Reversible denoising and lifting based color component transformation for lossless image compression
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Abstract
An undesirable side effect of reversible color space transformation, which consists of lifting steps (LSs), is that while removing correlation it contaminates transformed components with noise from other components. Noise affects particularly adversely the compression ratios of lossless compression algorithms. To remove correlation without increasing noise, a reversible denoising and lifting step (RDLS) was proposed that integrates denoising filters into LS. Applying RDLS to color space transformation results in a new image component transformation that is perfectly reversible despite involving the inherently irreversible denoising; the first application of such a transformation is presented in this paper. For the JPEGLS, JPEG 2000, and JPEG XR standard algorithms in lossless mode, the application of RDLS to the RDgDb color space transformation with simple denoising filters is especially effective for images in the native optical resolution of acquisition devices. It results in improving compression ratios of all those images in cases when unmodified color space transformation either improves or worsens ratios compared with the untransformed image. The average improvement is 5.0–6.0% for two out of the three sets of such images, whereas average ratios of images from standard testsets are improved by up to 2.2%. For the efficient imageadaptive determination of filters for RDLS, a couple of fast entropybased estimators of compression effects that may be used independently of the actual compression algorithm are investigated and an immediate filter selection method based on the detector precision characteristic model driven by image acquisition parameters is introduced.
Keywords
Reversible color space transformation Lifting technique Denoising Reversible denoising and lifting step Detector precision characteristic Lossless image compression1 Introduction
An undesirable side effect of reversible color space transformations employed by lossless image compression algorithms to reduce correlation among image components is that they contaminate transformed components with noise from other components. Reducing the correlation of components improves compression ratios, but the noise increase may reduce or thwart the expected transformation effects. Reversible color space transformations are constructed using Lifting Steps (LSs) because LSs have many practical advantages, like inplace computation or perfect reversibility [5], but it is these steps that propagate noise between components.
To solve the abovementioned problem, replacing LSs with reversible denoising and lifting steps (RDLSs) was proposed. RDLS prevents noise propagation by exploiting denoising filters while retaining other effects and advantages of LS. It was first applied to the simple RDgDb transformation [34], and then to three other more complex color space transformations [36]; the method of replacing LSs with RDLSs is general and may be used with any transformation exploiting LSs, for instance with the Discrete Wavelet Transformation (DWT) [35]. A transformation modified by applying RDLS (RDLSmodified transformation) retains the desirable effect of the original transformation (e.g., the correlation reduction) but avoids the unwanted side effect of noise propagation. The perfect reversibility of the RDLSmodified transformation is preserved despite the inherently lossy nature of denoising, which may be surprising. For this reason, the first application of RDLS is described in detail as a background in this paper and the forward and inverse transformation is illustrated with a stepbystep example. In addition, to facilitate understanding of the method, the notation for the transformed components is changed compared with other publications on RDLS.
By an imageadaptive denoising filter selection, the RDLSmodified transformation may be bettersuited to characteristics of the processed data. Filters might be selected by checking the actual compression ratio obtained by a compression algorithm for the image transformed using RDLSs exploiting different denoising filters. From a practical standpoint, however, the complexity of such a selection method would be too high. Here, research is reported in which, based on the transformation effects observed for real images contaminated to different degrees with noise, interesting properties of RDLS were identified that allowed proposing alternative filter selection methods. The first filter selection method exploits an entropybased estimator of compression effects that is applied directly to the transformed image components, or to components subjected to prediction (linear or nonlinear). This estimator is significantly faster than compression algorithms and is independent of the actual compression algorithm because the optimal filter selection depends more on the images than on the compression algorithm. Another filter selection method proposed in this study consists of building a model of an image acquisition device, which links the image acquisition parameters with suitable denoising filters. This method, for some types of images, allows for a virtually costless selection of filters (single table lookup using image acquisition parameters) and obtains good results in comparison to the abovementioned estimators.
