National Aerospace University “Kharkov Aviation Institute” SPIE Remote Sensing Performance prediction for 3D filtering of multichannel images Oleksii Rubel 1, Ruslan Kozhemiakin 1, Sergey Abramov 1, Vladimir Lukin 1, Benoit Vozel 2, Kacem Chehdi 2 1 Department of Transmitters, Receivers and Signal Processing, National Aerospace University, Kharkov, Ukraine 2 IETR UMR CNRS University of Rennes 1 – Enssat, Lannion, France

National Aerospace University “Kharkov Aviation Institute” Prediction of denoising efficiency 2 The DCT-based filters and, particularly, the conventional DCT-based filter have demonstrated high efficiency for a wide class of images and noise models. Component-wise DCT-based denoising often produces unsatisfactory results for multichannel imagery (vector processing is able to work better). Modifications of the DCT-based filter for multichannel images have been already proposed (MC-DCT). In practice, there are cases when filtering does not provide effective noise removal or even decrease visual quality. It could be good to predict denoising efficiency before starting image filtering. The main goal of prediction consists in obtaining some fast and accurate estimates of quality metrics without carrying out filtering itself. As an estimate of image quality and denoising efficiency, traditional peak SNR (PSNR) metric can be used. Since PSNR is expressed in dB, the contribution of filtering to quality increasing can be measured and expressed by improvement of this metric (IPSNR).

National Aerospace University “Kharkov Aviation Institute” Conventional DCT-based denoising and its modification for multichannel images 3 where is input transformed block, S in denotes an input block in spatial domain, n and m are indices of DCT coefficients in block, σ is standard deviation noise, β is thresholding parameter (the recommended value is 2.7). Afterward, inverse DCT is performed for the considered block, and then filtered values for overlapping blocks are collected together. Conventional DCT filter for grayscale images: The standard 2D DCT-based filter is a quite simple image processing procedure. It performs 2-dimensional DCT for full-overlapping blocks that are usually of fixed size 8x8 pixels. After performing 2D DCT in each block, hard thresholding procedure is commonly applied :

National Aerospace University “Kharkov Aviation Institute” Conventional DCT-based denoising and its modification for multichannel images 4 Multichannel DCT filter: An easy way to process corresponding blocks from different channels collaboratively is to use 3D DCT. It is possible and reasonable to split (separate) it into 2D and 1D transforms. 2D DCT is used for single-channel processing, so it can be applied to each block successively from all channels. Each transformed block can be stretched into a vector and all obtained vectors can be gathered into joint 2D array. Finally, this array can be transformed by 1D DCT with regard to certain DCT coefficients. Thus, the hard thresholding procedure would be the following: where k is index of DCT coefficient 1…64 for 8x8 block, l is index of channel.

National Aerospace University “Kharkov Aviation Institute” Image statistics in DCT domain 5 The DCT-based denoising removes DCT coefficients that are regarded to be noisy. The number of removed or preserved DCT coefficients strictly influences denoising efficiency. For prediction, probability that coefficients in image blocks will be removed or saved can be used. where q is an index of a analyzed block (q=1,…,Q), α is the thresholding parameter used for prediction), P ασ is a set of local probability estimates in a processed image blocks. The simplest way to use the set of local estimates of probabilities is to calculate their mean and to substitute it into the fitted curve. where k is index in block of some channel in vector representation, l denotes channel index, L is the number of channels. The adapted version of E out for multichannel images filters is based on analysis of DCT-coefficient amplitudes in 3D blocks:

National Aerospace University “Kharkov Aviation Institute” Basic prediction methodology 6 Prediction is based on assumption that there is an a priori obtained dependence of output parameter (metric) on input parameter (image/noise statistic).. For dependence obtaining, scatter-plots have been formed where a given probability serves as an argument (input) and a corresponding metric as an ordinate of a scatter-plot point. Each point is obtained for a test image corrupted by AWGN with a given variance, then filtered and analyzed (by analysis here we mean calculation of an analyzed metric, for IPSNR). Having a scatter-plot, a curve is fitted into it to describe dependence of a predicted metric on an input statistical parameter. For prediction performance characterization, traditional determination coefficient or goodness of fit R 2 is used. High values of R 2 indicate quite good predicting performance. It is enough to have about 300…500 local estimates of probability P ασ for effective prediction. A used statistical parameter is calculated for a given image under assumption of known variance of noise. Then, this parameter(s) is used as argument of a fitted curve, and a predicted value of an analyzed metric is obtained. Analyzing different approaches to fitting, it has been established that for P 0.5σ, the following expression of fitting function is suitable: where Metric pred is a predicted metric, a and b i are coefficients of fitting functions, O i is a statistical operator, i.e. a parameter characterizing the distribution of local estimates of P ασ.

