Image denoising example. Up: The noisy image and its denoised version at an automatically selected scale. Down: Model Error ME( t ) (ground truth) and Prediction Error PE + ( t ) (estimated by each of the two cross-validation algorithms). 

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... [8] and the references therein). For some model selection problems, like choosing the regularization parameter of smoothing splines, leave-one-out cross-validation (where K = M and T i only contains ( x i , y i ) ) can be approxi- mated analytically, leading to fast computations [7]. However, adopting leave-one-out cross-validation in our case would require building nonlinear diffusion scale-spaces of M images of size ( M − 1) -pixels each, which clearly is un- acceptably expensive. Hence we have explored two other data resampling configurations for our problem. Fig. 1 de- picts for each of these two alternative strategies the pixels D − T i used to build the scale-space model (in black) and the pixels T i used to estimate its prediction error (in white) for the case i = 1 . The first of these configurations (called quadruple-cv from now on), which was also used by Nason in the context of wavelet shrinkage [11], creates K = 4 nonlinear diffusion scale-spaces, each built up using ≈ 1 / 4 of the noisy image data and used to estimate the prediction error of the model on the remaining | T i | ≈ 3 / 4 of the data. The second (called double-cv in the following) selects the members of T i in a chessboard-like fashion, utilizing half for building the scale-space and half for testing it, repeat- ing K = 2 times. In more detail, in the case of quadruple-cv configuration, by selecting four different values { (0 , 0) , (0 , 1) , (1 , 0) , (1 , 1) } for the shift vector ( s i , s j ) , we get four subsampled by a factor of two in each direction versions of the noisy image y . Each consists of roughly M/ 4 pixels with coordi- nates { (2 i + s i , 2 j + s j ) : 0 ≤ i < M x / 2 , 0 ≤ j < M y / 2 } . We then build the nonlinear diffusion scale-space of each of these subsampled images, appropriately scal- ing the diffusion PDE. For example, if Eq.(1) is utilized, we must enforce τ = τ / 4 (for the time-step), g ( · ) = g ( · / 2) and σ = σ/ 2 . We subsequently use Eq. (6) to get PE CV quadruple − cv ( t ) , computing each of the four terms { PE + ( f t − T k ) : 1 ≤ k ≤ 4 } as follows: We take the t = t/ 4 snapshot of the corresponding auxiliary scale-space, with dimensions M / 2 by M / 2 and interpolate from it the values at the remaining 3 M/ 4 pixels p i ∈ T k . A nonlinear, edge-preserving procedure should ideally be utilized for the interpolation, although, in practice, we have noticed little difference when using simple bi- linear interpolation. We then penalize (by means of L ) the difference between the interpolated value and the initial noisy value y i at the same pixel p i and average over the 3 M/ 4 pixels of T k to get PE + ( f t − T k ) . The procedure just described adds automatic scale selection to nonlinear diffusion-based denoising procedures at roughly double the computational cost of plain nonlinear diffusion, since 4 auxiliary scale-spaces of size M/ 4 pixels each need to be built in parallel with the standard scale-space. The concept in the case of double-cv resampling is similar and won’t be described in detail. The main difference is that the pixels that build each of the auxiliary scale-spaces are not located on a rectangular lattice any more (see Fig. 1, right). Therefore, it is convenient to first interpolate the values at the M/ 2 “white” pixels of T k from the values at the remaining M/ 2 “black” pixels of the noisy image y i and then build the auxiliary scale-spaces. Since these auxiliary scale-spaces are full-sized, the PDE (6) doesn’t need any rescaling. The overall cost of the procedure is three times the cost of the standard scale-space, since two auxiliary full- sized scale-spaces evolve in parallel with the main one. An example of image denoising with automatic scale selection by cross-validation techniques can be seen in Fig. 2. At the first row one can see the noisy image and its denoised version at scale t ∗ double − cv