The research was done for RDLSmodified RDgDb (RDLSRDgDb) using several sets of test images (including standard sets as well as ones prepared especially for this research) and three significantly different standard compression algorithms (JPEGLS [42], JPEG 2000 [41], and JPEG XR [7]). The application of RDLS to the RDgDb color space transformation with simple denoising filters was found especially effective for images in the native optical resolution of acquisition devices. It results in improving the compression ratios of all these images and for all compression algorithms in cases when unmodified color space transformation either improves or worsens ratios compared with the untransformed image. The average improvement is 5.0–6.0% for two out of the three sets of such images, whereas average ratios of images from standard testsets are improved by up to 2.2%.
The remainder of this paper is organized as follows. In subsections of Section 2, after presenting methods of constructing color space transformations for image compression (Sect. 2.1) and characterizing LS (Sect. 2.2), RDLS is presented (Sect. 2.3) along with its application to RDgDb (Sect 2.4) and a detailed example of forward and inverse RDLSRDgDb (Sect. 2.5). The experimental procedure is described in Section 3. The effects of RDLSmodified transformation on the entropy and bitrates of transformed image components are evaluated and discussed in Section 4.1. In Section 4.2, from a practical standpoint, a fast and compression algorithmindependent entropybased estimator for selecting denoising filters for RDLS is investigated. Then, an immediate estimator based on image acquisition parameters is proposed in Section 4.3. Section 5 summarizes the research.
2 Background
2.1 Color space transformations for image compression
It is known that the red, green, and blue primary color components of the RGB color space are highly correlated for natural images [20, 24]. A common approach to RGB color image compression is the independent compression of the image components obtained using a transformation from RGB to a less correlated color space. This technique is employed by standards including JPEG 2000 [16, 41], DICOM incorporating JPEG 2000 [21], JPEGLS (part II of the standard) [11], and JPEG XR [7, 14].
For a specific image, by using Principal Component Analysis (PCA), the imagedependent Karhunen–Loève transformation (KLT) may be obtained that optimally decorrelates the image [24]. Note that the PCA/KLT is much better known for its use as a spatial transformation than for the transformation of color space. There are two important reasons why other color space transformations are used instead of it. The optimal decorrelation does not lead to the best lossless compression ratios [34] and, from a practical standpoint, PCA/KLT requires too much time to be computed each time an image gets compressed. Therefore, instead of directly using KLT, fixed transformations are constructed by performing PCA on a representative image set. It is then assumed that the obtained KLT transformation matches individual images both within and outside the set. Many such fixed transformations have been constructed and practically exploited. Established transformations, like the YCgCo from JPEG XR, were constructed for typical continuoustone photographic images (YCgCo was constructed using the Kodak set described in Section 3 [20]). Such transformations are only effective as long as the characteristic of the image to be transformed matches the assumed characteristic.
Images whose characteristic is different may require either color space transformations constructed especially for them or transformations that are imageadaptive. Two increasingly often exploited and diverse classes of such images are worth mentioning here: screen content images and medical images. Screen content images may be computergenerated (raytraced, drawn, etc.), composed from others (including natural ones), or be screenshots. There currently is a growing interest in compression of screen content images [22], the recent standard of video and image coding HEVC [17] includes an extension for these images [43]. The volume of medical images requiring storage is evergrowing, also the employed resolutions and depths tend to increase and these images often require longterm storage. Fortunately, PCA/KLT is not the only approach allowing adaptation of the color space transformation to a given image. A simple method named Adaptive Color Transform is used in the screen content coding extension of HEVC [43], where either the image block data is encoded directly in the RGB color space or the YCgCo transformation is applied before coding if it improves the coding effects. In [38, 39] it was shown that for a given image or a region of an image, an adaptive selection of a transformation from a large family of simple transformations may be done at the cost of a slight increase in color image transformation process complexity. An adaptive method based on Singular Value Decomposition is significantly more complex, yet it is simpler than computing KLT for the image [31].
