Goodness of fit (
Abstract
In processing of multichannel remote sensing data, there is a need in automation of basic operations as filtering and compression. Automation presumes undertaking a decision on expedience of image filtering. Automation also deals with obtaining of information based on which certain decisions can be undertaken or parameters of processing algorithms can be chosen. For the considered operations of denoising and lossy compression, it is shown that their basic performance characteristics can be quite easily predicted based on easily calculated local statistics in discrete cosine transform (DCT) domain. The described methodology of prediction is shown to be general and applicable to different types of noise under condition that its basic characteristics are known in advance or preestimated accurately.
Keywords
 Multichannel remote sensing data
 automatic processing
 denoising
 lossy compression
 performance prediction
 DCT
1. Introduction
Remotesensing (RS) data are widely used for numerous applications [1], [2]. Primary RS images acquired onboard of airborne or spaceborne carriers and intended for Earth surface monitoring are usually not ready for direct use and, thus, are subject to a certain preprocessing. This preprocessing can be carried out in several stages and includes the following operations: georeferencing and calibration, blind estimation of noise/distortion characteristics, prefiltering, lossless or lossy compression, [1], [2], etc. These operations can be distributed between onboard and onland computer means (processors) in different ways depending upon many factors [35].
Regardless of the distribution of functions, the operations onboard are usually performed in a fully automatic manner (although there can be some changes in algorithm parameters by command passed from Earth). In turn, the operations carried out on land can be, in general, performed in an interactive manner and labor of highly qualified experts is exploited for this purpose. However, a certain degree of automation of onland data processing is required as well. The need in processing automation is especially high if one deals with multichannel (e.g., hyperspectral) RS data [6], where the number of channels (components, subbands) can reach hundreds. Such RS images have become popular and widespread (available) currently due to their (potential) ability to provide rich information for various applications [6], [7].
Meanwhile, the multichannel nature of RS data results in new problems in their processing [3], [8]. The main problems and actual questions are the following:
How to manage large volumes of acquired data with maximal or appropriate efficiency (here, different criteria of efficiency can be used)?
Is it possible to skip some operations of data processing if their efficiency is not high and, consequently, if it is not worth performing them?
The latter question can be mainly addressed as mentioned below. It is strictly connected with other questions as follows:
Is it possible to predict the performance of some standard operations of RS data (image) processing?
What is the accuracy of such a prediction and is this accuracy high enough to undertake a decision to skip carrying out an operation or to set a certain value of some parameter used in the imageprocessing chain [9]?
This chapter will focus on two typical operations of multichannel RS data processing, namely, filtering and lossy compression. While considering them, the fact that the acquired images are noisy is taken into account. One can argue that noise is not seen in many RS images (or components of these images). This is true, and noise cannot be observed in approximately 80% of the visualized subband images of hyperspectral data. This is explained by the peculiarities of human vision, which does not see noise if peak signaltonoise ratio (PSNR) in a given singlechannel (component) image exceeds 32–38 dB. However, recent studies [7], [1012] have demonstrated that noise is present in all subband images and this is due to the principle of operation of hyperspectral imagers.
Moreover, it has been shown in [10], [11] that noise is (can be) of quite a complex nature and the noise acquired in multichannel RS images has specific properties. First, it is signaldependent [10], [11], [13]. Second, it is of essentially a different intensity (see Abramov et al., 2015 in [14]). More precisely, the wide variation of dynamic range and noise intensity in subband images jointly leads to wide limits of signaltonoise ratio (SNR) in components of multichannel images. This has led to the use of the term “junk bands” [15] and different strategies of coping with noisy channels in multichannel data. Some researchers prefer to use these subbands in further processing while others propose to remove them; it is also discussed whether they can be filtered or not [15]. It has been shown that if filtering of these junk bands is efficient, this can improve the classification of hyperspectral data [16]. However, the aforementioned questions concern the efficiency of image preprocessing and its prediction.
The questions raised can be partly answered with the results obtained in recent research. The objective is to show that important performance parameters of image denoising and/or lossy compression can be quickly and quite accurately predicted using simple input parameter(s) and dependences obtained in advance. The obtained results are divided into two parts. The first part deals with the prediction of filtering efficiency. This research has started in 2013 [17] and has its history in a study conducted in [18]. The second part relates to the compression of noisy images [19], [20]. In fact, the results obtained for predicting the parameters of lossy compression can be treated as based on the same principle as that for image filtering and for further research.
