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we consider the standard problem of denoising grayscale images that are corrupted with additive white gaussian noise @xcite . in this setup , we are given the _ corrupted _ ( or _ noisy _ ) image @xmath0 where @xmath1 is some finite rectangular domain of @xmath2 , @xmath3 is the unknown _ clean _ image , @xmath4 are independent and distributed as @xmath5 , and @xmath6 is the noise level . the goal is find a _ denoised _ estimate @xmath7 of the clean image from the corrupted samples . the denoised image should visually resemble the clean image . one way to quantify the resemblance is using the mean - squared - error ( mse ) which is defined to be @xmath8 later in the paper , we will use the peak signal - to - noise ratio ( psnr ) which is given by @xmath9 . the image denoising problem has been extensively studied and a survey of even a fraction of this literature is beyond the scope of this paper . we instead refer the interested reader to @xcite - @xcite and the references therein . our present interest is in the image denoising applications of the edge - preserving bilateral filter @xcite . the denoised image in this case is set to be @xmath10 where @xmath11 the gaussian kernels in are respectively referred to as the _ spatial _ and _ range _ kernels . in practice , the domain @xmath12 is restricted to some neighbourhood of the origin . typically , @xmath12 is a square neighbourhood , @xmath13 \times [ -w , w]$ ] , where @xmath14 @xcite . we refer the reader to @xcite for a detailed exposition on the working and in particular the edge - preserving property of the filter .    the bilateral filter has received renewed attention in the image processing community in the context of image denoising @xcite . it is well - known that , while the filter is quite effective in removing modest levels of additive noise , its denoising performance drops at large noise levels @xcite . it was demonstrated in @xcite that a patch - based extension of the filter can be used to bring the denoising performance of the filter at par with state - of - the - art methods . these , and other advanced patch - based methods @xcite , are however much more computation - intensive than the bilateral filter .      in this work , we present a couple of ideas for improving the denoising performance of the classical bilateral filter , and give fast algorithms for the same . the contributions ( and the organization ) of the paper are as follows :    @xmath15 in section [ sec : rbf ] , we demonstrate how the denoising performance of the bilateral filter can be improved at large noise levels ( at almost no additional cost ) by incorporating a simple pre - processing step into the framework . @xmath15 the modified filter is designed to be robust at large noise levels , and tends to perform poorly below a certain noise threshold . to get the best of the either filters at all noise levels , we propose to optimally combine them in a weighted fashion in section [ sec : owf ] . the optimality is in terms of a certain surrogate of the mean - squared - error ( mse ) known as stein s unbiased risk estimate ( sure ) @xcite . this mse - estimate is known to be quite accurate in practice @xcite and , as a result , the combined filter is almost always guaranteed to provide a lower mse than the original filter . @xmath15 following @xcite , we present an approximation for the proposed filtering in section [ sec : fastsure ] that has a fast implementation . we derive the sure estimator for this approximation , and demonstrate how it can be efficiently computed in the process of approximating the bilateral filter . the overall cost of computing the estimator and the optimally - weighted filters is about twice the cost of computing a single bilateral filtering using the fast algorithm in @xcite . we provide both visual and quantitative denoising results on standard test images in section [ sec : experiments ] which demonstrate that the improvement over the original filter is significant both visually and in terms of psnr . notice that the range filter in operates on the noisy samples . in other words , the corrupted image is used not just for the averaging but also to control the blurring via the range filter . what if the range filter could directly operate on the clean image ? it can be verified that the denoising image obtained using this `` oracle '' filter is visibly better and has higher psnr , which is not surprising . of course , we do not have access to the clean image in practice , and thus the oracle bilateral filter can not be realized . nevertheless , one could consider some form of proxy for the clean image . our proposal is simply to use the box - filtered version of the noisy image as a proxy . in particular , we propose the following robust bilateral filter ( rbf ) : @xmath16 where @xmath17 the amount of smoothing induced by the box filter is controlled by @xmath18 . we performed exhaustive simulations in this direction . the simulation results suggest that @xmath19 ( @xmath20 blur ) is optimal for most settings . a possible way to explain this is that this small blur is able to suppress the noise without excessively blurring the image features . the denoising results obtained using the standard and the robust filter a couple of test images are shown in table [ table1 ] . we note that the filters have been independently tuned with respect to @xmath21 and @xmath22 to get the best psnr . these results ( and the results on other test images that are not shown here ) suggest that the robust filter starts to perform better beyond a certain noise level depending on the type of image . this is not surprising since , at low noise levels , the box filtering introduces more blurring than noise suppression which brings down the overall signal - to - noise ratio . indeed , the corrupted image is already a good proxy for the clean image when the noise floor is small . on the other hand , notice that the improvement in psnr is often as large as @xmath23 db at large noise levels . the results in table [ table1 ] suggest that we could possibly perform better denoising by combining and . a particularly simple idea would be to take a linear combination of these estimates . that is , we could set the denoised image to be @xmath24 notice that this includes and as special cases . this bring us to the question of setting the weights @xmath25 and @xmath26 . this can hypothetically be done by minimizing the mse between and the clean image . however , we do not have access to the clean image . this is precisely where the classical stein s unbiased risk estimate ( sure ) @xcite is useful . this is given by @xmath27 sure has the property that its expected value equals that of for the gaussian noise model in @xcite . this makes it an useful surrogate for the mse , which can be computed without the knowledge of the clean image . we note that the idea of taking linear ( or affine ) combinations of estimators and tuning the weights