RDgDb was found to be the most universal because it resulted in the best average bitrates for two out of the three compression algorithms tested (JPEG 2000, JPEG XR) and the secondbest bitrate for the third (JPEGLS [12, 42]). The above transformation was independently proposed and investigated in [39] as part of a large family of reversible transformations. However, for images used in that research, its performance evaluated with a different methodology was not the best.
An unwanted side effect of established color space transformations, like RCT, is that while removing correlation they contaminate all transformed components with noise from other components. Components of images acquired by consumer acquisition devices operating at high photon flux are mainly affected by additive white Gaussian noise [2]; however, noise parameters may differ among components due to differences among primary color detector cells or detector filters. The luminance component of RCT (Y) contains a fraction of noise from all the primary color components, and each chrominance component (Cu and Cv) contains noise from two primary colors. The key difference between RCT and RDgDb is in replacing the luminance component by the red primary color component. Now only the chrominance components may contain more noise than the primary color components. This may explain the good performance of the simpler transformation that indeed decorrelates images worse than established ones (the average correlation of RDgDb transformed image components was the second largest of the transformations examined in [34]). However, bitrates of chrominance components usually are significantly better than bitrates of individual primary color components despite an increased amount of noise. The method presented in this study is aimed at constructing a reversible transformation that would remove the correlation but without increasing the amount of noise.
2.2 Lifting step
LS exploited by a color space transformation is a simple operation on a pixel’s components that modifies one of the components by adding to it a linear combination (a weighted sum) of some or all of the other components of the same pixel; the sum may be negated. A practical advantage of employing LSs is that the transformation may be computed in place, i.e., the transformed components overwrite their untransformed counterparts and, therefore, do not require any additional memory space. A typical reversible color space transformation is constructed as a sequence of LSs and, because of the inplace operation, is implemented as such a sequence. However, alternative methods of implementation are possible—for instance, RCT components might be obtained by directly implementing formulas from Eq. 2 and treating untransformed components as readonly data. In RCT implemented using LSs, the Cu component, after computing it as Cu = B − G, is stored in place of the B component, overwriting it. In other words, the B component is modified by subtracting G from it; after this modification, B is renamed as Cu. Next, the modified R becomes Cv, and then the modified G becomes Y. For further information on LSs, the lifting scheme, and their use in color space transformations the Reader is referred to [5, 9, 20, 40].
2.3 Reversible denoising and lifting step
LSbased color space transformation may be performed for each pixel independently of other pixels. The RDLS sequence, constructed on the basis of a color space transformation, is a transformation of the entire image components, not of the color space because denoising of a pixel’s component usually requires access to the same component of neighboring pixels. Now, let us recall that in Eq. 4 we could use any components as arguments of function f, except for C_{x}. In Eq. 5, we denoise these arguments, thus in denoising them we may use any component of any pixel, except for C_{x} of the pixel to which RDLS is being applied because otherwise this C_{x} would be indirectly used by f. For RDLS, not only the function f has to be deterministic but also all the denoising filters must be deterministic.
In a general case, denoising (i.e., computing the temporarily denoised components to be used as arguments of function f) cannot be performed before or after a transformation that is a sequence of RDLSs but has to be done within it because the same component may be used in more than one step of the transformation. Denoising may be applied to both the untransformed and the already transformed component. In consequence, different filters may be used not only for different components but also for the same component in different transformation steps. Furthermore, because the filter should match the characteristic of the data to be denoised, filters for each RDLS should be selected in an imageadaptive way.
Actual denoising filters are not perfect because they remove noise only partially. Consequently, RDLS transfers a fraction of noise from the component being denoised to the transformed component. On the other hand, they may introduce distortions, e.g., blurring, to the component being denoised, making it less suitable for removing correlation. Thus, finding the right denoising filter is crucial because the introduced distortions might affect the bitrate of the transformed component more than the noise reduction, resulting in bitrate worsening.