Before taking the image performance criteria and preprocessing techniques into consideration, it is important to note the following: first, there are two hypotheses. It is supposed that noise type is known or determined in advance. It is also assumed that its parameters are either known or accurately preestimated. It is to be noted that, currently, there are quite a few efficient methods for estimating the parameters of pure additive noise [8], [2125], speckle noise [26], and different types of signaldependent noise [1012], [27], [28]. The noise parameters are taken into account by the most modern filtering techniques that belong to the families of orthogonaltransformbased filters [2933] and nonlocal filters, for example, blockmatching and threedimensional filtering (BM3D) [34]. The same relates to modern methods of lossy compression of noisy images [19], [35].
Second, we restrict ourselves to consider the image filtering and compression techniques based on discrete cosine transform (DCT). This is explained using several reasons. DCT is a powerful orthogonal transform widely exploited in image processing. Filters and compression techniques based on DCT are currently among the best [34]. They can be quite easily adapted to the signaldependent noise directly [32], [36] or equipped with proper variancestabilizing transformations (VST) [19], [32], [37]. This restriction does not mean that the approach to prediction cannot be applied to other filtering and lossy compression techniques. This approach should be applicable (with certain modifications) but is yet to be thoroughly checked.
Third, in the analysis of the prediction approach, traditional quality metrics are employed such as mean square error (MSE) and peak signaltonoise ratio (PSNR), as well as some visual quality metrics such as PSNR human visual system masking metric (PSNRHVSM) [38]. Behavior and properties of traditional metrics are understood well by those dealing with image processing. Although PSNRHVSM is less popular, this is one of the best metrics that takes into account the peculiarities of human visual system (HVS) and that can be calculated for either one component of a multichannel image or a group of components of a multichannel image. It is expressed in dB, and it is usually either slightly smaller than PSNR (for annoying types of distortions like spatially correlated noise) or larger than PSNR (if distortions are masked by texture). This is important since we assume that the processing of multichannel images is carried out either componentwise or in groups of channel images, where a group includes the entire image in marginal case.
Fourth, other criteria of imageprocessing efficiency, such as classification accuracy, object detectability, etc., are important for the preprocessed RS data. We are unable to predict them, but recent research shows [39] that these criteria are connected with the traditional criteria of image processing. Thus, it is expected that if good values of conventional and HVS metrics are provided due to preprocessing, appropriate classification accuracy and other criteria will be attained.
2. The considered imageperformance criteria and preprocessing techniques
This chapter considers the following model of an observed multichannel image:
where
and input PSNR
The same assumptions are valid for input
After applying a considered filter, one obtains a filtered image
Output
Then, one has to characterize the efficiency of filtering. One way to do this is to use
Small values of the ratio in expression (6) and large values of expressions (7) and (8), both expressed in dB, are evidence in favor of efficient filtering.
Similarly, after lossy compression, one obtains
occurs to be less than
on QS for the lossy DCTbased coder AGU [42] for two known grayscale test images Airfield (Fig. 1(b)) and Frisco (Fig. 1(c)) corrupted by additive white Gaussian noise (AWGN) with variance
The lossy compression in the neighborhood of OOP has obvious advantages. Compressed images have high quality, and, at the same time, they have CR considerably larger than for lossless compression [9], [44]. Because of these benefits, the lossy compression of noisy images in the OOP neighborhood is considered. If OOP does not exist, nevertheless, the recommended setting
where
Certainly, there are also other valuable performance criteria. For image prefiltering, it is important to know the computational efficiency of the denoising method and how easily it can be implemented, especially onboard. For image lossy compression, it is important to know CR provided and how easily it can be attained. To partly address these issues, the filtering and compression techniques are briefly described.
DCTbased filtering [18], [30] is performed in a blockwise manner, where 8 × 8 pixels are a typically set block size. Filtering can be performed with nonoverlapping, partly overlapping, and fully overlapping blocks. In the latter case, filtering efficiency (expressed in improvement of PSNR (
There are three main steps in processing: direct 2D DCT in each block; thresholding of DCT coefficients; inverse DCT applied to thresholded DCT coefficients; then, the filtered data from overlapping blocks are aggregated. Within this structure, different variants of thresholding are possible but employing hard thresholding is preferred, where DCT coefficient values remain unchanged if their amplitudes exceed a threshold or are assigned zero values otherwise. If one deals with AWGN, the threshold is set fixed as
