on the tensor least squares (TLS) framework and the statistical (inherent) feature of natural color images. This TLS framework is the extension and evolution of the WLS used in [15]–[19]. The most difference is the introduction of tensor [22] in our model (the color image is taken as an inseparable indivisibility, but not three ones of RGB respectively), which can deal with the color images from the physical nature. As for color images,the most traditional methods only treat RGB channels as three independent “gray-scale” images and process them in a monochrome way [1]–[23], [26]. These works completely ignore the inter-relationship among the multiple channels,which is likely to produce hue distortions in the reconstruction results. Hence, in essence, the three color channels (RGB) are related and not independent. Therefore, it is natural that this paper takes the color image as an inseparable indivisibility in processing via tensor.

The remainder of this paper is organized as follows. In Section II we discuss the background of sandstorm in outdoor images, and explain the causes of poor contrast and unpleasing vision. Next, in Section III we show how the TLS framework works and how it is used to perform edgepreserving decomposition in the form of tensor, and describe the multi-scale decomposition construction process. Section IV presents a detailed TLS-based multi-scale decomposition with the application in enhancement, including its connections with previous schemes. In Section V, the experiments and comparisons with previous methods on real degraded natural images by sandstorm weather are shown to verify our method’s efficiency. Finally, we conclude this paper and show our future work.

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II. Background

Fig. 1. The comparison of different methods for sand removal. (a) the original image in sandstorm; (b) the result by He et al. [1] using dark prior based haze removal algorithm (the far buildings become visible, but the unpleasant sand color is aggrandized); (c) the WLS based result by Farbman et al. [15] using contrast enhancement algorithm in RGB space with the same operation in every color channel (the far buildings become salient but without color correction); (d) the GIF decomposition based result [14] in HSV space,the contrast enhancement is operated in V channel and the saturation enhancement is made in S channel by 2.5 times of original saturation (the far buildings become salient, but the unpleasant sand color is aggrandized greatly and near the edges the artificial halos become salient); (e) the result by the method of [23] (the far buildings are still invisible, but the color is corrected greatly); (f) the result by the histogram equalization method of the V-channel in HSV space [26] (the far buildings are still invisible and there has no pretty color correction); (g) our result (the far buildings become salient, and the color is also prettily corrected. Moreover, near the edges nearly no artificial halos are salient).

The sandstorm is one of the common weathers in many places in some season, such as Beijing in the spring. In sandstorm weather, the bad visibility (e.g., Fig. 1(a) and Fig. 2)is crucial to surveillance, navigation, traffic safety and consumer/computer photography. Therefore, the restoration of color images in sandstorm is imperative and in great need nowadays. As shown in the first section, there have been lots of methods [1]–[15], [19] for haze removal. However, to our best knowledge, there has been no well-known reported work related to sandstorm/sand removal for outdoor scene images except the work in [23] up till now. Unfortunately, how to determine the important parameters in color correction is hard,in addition the contrast enhancement for heavy sand/sandstorm is not ideal (see Fig. 1(e)) and how to perform haze removal is unknown in [23].

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Fig. 2. An outdoor image under heavy sandstorm weather.

A. Model of Sand Images Under Sandstorm

In computer vision and computer graphics, the model widely used to describe the formation of a bad weather degraded image is as follows [2], [8], [9], [20]:

whereIis the observed intensity,Jis the scene radiance,Ais the global atmospheric light,Sdenotes the color image tensor,andtis the medium transmission describing the portion of the light that is not scattered and reaches the camera. The goal of haze removal is to recoverJ,A, andtfromI. The first termJ(x)t(x) on the right hand side of (1) is called direct attenuation [20], and the second termA(1 −t(S)) is called airlight [20]. Direct attenuation describes the scene radiance and its decay in the medium, while airlight results from previously scattered light and leads to the shift of the scene color. When the atmosphere is homogenous, the transmissiontcan be expressed as

whereβis the scattering coefficient of the atmosphere. It indicates that the scene radiance is attenuated exponentially with the scene depthd.

However, under the sandstorm weather the physical model in (1) does not hold any longer because the radius of particles in sandstorm is nearly 25 μm [21], which is much larger than haze (0.01–1 μm) and fog (1–10 μm) [9]. Therefore, the sandstorm should be considered as the part of the natural scene radiance, not like the haze and fog taken as the global atmospheric light. In many cases, the sand floated in the atmosphere not only attenuates (more exactly to say, shade or hide) the object radiance, but also radiates its radiance whose color is yellow/red or khaki in most cases. That is

whereBis the sand veil radiance by the sand suffusing in the air, and others are the same as (1).