RDLS was proposed for transformations for noisy images, but it appeared more general. Two interesting special filters for RDLS were found effective in improving the compression ratios in practice: the None filter for which \( {C}_i^d={C}_i \) [35] and the Null filter, for which \( {C}_i^d=0 \) [36]. The None filter turns RDLS into a regular LS if it is applied to all arguments of function f (see Eqs. 4 and 5). Thanks to the None filter, the original transformation becomes a special case of the RDLSmodified transformation. The Null filter, if applied to all arguments of function f, may result in effective skipping of the step (the steps of color space transformations and DWT are by the Null filter turned into either C_{x} ← C_{x} or C_{x} ← − C_{x}). These filters may turn the RDLSmodified transformation into a lifting transformation or cause skipping it as a whole or partially—because different filters may be selected for different steps by the filter selection method employed for a specific image. The Null filter provides an additional mechanism of adaptation of the RDLSmodified transformation in the case when overall transformation effects (both unwanted side effects and desirable ones) result in worsening of the compression ratio. Thanks to this filter, RDLS may be effective also for the noisefree data.
The method of replacing LSs with RDLSs is of a general nature and is applicable to other transformations that are sequences of LSs. RDLS has already been found effective for color space transformations more complex than RDgDb (RCT, YCoCgR, and LDgEb) [36]. Significant bitrate improvements were obtained for grayscale nonphotographic images by the application of RDLS to a multiplelevel multidimensional DWT in JPEG 2000 lossless compression [35, 37].
2.4 RDLSmodified RDgDb transformation
This notation shows explicitly how the inplace computations are performed and uses the same symbol for a pixel’s component before and after modifying its value by the step. For this reason, the C_{1}, C_{2}, and C_{3} components have dual meaning compared with the traditional notation used in Eqs. 2 and 3—they denote R, G, and B components of the untransformed image, respectively, and R, Dg, and Db components of the transformed image, respectively. In a general case, the steps must be performed in a specified order. The R = R formula in Eq. 3 (and its equivalent C_{1} ← C_{1} in Eq. 7) may be seen as just additional information that R is a component of the transformed pixel, or as a special case of LS, where a ⊕ b = a ⊗ b = a (recall the reversible operations ⊕ and ⊗ defined in Section 2.2). In the other steps, a ⊕ b = a ⊗ b = − a + b.
C_{1}, C_{2}, and C_{3} denote R, G, and B components of the untransformed image, respectively, and R, dDg, and dDb components of the transformed image, respectively. To distinguish between components of RDLSRDgDb and RDgDb, by dDg and dDb denoted are RDLSRDgDb components obtained using RDLSs constructed based on LSs of RDgDb resulting in obtaining Dg and Db components, respectively.
The third steps of the forward RDgDb and RDLSRDgDb, as well as their inverses (the first steps of the inverse transformations), do not require any computations. They are presented in Eqs. 7 and 8 for the sole purpose of illustrating the application of RDLS to a sample liftingbased transformation.
The liftingbased color space transformation of an image may be performed in a pixelbypixel regime, for which all transformation steps are applied to each pixel in turn, or stepbystep, meaning a lifting step is applied to all pixels before the next step is applied. These regimes are equivalent. For the RDLSmodified transformation, the filter depends on the regime. A typical denoising filter applied to a component of a pixel does not exploit other components and calculates the denoised component based on the same component of pixels inside a window whose center is the pixel being denoised. Naturally, if we process an image pixelbypixel, then all components of all the already processed pixels are transformed. Assuming the raster scan order of processing, except for the top left and bottom right corner, the filter window contains both transformed and untransformed pixels. Thus, a sophisticated denoising filter should be employed that is either able to take advantage of both transformed and untransformed components or uses only a part of the window that contains either only transformed or only untransformed pixels. The stepbystep regime is in practice simpler to apply. In this regime, for each image component, except for the component being modified in the current step, either all pixels are transformed or all are untransformed. In the remainder of this study, the stepbystep regime is employed.