For spatially uncorrelated signaldependent noise with
Finally, for spatially correlated and signaldependent noise with
In expressions (14–16),
Conventional BM3D [34] is a more sophisticated denoising method. It presumes search for similar patches (blocks), with their joint processing in a 3D manner using DCT and Haar transform, and postprocessing stage. This filtering principle, originally designed to cope with AWGN in grayscale images, has been later adapted to the cases of signaldependent noise after a proper VST [37], spatially correlated noise [45] and color (threechannel) images corrupted by AWGN [46]. The BM3D and its modifications provide a slightly better performance than the corresponding modifications of the conventional DCTbased denoising by the expense of considerably more extensive computations.
The lossy compression technique called AGU [42] is based on DCT in 32 × 32 pixel blocks, a more efficient (compared to JPEG) coding of quantized DCT coefficients and postprocessing to remove the blocking artifacts after decompression. This coder is quite simple but slightly more efficient than JPEG 2000) or set partitioning in hierarchical trees (SPIHT) in rate/distortion sense. This coder has 3D version [19] and CR for both 2D and 3D versions is controlled (changed) by QS.
3. Prediction of filtering efficiency
The main idea of filtering efficiency prediction is the following [17]. Suppose there is some input parameter(s) able to jointly characterize image complexity and noise intensity and also there is some output parameter(s) capable of adequately describing the image denoising efficiency. Assume that there is a rather strict connection between these input and output parameters that allows predicting output value(s) having input value(s).
An additional assumption (and requirement to prediction) is that input parameter(s) have to be calculated easily and quickly enough, faster than denoising itself (otherwise, the prediction becomes useless). If all these assumptions are valid, it becomes possible to determine a predicted output value before starting image filtering and to decide whether it is worth filtering a given image (component) or not. Another decision can relate to setting parameter(s) of a used filter. For example, if a processed image seems to be textural (having high complexity), parameter(s) of a used filter can be adjusted to provide better edge/detail/texture preservation. For example, the parameter
Keeping these general principles in mind, we have to address several tasks:
What is a good (in the best case, optimal) input parameter (or a set of parameters)?
What is a good (proper, acceptable) output parameter (or a set of parameters) that allows to characterize the filtering efficiency adequately and to undertake a decision (on using filtering or not, on setting a filter parameter, etc.)?
How to get dependence between output and input parameters and how accurate it is?
These questions are partly answered below and the outcomes obtained in design and performance analysis of prediction techniques are described. We believe that a partial answer to the second question is the following. The ratio in expression (6) as well as the parameters
3.1. Input and output parameter sets testing and comparison
Based on the outcomes of the study [18], Abramov et al. in 2013 [17] observed that there is dependence between efficiency of filtering expressed by (6) and simple statistics of DCTcoefficients determined in 8 × 8 blocks. Two probability parameters have been considered. The first one denoted as
The results of the study conducted in [17] have also shown the following. First, quality of fitting has to be characterized quantitatively. For this purpose, the approach [50] works well. It provides the parameter (coefficient of determination)
The conclusions drawn in [17] can be recalled here. First, the prediction of filtering efficiency for BM3D is less accurate than for the conventional DCTbased filter. This conclusion has been confirmed in later studies. This is associated with the use of two denoising mechanisms (DCT denoising and similar block search with their joint processing), where the latter mechanism has no connection to DCT statistics. Second, although the prediction accuracy for both
There are also observations understood later (in two recent years). First, there should be some restrictions imposed on the approximating function. For example, it is clear that the ratio in expression (6) cannot be negative. It is also clear that an approximating (fitting) function should be determined for all possible values of its arguments. Since the probabilities serve as arguments, they can vary from zero to unity. Meanwhile, arguments in both scatterplots in Fig. 2 vary in narrower limits. Besides, it could be good for curve fitting to have point arguments with approximately uniform density.
These requirements have been satisfied by using considerably more test images (including highly textural ones) and a wider set of noise standard deviations (including quite small ones). This has allowed obtaining scatterplot points for small
Examples of the obtained scatterplots and fitted curves for the DCTbased denoising are shown in Fig. 3. As it is seen, fitting is rather good and coefficient of determination is approximately 0.95 (see the details below). We believe these are already good results that allow practical recommendations. For example, it is clearly seen that there is no reason to carry out filtering if
Expressions for the obtained approximations for the DCT filter are as follows (we give only the functions of
The values of