Clearly, when there is haze/fog mainly without sand or with very little sand in the air, (2) reduces to (1), whileB→ 0.When there is much sand in the air,Bwill play an important role orBplays the most important role inI(S) in (2). IfBJwhile there has heavy sandstorm, maybe we only can see the khaki sand suffusing nearly everywhere in the image such as Fig. 2.

We note that, the sandstorm weather often arises from great wind in dry seasons [24] such as in the spring in some place of the north China. That is to say, when there has heavy sandstorm, the haze or fog is trivial generally. Therefore,when in heavy sandstorm, the dark channel prior based haze removal algorithm by Heet al. [1] and other haze removal algorithms [2], [6], [7] based on the physical model defined in(1) cease to work (they often enlarge the sand veil radiance with worse vision as shown in Fig. 1 (b), (d)). Meanwhile,sinceBnot only attenuates the scene radiance, but also radiates its radiance whose color is yellow/red or khaki in most cases, the contrast enhancement based methods such as[15] will result in unpleasant vision as shown in Fig. 1 (c), (d)because they cannot correct the color to normal case although they magnify the detail/edge sharpness and sometimes the saturation.

As shown in above, the haze mainly attenuates the scene contrast and saturation. Unlike the haze/fog, the sandstorm not only attenuates the scene contrast and scene saturation, but also distorts the scene color/hue greatly. Compared with the haze removal, the sand removal in outdoor images under sandstorm is very different and challenging.

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B. Statistical Features of Outdoor Natural Images

Since the sandstorm not only attenuates the scene contrast and scene saturation in images, but also distorts the scene color/hue greatly, we must adopt a new scheme to solve this problem.

Through widely studying the natural images of the sandstorm-free landscape and cityscape, we find an interesting feature that the RGB channels’ intensities have the similar/close outlines (wave shape or fluctuation) as shown in Fig. 3, and on the other hand, this similarity is only coarse and part-wise different. To verify how good the statistical feature of natural images prior is, we collect an outdoor image set from some image search engines using the most popular tags.Since sandstorm usually occurs in outdoor landscape and cityscape scenes, we manually pick out the sandstorm-free landscape and cityscape ones from the downloaded images.Among them, we randomly select 2000 images and test them.

In order to measure the intensity outline similarity and partwise difference between the RGB channels, we employ two parameters (the generalized RGB difference mean (shorted as RGBM)eand the generalized RGB difference standard deviation (STD) (shorted as RGBSTD)v) of the three channels in the color imageSwhich are defined as follows:

Fig. 3. The similar/close outlines in three images. (a) the three randomly selected sandstorm-free outdoor images with the randomly selected lines labeled by dark; (b) the intensity lines in the RGB channels respectively positioned by the dark line in the left sandstorm-free image, from left to right: the original RGB lines, the coarse scale RGB lines, the middle scale RGB lines and the fine/detail scale RGB lines; (c) the intensity lines in the RGB channels respectively positioned by the dark line in the middle sandstorm-free image, from left to right: the original RGB lines, the coarse scale RGB lines, the middle scale RGB lines and the fine/detail scale RGB lines; (d) the intensity lines in the RGB channels respectively positioned by the dark line in the right sandstorm-free image,from left to right: the original RGB lines, the coarse scale RGB lines, the middle scale RGB lines and the fine/detail scale RGB lines (Horizontal ordinate denotes the pixel position and vertical ordinate denotes the pixel gray value).

In Fig. 4(a)–(d), we respectively give the statistical histograms of the RGBM, the generalized RGBSTD for all of the 2000 sandstorm-free landscape and cityscape images and all of the 2000 landscape and cityscape images under sandstorm weather. We can see that about 80% of the RGBMs are less than 0.2 in Fig. 4(a), but about 80% of the RGBMs are less than 0.4 in Fig. 4(c). This shows that the sandstorm images will clearly enlarge the mean. We can also see that about 95%of the RGBSTDs are less than 0.4 in Fig. 4(b), but about 95%of the RGBSTDs are less than 0.2 in Fig. 4(d). This shows that the sandstorm images will clearly shrink the STD.

Fig. 4. Statistical histograms. (a) the statistical histogram of the RGBM for all of the 2000 sandstorm-free landscape and cityscape images; (b) the statistical histogram of the RGBSTD for all of the 2000 sandstorm-free landscape and cityscape images; (c) the statistical histogram of the RGBM for all of the 2000 landscape and cityscape images under sandstorm weather; (d) the statistical histogram of the RGBSTD for all of the 2000 landscape and cityscape images under sandstorm weather.