determined by the double- cv cross-validation algorithm. The corresponding plots de- picting the ME( t ) (ground truth) and PE + ( t ) (as estimated by the two cross-validation algorithms) can be seen at the second row. Notice that, as we discussed in Sec. 2, ME( t ) is smaller than PE + ( t ) . Nonetheless, both quantities attain their minimum at roughly the same scale (after about 8 iter- ations). Some further examples of automatic denoising, one of an MRI scan and one of an aerial image, utilizing cross- validation scale selection techniques, can be seen in Fig. 3. In order to systematically assess the performance of the proposed algorithms and compare them with existing techniques, we run a series of denoising experiments on a dataset of 39 natural grayscale images (the ko- dak, aerial and misc1 collections available from http: //www.cipr.rpi.edu/resource/stills/ ), cor- rupted by artificial noise so that the ground truth f ∗ would be available. Apart from the two cross-validation algorithms we have proposed ( double-cv and quadruple-cv ), we have also implemented the snr-based method of We- ickert (with ground-truth snr value), and the decorrelation method of Mrazek, in the sequel dubbed snr and dec , respectively. The nonlinear diffusion scale-space used in the experiments reported here was generated by the PDE of Eq. (1) with σ = 0 . 1 pixels. We employed the diffusivity function g ( r ) = 1 / [1 + ( r/λ ) 2 ] [12], with λ = 0 . 01 in all of the tests (the intensity values of the images were in [0 , 1] ) and the AOS numerical scheme [16]. Bilinear interpolation was utilized for upsampling. We experimented with three different noise types nt ∈ { gaussian , salt&pepper , speckle } . The degraded images were respectively generated by y i = f ∗ ( x i ) + i (gaussian), y i = (1 + i ) f ∗ ( x i ) (speckle), and y i = f ∗ ( x i ) , with probability 1 − p and y i = 0 or 1 with probability p/ 2 each (salt&pepper). In the gaussian and speckle cases, the i were i.i.d. sampled from N (0 , σ 2 ) . We conducted tests for varying noise levels nl ∈ { 0 . 05 , 0 . 1 , 0 . 15 , 0 . 2 , 0 . 3 , 0 . 4 } , where nl = σ in the case of gaussian or speckle noise and nl = p in the case of salt&pepper noise and for two different choices of the loss function L ∈ { L 1 , L 2 } . To evaluate the performance of the algorithms under consideration for each of the 2 · 3 · 6 = 36 combinations of ( L, nt, nl ) we run experiments on all 39 images in the database and averaged the performance results of each algorithm on the different images. We totally performed 36 · 39 = 1404 experiments. At each experiment and for each scale selection algorithm alg ∈ { double − cv , quadruple − cv , dec , snr } , we eval- uated two quantities, namely the relative increase in model error , defined as ME = ME( t ∗ ) and the rela- ME tive scale error , given by ∆ t ∗ t ∗ = | t ∗ alg t ∗ − t ∗ ME | . ME In Figs. 4 and 5 we present average benchmark scores for ∆(ME) ME and ∆ t ∗ t ∗ , respectively, acquired by the experimental process just described. In these figures one can see that the two algorithms based on cross-validation consistently out- perform the dec and snr methods. The robustness of both cross-validation algorithms, irrespectively of the noise type or the utilized loss function is particularly noteworthy. ∆(ME) In more detail, as far as is concerned (Fig. 4), ME double-cv and quadruple-cv give results that sometimes are one order of magnitude better than the results given by the snr and dec algorithms. This is particularly true in the ∆(ME) case of gaussian noise, where the of the two cross- ME validation algorithms is almost always less than 1% , ap- proaching sometimes 0 . 1% , practically meaning that the automatically selected scale t ∗ CV in these cases coincided with the ground truth optimal scale for almost every image in the dataset. The performance of double-cv and quadruple-cv in the two cases of non-gaussian noise is less impressive, although the error still remains under 10% . Among the other two algorithms, snr seems to perform better than dec . It is noteworthy in Fig. 4 that the error made by the snr algorithm consistently lies around 10% . The decorrelation ...