2.5 RDLSRDgDb example
2.5.1 Forward transformation
We apply step 1 of the forward RDLSRDgDb transformation (from Eq. 8) to the pixel surrounded in Fig. 1 a–c by a solid line. Step 1 (\( {C}_3\leftarrow {C}_3+{C}_2^d \)) modifies the C_{3} component of the pixel by assigning to it a difference between the \( {C}_2^d \) (the denoised C_{2}) and the intensity of the C_{3} component before applying step 1. We calculate the \( {C}_2^d \) as the average of component intensities within the area surrounded in Fig. 1b by a dashed line (in this case the average is 80). Therefore, the value of the transformed C_{3} component, whose intensity before the transformation was 75, is calculated as \( {C}_3+{C}_2^d=75+80=5 \). In the same way, we apply step 1 to all the other image pixels; Fig. 1d presents \( {C}_2^d \) of all pixels and Fig. 1e presents the C_{3} component after applying step 1 to all pixels. While applying step 1 to a certain pixel, the denoised C_{2} component \( {C}_2^d \) is needed only temporarily; denoising may be performed “on the fly” because \( {C}_2^d \) is used only once and only in this step, so there is no need to store it. After applying step 1 to all pixels, the image consists of the transformed C_{3} component (i.e., the dDb component of RDLSRDgDb transformed image, Fig. 1e) and the untransformed C_{1} and C_{2} components (Fig. 1 a and b, respectively); the untransformed C_{3}, as well as the denoised \( {C}_2^d \), are not available in further steps.
Similarly, we apply step 2 (\( {C}_2\leftarrow {C}_2+{C}_1^d \)), which modifies the C_{2} component. For each pixel, the transformed C_{2} (i.e., dDg of the transformed image, Fig. 1g) is calculated as the difference between the denoised C_{1} (\( {C}_1^d \), Fig. 1f) and the untransformed C_{2} (Fig. 1b). Step 3 (C_{1} ← C_{1}) does not require any computations, so we may skip it. The RDLSRDgDb transformed image thus consists of components R, dDg, and dDb (Fig. 1 a, g, and e, respectively).
2.5.2 Inverse transformation
During the inverse RDLSRDgDb, we apply the inverses of the forward transformation steps in reverse order. The C_{1}, C_{2}, and C_{3} components of the transformed image are R, dDg, and dDb, respectively. We skip step 1, which is an inverse of the skipped step 3 of the forward transformation. Then, we apply step 2 of the inverse RDLSRDgDb (\( {C}_2\leftarrow {C}_2+{C}_1^d \), the inverse of the forward transformation step 2:\( {C}_2\leftarrow {C}_2+{C}_1^d \)), which modifies the C_{2} component. For each pixel, the inversetransformed C_{2} equals the difference between the denoised C_{1} and the transformed C_{2}. Because denoising is performed using exactly the same data as in step 2 of the forward transformation (i.e., the same C_{1} shown in Fig. 1a), it results in the same denoised component \( {C}_1^d \) as in the forward transformation (Fig. 1f), which in turn allows the perfect reconstruction of the untransformed C_{2} component of the original image, i.e., the G component (Fig. 1b), despite the inherently lossy nature of the denoising employed. After applying step 2 to all pixels, the image consists of the untransformed C_{1} and C_{2} components (i.e., R and G, respectively) and the transformed C_{3} component (dDb).
Finally, for all pixels, we apply step 3 (\( {C}_3\leftarrow {C}_3+{C}_2^d \), the inverse of the forward step 1:\( {C}_3\leftarrow {C}_3+{C}_2^d \)), which modifies the C_{3} component. Denoising is based on the same data as in step 1 of the forward transformation (i.e., on the already untransformed C_{2}, Fig. 1b), so we obtain the same denoised \( {C}_2^d \)component (Fig. 1d) and reconstruct the untransformed C_{3} (B, Fig. 1c).