C. Tensor for Color Images

In Section II-B, the statistics tells us that the natural color image has the close relations between the three color channels(R, G, B) that cannot be processed independently/separately,and the color image should be considered carefully as an inseparable indivisibility. Therefore, in this paper we employ the tensor [22] rather than the matrix [25] to analyze the sandstorm/sand images. A tensor is a multidimensional array.More formally, anNth-order tensor is an element of the tensor product ofNvector spaces, each of which has its own coordinate system. This notion of tensors [22] is not to be confused with tensors in physics and engineering (such as stress tensors), which are generally referred to as tensor fields in mathematics. A third-order tensor has three indices as shown in Fig. 6.

Fig. 5. Statistical histograms of the Mean/STD. (a) the statistical histogram for the 2000 sandstorm-free landscape and cityscape images; (b) the statistical histogram for the 2000 landscape and cityscape images under sandstorm weather.

Fig. 6. Fibers of a 3rd-order tensor.

The color image, strictly speaking, should be considered as a 3rd-order tensor. AnM×N(size) color image is anM×N×3 3rd-order tensor in essence. We take the RGB channels as the tubes as shown in Fig. 6(c) and Fig. 7. In traditional image processing, the color images are often processed in RGB or HSV space, i.e., the R, G, B or H, S, V channels are respectively considered. In this paper, the most difference is the introduction of tensor (the color image is taken as an inseparable indivisibility, but not three ones of R, G, B respectively) in our model, which can deal with the color images in the more physically sense. In essence, the three channels (R, G, B) are related and not independent (the conclusions shown in Figs. 3–5 strongly support this idea).Therefore, we take the color image as an inseparable indivisibility in the form of tensor.

Fig. 7. X, Y, Z of the color image.

III. Tensor Least Squares for Color Image Smoothing

In this section, we first give the tensor least squares model for color images, then we show how to simplify this model to improve the computation efficiency.

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A. The Prototype of the Model

From (7) we can see that the third term strives to achieve three components whose outlines are much close to each other globally. In this term, there are no weights defined by gradients like the second term. That is to say, the third term makes no difference of smoothness inXandYdirections. So,in the next section, we will introduce the simplified version and furthermore efficient computation of (7).

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B. Simplified Version and Efficient Computation

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IV. Edge-Preserving Multiscale Decomposition and Color Image Enhancement

A. Edge-Preserving Multiscale Decomposition

Using the TLS model described above, it is easy to construct a multi-scale edge-preserving tensor image decomposition. The decomposition consists of a coarse,piecewise smooth, version of the image, along with a sequence of difference images, capturing detail at progressively finer scales. In fact, the proposed TLS method is the extension of WLS in [15]. If we only consider one single channel, our method will reduce to the method of [15]. That is to say, our method has the similar actions of edge-preserving[15]. On the other hand, since our method includes three channels (R, G, B), the new terms added by our method mainly make the three channels as close as possible so that we can recolore the images to make them more natural in color.

More specifically, let S denote the input image for which we would like to construct a (k+1)-level decomposition. LetU1,U2, …,Ukdenote progressively coarser versions ofS. The coarsest of these versions,Ukwill serve as the base layerB,with the k detail layers defined as follows:

B. Color Image Enhancement for Sandstorm Weather Images

V. Experiments and Comparisons

In order to show our method’s efficiency, it is hard to compare with all methods related to this topic. We only compare with a few well-known methods in the field of image enhancement and haze removal. These methods cannot stand for all the related methods, but they nearly cover the few main branches of degraded image restoration in bad weather conditions. These methods mainly include the currently fashionable dark channel prior based method by Heet al. [1],the WLS based enhancement method in RGB space by Farbmanet al. [15], the WLS based enhancement method in HSV space via adjusting S-channel and V-channel, the classical histogram equalization method [13], [26] and the method in [23] and so on. We not only compare with these methods for sand/sandstorm removal, but also we will compare with these methods for haze removal to show our method’s extensive and valuable applications.