0.978  0.955 

0.963  0.935 

0.82  0.78 
It has been discovered that not only the mean of local (block) estimates of probability
where
The results of using multidimensional regression are presented in Table 2. The abbreviations used are the following:




DCT filter 


0.963 

0.971  

0.974  

0.976  

0.977  


0.848  

0.923  

0.926  

0.927  

0.928  
BM3D 


0.95 

0.955  

0.959  

0.961  

0.961  


0.845  

0.905  

0.905  

0.909  

0.917 
The conclusions are the following. The use of more input parameters leads to larger (better)
More input parameters provide better prediction. At the same time, more time is needed for calculation of input parameters (although their calculation is not difficult). Then, a compromise solution could be the use of the dependence of the type
where





DCT filter 

0.023  6.338  7.459 

2.225*10^{−4}  10.81  37.14  
BM3D 

0.019  6.591  6.849 

5.324*10^{−5}  12.42  41.36 
The expression (20) is not the only way to combine several input parameters into a joint output. Neural networks (NN) are known to perform this task rather well and to be good approximators [52]. This property has been used by us in [53] to make the neural network predict the considered metrics based on multiple input parameters. The obtained results are practically the same as in Table 3. Therefore, there is no need to use a more complex NN approximator instead of expression (20).
A more reasonable solution is to look for better input parameters. Such a study has been conducted in [51]. It has been shown that the probability
The obtained results for multiparameter fitting are presented in Table 4. The abbreviations are the same as in Table 2. The first observation is that even for one parameter (mean of local probabilities), the values