National Aerospace University “Kharkov Aviation Institute” Hyperspectral test images: EO-1 Hyperion 7 For fitting functions obtaining, five relative noise levels (input PSNR = 20, 25, 30, 35 and 40 dB) and 20 Hyperion test images (with low cloud-cover, less than 10% from the overall image covered area) have been used. There are 220 unique spectral channels covering spectrum from 357 to 2576 nm. The images of radiometric product have 242 bands, 256 samples and approximately 3130 rows. For multichannel processing, the following groups of sub-bands: , , and 211…220 have been used. EO1H KF.L1R EO1H K2.L1R EO1H KF.L1R EO1H KF.L1R

National Aerospace University “Kharkov Aviation Institute” Hyperspectral test images: EO-1 Hyperion 8 where S is the estimated dynamic range of an image. As the test set, Hyperion images divided into small parts with the size 256x256 pixels (voxels) have been used. EO1H KF.L1R Due to different dynamic ranges of sub-band images, relative noise levels have been used. To obtain an estimate of dynamic range, the difference between 0.99 and 0.01 quantiles in each sub-band was calculated. Standard deviation of artificially added noise has been calculated to produce PSNR values: 20, 25, 30, 35 and 40 dBs for noisy images:

National Aerospace University “Kharkov Aviation Institute” Multichannel DCT-based denoising performance 9 The improvement of PSNR provided by 5- and 10-channels can be better than one-channel denoising by about 3-5 dB. As result, the output image is considerably better, many fine details are preserved. PSNR = 20 dB (noisy) PSNR = dB (1-channel denoising) PSNR = dB (5-channels denoising) PSNR = dB (10-channels denoising)

National Aerospace University “Kharkov Aviation Institute” Predicted IPSNR performance using mean of P 0.5σ for multichannel denoising 10 Multichannel DCT-based prediction produces lighter and more homogenous maps that means more efficient joint denoising than component-wise. P 0.5σ = channel denoising P 0.5σ = channels denoising P 0.5σ = channels denoising

National Aerospace University “Kharkov Aviation Institute” IPSNR prediction using mean of P 0.5σ for 1-channel denoising 11 The dashe lines correspond to the same input PSNRs for images from different bands. It is well seen that all sub-band results are quite compact and form some trend. Totally, the scatter-plot contains almost points provided from 5 relative noise levels, 20 test images and about 12 sub-images obtained from them. The obtained set of P 0.5σ covers full range of possible values which is important for fitting and prediction. The values of IPSNR are not much scattered with respect to fitted curve. That means that high prediction performance (accuracy) can be provided for IPSNR. The presented scatter-plot clearly shows strong and obvious dependence of IPSNR on P 0.5σ.

National Aerospace University “Kharkov Aviation Institute” IPSNR prediction using mean of P 0.5σ for multichannel denoising 12 If more channels are used in joint denoising, the obtained results (scatter-plot points) become less scattered and most of points move to higher P 0.5σ.

National Aerospace University “Kharkov Aviation Institute” IPSNR prediction using mean of P 0.5σ for multichannel denoising 13 The difference in increasing IPSNR for the cases of 7-10 channels processed jointly is not essential. That means that often there is no necessity to use maximally possible number of channels.

National Aerospace University “Kharkov Aviation Institute” Universal IPSNR prediction for multichannel denoising 14 It is seen well that results from different sub-ranges are compatible for their joint use for prediction.

National Aerospace University “Kharkov Aviation Institute” Goodness of fit for fitting functions 15 Goodness of fit parameter values are very high, R 2 > 0.98 for all functions. Note that R 2 > 0.9 means that two variables are strictly connected. Estimated parameters a and b for all functions are very close. As a result, the obtained functions are similar, have close goodness of fit and, potentially, could be used for other data. 3D filtering provides by 1…6 dB better IPSNR compared to component-wise processing. Due to this, it occurs that it is almost always expedient to apply 3D denoising to improve multichannel RS images. Prediction method can be applied to denoising results obtained from filtering of different bands or groups of neighbor bands filtered jointly. The obtained prediction functions for different cases are very similar; then, it is possible to obtain generalized function with high goodness of fit that can be used for different cases effectively. ChannelsabR2R All

National Aerospace University “Kharkov Aviation Institute” Conclusions 16 The proposed prediction method uses image statistical parameters in DCT domain that can be easily calculated. The estimated statistical parameters are strictly connected with DCT-based denoising mechanism and efficiency. Prediction method and DCT-based denoising are easily extended to multichannel images. It is possible to predict improvement of PSNR for multichannel-DCT filter by fitting functions obtained in offline mode. Similar ranges of denoising results (IPSNR) and prediction parameter P 0.5σ for multichannel filtering with different number of jointly processed channels make procedure universal respect to data nature. The fitted functions have high goodness of fit values that means high prediction performance. The proposed prediction method is very fast and can be extended to other filters and noise models.

National Aerospace University “Kharkov Aviation Institute” Performance prediction for 3D filtering of multichannel images 17 Thank you! Benoit Vozel - Oleksii Rubel - Ruslan Kozhemiakin - Sergey Abramov - Vladimir Lukin - Kacem Chehdi -