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A. The TLS based Color Image Decomposition

Fig. 8. An example of TLS based decomposition/smoothing. (a) the original image; (b) left: the coarse base layer with edge-preserving; right: the fine details, the decomposition under the parameters: δ11 = 1, δ12 = 0, δ13 = 0,δ21 = 0, δ22 = 0, δ23 = 0, δ31 = 0, δ32 = 0, δ33 = 1, λ1 = 1.8, [ALx]xy = 1.2, [ALy]xy= 1.2, [Ck]xy = 0.4, ξ = 0.25; (c) left: the coarse base layer with edgepreserving; right: the fine details, the decomposition under the parameters: δ11= 0.6, δ12 = 0.2, δ13 = 0.2, δ21 = 0.2, δ22 = 0.6, δk3 = 0.2, δ31 = 0.2, δ32 = 0.2,δ33 = 0.6, λ1 = 2.8, [ALx]xy = 1.2, [ALy]xy = 1.2, [Ck]xy = 0.4, ξ = 0.25; (d) left:the coarse base layer with edge-preserving; right: the fine details, the decomposition under the parameters: δ11 = 0.2, δ12 = 0.6, δ13 = 0.2, δ23 = 0.6,δ21 = 0.2, δ22 = 0.2, δ31 = 0.6, δ32 = 0.2, δ33 = 0.2, λ1 = 2.8, [ALx]xy = 1.8,[ALy]xy = 1.8, [Ck]xy = 0.6, ξ = 0.25 (note the color transmission in this case).

In Fig. 8, these are some decomposition/smoothing results by our TLS based method with the given parameters. Note that all the details are added by 0.5 in every color channel so that we can see the negative-valued details. This figure shows that our TLS based decomposition has two effects: one is the decomposition of edge-preserving base layers and fine details that can be employed for contrast enhancement without or with less artificial halos (e.g., in Figs. 1 and 9, the comparison with other methods), and the other one is the color driving ability that will be employed by us for color correction in sand/sandstorm images (see our results of sand removal in Fig. 1 and Figs. 9–11).

Fig. 9. The comparison of different methods for heavy sand/sandstorm removal. (a) the original image in sandstorm; (b) the result by He et al. [1]using dark prior based haze removal algorithm (nearly no difference); (c) the WLS based result by Farbman et al. [15] using contrast enhancement algorithm in RGB space with the same operation in every channel (the objects become somewhat salient); (d) the result by the method in [23] (the vision is somewhat improved, but some main objects are nearly invisible); (e) our result (the vision is improved greatly and the main objects are salient).

B. Comparison in Sand/Sandstorm Removal

Fig. 10. The comparison of different methods for heavy sand/sandstorm removal. (a) the original image in sandstorm; (b) the result by He et al. [1] using dark prior based haze removal algorithm (the far buildings become more salient, but the sand color is aggrandized); (c) the WLS based result by Farbman et al. [15]using contrast enhancement algorithm in RGB space with the same operation in every channel (the far buildings are salient, but the color correction is no difference); (d) the result by the histogram equalization method in RGB space (the far buildings are still invisible, and the vision is unpleasant); (e) our result with the salient far buildings and pretty color correction.

In Fig. 1, it is the comparison of different methods for sand removal of a cityscape image under sandstorm weather. In Fig. 1(a), we find that the original image has both unpleasant vision (full of sand color) and invisible far buildings because of the sandstorm. In Fig. 1(b), it is the result by Heet al. [1]using dark prior based haze removal algorithm. Unfortunately,although this method successfully obtains the haze removal and has been widely used in many kinds of settings full of haze/fog, it fails here. We can find that the far buildings become visible, but the sand color is aggrandized greatly. That is to say, this method introduces unpleasant vision in sand/sandstorm images because the sand veil is taken as the object/scene by this method as shown in Section II-A.Fig. 1(c) is the result of the WLS based method by Farbmanet al. [15] using contrast enhancement in RGB space with the same operation in every channel. Clearly, we can find that the far buildings are salient, but the sand color is preserved without correction. Like Fig. 1(c), Fig. 1(d) is also resulted by the contrast based method. Differently, here the WLS based decomposition is replaced by the GIF that is originally employed for the acceleration of image decomposition and other applications. In addition, this result is derived in HSV space, i.e., the contrast enhancement is operated inV-channel and the saturation enhancement is made inS-channel by 2.5 times of original saturation. We can see that the far buildings are salient, but the sand color is aggrandized greatly.Moreover, near the edges some artificial halos are salient.Fig. 1(e) is the result by the method in [23]. Fig. 1(f) is resulted by the histogram equalization method. In addition,Fig. 1(f) is the result by the histogram equalization method of theV-channel in HSV space. In Fig. 1(e), the far buildings are still invisible, but the vision is enhanced greatly. However, in Fig. 1(f), the far buildings are still invisible without color correction. Finally, Fig. 1(g) is our result using the proposed method in this paper, and we can see that the far buildings are salient, and the vision is enhanced greatly as well with pretty color correction.