DCT filter 


0.986 

0.989  

0.989  

0.989  

0.99  


0.844  

0.944  

0.949  

0.951  

0.952  
BM3D 


0.975 

0.977  

0.978  

0.978  

0.978  


0.852  

0.935  

0.939  

0.941  

0.941 





DCT filter 

0.168  10.8  19.28 

0.01  15.66  144.3  
BM3D 

0.148  11.33  17.7 

0.004  18.25  161.7 
3.2. Analysis for signaldependent and spatially correlated types of noise
Let us define the models of signaldependent noise used. According to a first model [7], [11], the expression (1) transforms to
where
As mentioned in Section 2 (expression no. 15), the local threshold is set as
where
Some of the results of studies in our papers [54], [55] are presented next. One aspect that was specially addressed in these studies was to check the influence of an image set used in forming a scatterplot. In fact, two scatterplots have been formed separately: for the set of standard images used in optical image processing as Baboon, Barbara, Lena, etc., and for the set of images called “Remote Sensing” as Frisco, Diego, etc. The reason for such study was the following fact. Some people from RS community are categorically against using standard grayscale test images in their studies although there are no commonly accepted sets of test RS images.
The methodology of obtaining scatterplot was modified a little. For the noise expression model (22), three different cases were modeled: prevailing influence of SI noise, dominant influence of SD noise, and comparable contribution of both components. As a result, a wide range of mean
Two examples of image processing are presented here. Fig. 6(a) represents the noisy image Frisco, where noise parameters are σ_{0}^{2}=100;
For a reallife data, it is impossible to determine true values of the considered metrics characterizing filtering efficiency. However, it is possible to analyze the predicted values and denoising results visually. For fragments of subband images of hyperspectral sensor, Hyperion, such analysis was done. For example, noise parameters of the expression model (22) have been blindly estimated [11]. The noisy image for the 13th subband of the set EO1H1800252002116110KZ is depicted in Fig. 7(a). Noise is clearly seen. The prediction of
The subbands 13...22 are considered for two sets of Hyperion data. The values
Considering certain benefits achieved due to using
Additional studies concentrated on the multilook SAR images that were corrupted by pure multiplicative noise [57]. Analysis has been done for speckle variance
Understanding that, in practice, noise can be spatially correlated [33], the case of spatially correlated noise – additive in [45] and multiplicative in [57] – are also studied. A difficulty of dealing with spatially correlated noise is that there are numerous shapes (and parameter sets) of 2D autocorrelation function or spatial spectrum of such a noise. Thus, studying a particular case of spatially correlated noise gives only limited information on general dependences. Hence, two models of spatially correlated noise (called middle correlation and strong correlation) have been considered [45]. A peculiarity of prediction is that the local estimate of probability
The scatterplots and fitted curves are presented in Fig. 10. The fitted curves are similar and they clearly show that there is no reason to filter images if
We have also studied the case of spatially correlated speckle [57]. It has been shown that the prediction seems possible for a spatially correlated noise. However, more research is needed to understand how to select a parameter or several parameters to characterize spatial correlation and how it can be involved in prediction.
Finally, a preliminary research has been carried out for denoising color images corrupted by AWGN with equal variance values in channels [58]. There are two differences in prediction. First, all DCT coefficients in 3D block are subject to analysis for estimating the local probabilities. Second, the metric PSNRHMA [59], which is a color extension of PSNRHVSM, and improvement of this metric due to filtering similar to expression (8) have been used. In addition, instead of BM3D, its color version called CBM3D has been analyzed [46].
The scatterplots have been obtained and curves were fitted to them (see examples in Fig. 11). As mentioned earlier, filtering is useless for
Taking into account our previous experience, the multiparameter input was analyzed with exponential function expressed in (20). Considerable improvement has been reached, especially for
4. Prediction in lossy compression of noisy images
In this section, the compression of images corrupted by AWGN is considered. Lossy compression is carried out by the aforementioned coder AGU with
4.1. Prediction of OOP existence and metrics’ values in it
This section shortly describes how the scatterplots were obtained. As in the filtering case, a set of grayscale test images of different content and complexity was used. AWGN of different intensity has been added and then the obtained images have been compressed by AGU. After this, the parameters (12) and (13) have been calculated as well as
The obtained scatterplot is presented in Fig. 12. A specific feature of this scatterplot is that it has negative values and they seem to be approximately −3.5 dB for
The scatterplot for the metric
Although prediction has been studied by simulations only for images corrupted by AWGN, it can also be applied to images corrupted by a signaldependent spatially uncorrelated noise under condition that a proper VST is applied to them before compressing. Such VST (a generalized Anscombe transform in this case) provides approximately constant noise variance that usually equals to unity. Thus, QS = 4 is used. This approach has been used for Hyperion data and the results are presented in Fig. 14. There are two groups of subbands that are usually not analyzed in Hyperion data since they are too noisy. Thus, the prediction values are not given for all subbands. Analysis of the presented values shows that there are only a few subbands where it is worth expecting OOP. For most other subbands,