to get the optimal sure has been tried in different contexts in the literature ; e.g. , see @xcite . note that the added computation required for sure are the computation of the partial derivatives in . we propose to find the optimal weights in by minimizing sure . in particular , on substituting in , we see that sure is quadratic in @xmath25 and @xmath26 . thus , by convexity , a necessary and sufficient condition for optimality is that that the gradient of must vanish at the optimal weights . in particular , it can be verified that the resulting gradient equations can be written as @xmath28 , where @xmath29   \sum_{\i \in i } \hat{f}_1(\i)\hat{f}_2(\i )   &   \hspace{2 mm } \sum_{\i \in i } \hat{f}_2(\i)^2   \end{pmatrix},\ ] ] and @xmath30      \sum_{\i \in i } f(\i ) \hat{f}_2(\i ) - \sigma^2 \sum_{\i \in i } \partial_{\i } \hat{f}_2(\i )   \end{pmatrix},\ ] ] and @xmath31 are the optimal weights . to simply notations , we will henceforth use the shorthand @xmath32 in place of the operator @xmath33 appearing in . in summary , by computing @xmath34 and @xmath35 , and solving the @xmath36 linear equation @xmath28 , we get the optimal weights . the weights are then plugged into to get the final denoised image . .comparison of the standard bilateral filter ( sbf ) and the robust bilateral filter ( rbf ) at @xmath37 . for a fixed image and noise level , we tuned @xmath21 and @xmath22 to individually optimize the psnr obtained using both methods . [ cols="<,^,>,>,>,>,>,>",options="header " , ]     [ table2 ]    we now present some denoising results on standard test images . our objective is to compare the denoising results obtained using the proposed modification with the standard bilateral filter , both visually and in terms of psnr . in this work , we assume that @xmath6 is provided ; this has to be estimated in practice from the noisy image . in this regard , we note that it has been demonstrated in @xcite that the sure is quite robust to standard data - based estimates of @xmath6 . we remark that one can optimize the sure for the proposed filter with respect to @xmath21 and @xmath22 as done in @xcite ; however , we do not investigate this possibility in this paper . l*6cr & @xmath38 & ( 4,20 ) & @xmath39 & @xmath40 & @xmath41 & @xmath42 + direct & 32s & 120s & 70s & 185s & 74s & 125s + fast & 1.2s & 0.8s & 0.7s & 0.6s & 0.5s & 0.6s +    [ table3 ]    in table [ table2 ] , the denoising performance of the proposed weighted filter at different noise level is compared with the standard and the robust bilateral filter , as well the patch - based non - local means filter @xcite . we have used @xmath19 in for all the experiments . to have a fair comparison for a given image and noise level , we independently tuned @xmath21 and @xmath22 to optimize the psnrs of the respective filters . we also tuned the parameters of nlm to get the optimal psnr at each noise level . as remarked earlier , the robust filter starts to perform better than the standard filter beyond a certain noise level ( @xmath43 ) . notice that the psnr obtained using the optimally - weighted filters is consistently higher than that of the constituent filters at all noise levels ; the improvement is often as large as @xmath44 db . on the other hand , the improvement in psnr over the standard bilateral filter is often as high as @xmath23 db . this does not come as a surprise since it is well - known that the bilateral filter has a lot of scope of improvement at high - noise levels @xcite . however , what does come as a surprise is that proposed filter is competitive with the non - local means filter @xcite that uses patches ( groups of neighboring pixels ) instead of single pixels for denoising . for a visual comparison , the results of a particular denoising experiment are shown in figure [ images ] . notice that the denoised image obtained using the proposed filter looks much more sharp than that obtained using the standard filter , and has a significantly higher psnr . also , notice that the noise in the background is much less in ( d ) compared to ( c ) . in table [ table3 ] , we compare the run times of the direct and the fast implementation of the proposed filter for different parameter settings . notice that the fast implementation is significantly faster than the direct implementation . we demonstrated how a simple pre - processing step can substantially improve the denoising performance of the bilateral filter . to consistently get the best of the standard and the pre - processed filter at all noise levels , we proposed to optimally weight them using stein s unbiased estimate of the mse . the optimal weights were found by solving a small linear system . a fast algorithm for implementing the optimally - weighted filters was also described . we reported visual and psnr results on test images which confirmed the improvement over the original bilateral filter . an interesting finding was that the weighted bilateral filter is competitive with the non - local means filter . j. portilla , v. strela , m. j. wainwright , and e. p. simoncelli , `` image denoising using scale mixtures of gaussians in the wavelet domain , '' _ ieee transactions on image processing _ 1338 - 1351 , 2003 . s. p. awate and r. t. whitaker , `` unsupervised , information - theoretic , adaptive image filtering for image restoration , '' _ ieee transactions on pattern analysis and machine intelligence _ , vol . 3 , pp . 364 - 376 , 2006 .
the bilateral filter is known to be quite effective in denoising images corrupted with small dosages of additive gaussian noise . 
 the denoising performance of the filter , however , is known to degrade quickly with the increase in noise level . 
 several adaptations of the filter have been proposed in the literature to address this shortcoming , but often at a substantial computational overhead . in this paper , we report a simple pre - processing step that can substantially improve the denoising performance of the bilateral filter , at almost no additional cost . 
 the modified filter is designed to be robust at large noise levels , and often tends to perform poorly below a certain noise threshold . to get the best of the original and the modified filter , we propose to combine them in a weighted fashion , where the weights are chosen to minimize ( a surrogate of ) the oracle mean - squared - error ( mse ) . 
 the optimally - weighted filter is thus guaranteed to perform better than either of the component filters in terms of the mse , at all noise levels . 
 we also provide a fast algorithm for the weighted filtering . visual and quantitative denoising results on standard test images 
 are reported which demonstrate that the improvement over the original filter is significant both visually and in terms of psnr . 
 moreover , the denoising performance of the optimally - weighted bilateral filter is competitive with the computation - intensive non - local means filter . 
 image denoising , bilateral filter , unbiased risk estimator , fast algorithm .