In Fig.9, it is a comparison of different methods for heavy sand/sandstorm removal. It shows that our method’s advantage over other methods is very clear. Figs. 10 and 11 are the other comparisons between the few methods and ours,and we can obtain the same conclusion: our method not only improves the vision greatly with more pleasing scene and better color correction, but also results in more visible contrast.

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C. Comparison in Haze Removal

Our method is well suitable for both sand/sandstorm removal and haze removal. In this section, we give the comparison with other methods in haze removal. Figs. 12 and 13 are the comparisons with other methods in haze removal.Clearly, our results yield the perfect vision for human eyes although it may be not the best method.

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VI. Conclusions and Future Work

Fig. 11. The comparison of different methods for heavy sand/sandstorm removal. (a) the original image in sandstorm; (b) the result by He et al. [1]using dark prior based haze removal algorithm; (c) the WLS based result by Farbman et al. [15] using contrast enhancement algorithm in RGB space with the same operation in every channel; (d) the result by the histogram equalization method in RGB space; (e) our result with the salient objects and pretty color correction.

Fig. 12. The comparison of different methods for haze removal. (a) the original image with haze; (b) the result by He et al. [1] using dark prior based haze removal algorithm (the far objects become salient); (c) the GIF based result by He et al. [14] using contrast enhancement algorithm in RGB space with the same operation in every channel (some artificial halos near the edges are salient); (d) the result by the method in [23]; (e) our result with the salient far objects and pretty color correction (no salient artificial halos near the edges).

Fig. 13. The comparison of different methods for haze removal. (a) the original image with haze; (b) the result by He et al. [1] using dark prior based haze removal algorithm (the far objects become salient); (c) the GIF based result by He et al. [14] using contrast enhancement algorithm in RGB space with the same operation in every channel (some artificial halos near the edges are salient with bad color); (d) the result by the method in [23]; (e) our result with salient far buildings, pleasant hue and pretty contrast (no artificial halos near the edges are salient).

Fig. 14. A failed example for super heavy sand/sandstorm removal. (a) the original image in the very heavy sand/sandstorm; (b) the result by He et al. [1]using dark prior based haze removal algorithm; (c) the WLS based result by Farbman et al. [15] using contrast enhancement algorithm in RGB space with the same operation in every channel; (d) the result by the histogram equalization method in RGB space; (e) the result by the method in [23]; (f) our result (although the far buildings become salient through enhancement, the color correction is not ideal).

Since the light reaching a camera is severely scattered and distorted by the atmosphere with color distortion by sand or dust under the bad weather of the sandstorm, the images of outdoor scenes captured will suffer from poor contrast and unpleasing vision. So, we give a method to enhance the outdoor images under the sandstorm here. In this paper, the main contributions are threefold. First, an important natural feature of outdoor scenes is found that the outlines in RGB channels are similar/close, which discloses the physical validation using the tensor instead of the matrix. Second, a tensor least squares optimization model is presented for the decomposition of edge-preserving layers and details. This model takes the color image as an inseparable indivisibility,and it decomposes the color image not only inX,Ydirections,but also inZdirection, which fits into the statistical feature of natural scenes. The model’s biggest advantage is to make the enhanced images as close to natural images as possible. Third,the tensor least squares optimization model based image enhancement scheme is discussed for the sandstorm weather images. Furthermore, our methods can be applied to other bad weather such as haze, fog, mist. Finally, the experiments and comparisons with other methods on real degraded natural images in sand/sandstorm and haze are shown to verify our method’s efficiency. These experiments and comparisons show that our method not only improves the vision greatly with more pleasing scene and perfect color correction, but also results in more salient contrast.

However, for the outdoor images under very heavy sand/sandstorm weather such as Fig. 14(a), both our method and others fail to perform sand removal. Fig. 14 shows the different results by the few methods under the very heavy sandstorm weather, which shows that all the methods fail to perform sand removal. Since under this case there is too little intrinsic natural color information that can be used for perfect recovering of hue, only the details can be magnified to enhance images (see our result in Fig. 14(e), although the far buildings become salient through enhancement, the color correction is not ideal). How to deal with these images will be our future work by sparse representation [29], [30] and machine learning [31], [32].