Fig. 15 shows the original and the decompressed images in 110th subband, where decrease of visual quality according to quantitative criteria is predicted. Noise is not seen in the original image and the compression practically does not influence the image quality (in our opinion, both images look the same).
A study [44] also presents data for three other DCTbased coders, where two of them are specially suited for providing better visual quality. It is demonstrated that the coder adaptive DCT (ADCT), which exploits the optimized partition schemes [60], provides certain improvements compared to AGU. Meanwhile, DCT coders oriented on improving the visual quality being applied to noisy images do not offer substantial benefits and, moreover, are even less efficient in many practical situations.
4.2. Prediction of compression ratio in OOP
The methodology of predicting CR in OOP is the same as that for filtering. It is based on the scatterplot obtaining and curve fitting. The only difference is that the vertical axis relates to CR, while the horizontal axis, as earlier, corresponds to mean probability. Two mean probabilities
Two lossy compression methods, namely, the coders AGU and ADCT, have been studied. Their scatterplots are presented in Fig. 16. Contrary to other cases considered above, fitting is performed using a sum of two weighted exponential functions. As can be seen, fitting in both cases is very good with
We did not have reallife multichannel images corrupted by AWGN. But the hyperspectral data for the sensors Hyperion and airborne visible/infrared imaging spectrometer (AVIRIS) were available. Noise in them is signal dependent [14] with prevailing SD component for the model (22). The parameters of this noise were estimated in an automatic manner [11] and, thus, it became possible to apply VST (a generalized Anscombe transform with properly adjusted parameters) with converting noise into pure additive with unity variance.
Lossy compression in OOP neighborhood has been applied after VST. After decompression, inverse transform has to be applied, respectively. The obtained and predicted values of CR for Hyperion data are depicted in Fig. 17(a). As can be seen, the curves are in good agreement. There are some channels where predicted CRs are slightly larger than attained ones. This is explained by the imperfectness of VST and blind estimation of noise parameters for channels with high signaltonoise ratio. The largest CRs take place for subbands with low SNR (these are the subbands with indices 13–20, 125–130, and 175–180).
The results for the AVIRIS test image Lunar Lake are given in Fig. 17(b). Here, the agreement between the predicted and the attained values is even better than for the Hyperion data. Again, the largest CR is observed for subbands with low SNR. There are considerable differences in maximal and minimal values of CR. The main reason is the different SNR and different dynamic range in subband images. Certainly, CR also depends upon the image content.
5. Conclusions and future work
It is demonstrated that it is possible to predict the efficiency of image filtering as well as the parameters of lossy compression of a noisy image in OOP neighborhood. As opposed to the earlier known approaches that allow predicting potential efficiency of filtering, the present approach predicts practically a reachable performance and makes this very rapidly, by one or more orders faster than filtering or compression itself.
Certainly, a limited number of quality metrics, filtering, and compression techniques have been considered. However, it is important that a general methodology of prediction is proposed, and it is shown there are somewhat strict connections between simple input parameters (that can be easily and quickly calculated) and output parameters that are able to adequately characterize the efficiency of filtering or lossy compression techniques. In favor of this methodology, there are certain facts. First, there are many modern filters that have filtering efficiency of the same order as the DCTbased filter and BM3D. Thus, predicting denoising efficiency for the filters mentioned above, it is possible to approximately predict performance for other modern filters (although such prediction would be less accurate). Second, the same holds for lossy compression methods. For example, AGU and JPEG2000 provide similar performance characteristics. Then, by predicting compression parameters for AGU, they are, in fact, estimated for JPEG2000 as well.
Concerning the decision making, whether to perform filtering or not, strict recommendations have been given for probabilities
The results of this research show that although sometimes the prediction of performance characteristics based on one input parameter is appropriately accurate, there are several means to improve the prediction accuracy. One way that deals with multiparameter input has been already used for particular cases. The use of mean
There are also other possible directions for future research. 3D filtering warrants a more thorough study, at least, for the case of more than three channels. The same relates to 3D lossy compression performance, which has not been tried to predict yet. Compression parameters for QS other than the one recommended for OOP is also of sufficient interest in DCTbased lossy compression. Influence of errors in
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