3 Experimental procedure and implementations
The effects of RDLS were evaluated by comparing bitrates of lossless image compression algorithms as well as the entropy of components of RDgDb (Eqs. 3 and 7) and RDLSRDgDb (Eq. 8). Besides the Null and None special case filters described in Section 2.3, simple Smoothing filters were used, i.e., lowpass linear averaging filters with a 3 × 3 pixel window. The filtered component intensity was calculated using the Smoothing filter as a weighted arithmetic mean of intensities of the same component of pixels from the window. At the edges of the images, the number of pixels used was smaller than the window size. Eleven Smoothing filters of different denoising strengths were employed. The denoising strength was controlled by setting a weight w of the center point in the weighting mask (in the range 1 to 1024; integer powers of 2 only), while the weights of neighboring points were fixed to 1; therefore, greater w results in weaker denoising.
Three significantly different standard image compression algorithms in the lossless mode were used in experiments: JPEGLS, JPEG 2000, and JPEG XR. JPEGLS is a standard of ISO/IEC and ITUT for primarily lossless compression of still images which originates from the LOCOI algorithm [12, 42]. The baseline standard defines a lowcomplexity predictive image compression algorithm with entropy coding using the modified Golomb–Rice code family. JPEG 2000 is an ISO/IEC and ITUT standard lossy and lossless image compression algorithm based on DWT image decomposition and arithmetic coding [16, 41]. JPEG XR is an ISO/IEC and ITUT lossy and lossless standard algorithm that is designed primarily for highquality, high dynamic range photographic images and is based on Discrete Cosine Transformation image decomposition and adaptive Huffman coding [7, 14]. The SPMG/UBC JPEGLS implementation, version 2.2 [30], JasPer implementation of JPEG 2000 by Adams, version 1.900 [18], and a standard reference implementation of JPEG XR, version 1.6 [15] were used.
To determine quickly the right filter for denoising an image component, we may simply use the transformed component entropy as an estimator of component compression effects. However, because image compression algorithms exploit the fact that nearby pixels are correlated, the effects of transformation on component compressibility may be better estimated by the entropy of the component residual image, which can be obtained by subtracting from each pixel the predicted pixel value. Two predictors were investigated: AVG, which is a simple linear predictor predicting that the pixel intensity equals the average of its upper and left neighbor; and MED, which is a simple nonlinear edgedetecting predictor used, among others, in the JPEGLS algorithm [42]. When the acquisition device characteristic and parameters of the acquisition process are known, then the denoising filter parameters may be estimated directly from the image acquisition process parameters—e.g., see [1, 2]. Inspired by the above approach, an acquisition process model for images of which the exposure parameters are available (sets A3 and A3red.3 described below) is proposed in this study. This model determines the denoising filters based on the image resolution and ISO speed used for image acquisition; it is described in detail in Section 4.3.

Waterloo – a set of eight color images from the University of Waterloo, Fractal Coding and Analysis Group repository, used for a long time in image processing research (Fig. 2 a–c). The set contains eight natural photographic and artificial images, including the wellknown “lena” and “peppers.” Image sizes vary from 512 × 512 to 1118 × 1105, available from [26].

Kodak – a set of 23 photographic images released by the Kodak Corporation. The set is frequently used in color image compression research. All images are of size 768 × 512, available from [27].

EPFL – a set of 10 highresolution images used at the École Polytechnique Fédérale de Lausanne for subjective JPEG XR quality evaluation [6]. Image sizes range from 1280 × 1506 to 1280 × 1600, available from [28].

A1 – these images (Fig. 2 d–f) were originally acquired from a 36 mm highquality diapositive film using a Minolta Dimage 5400 film scanner. After size reduction and conversion to grayscale, they were used in research on grayscale image compression algorithms [32]. Set A1 contains three images in the device’s optical resolution. Image sizes vary from 7376 × 4832 to 7424 × 4864, available from [29].

A2 – a set of 17 images acquired from negatives using a generalpurpose Agfa e50 scanner (see Fig. 2g–i) in the device’s optical resolution. Image sizes range from 1620 × 1128 to 1740 × 1164, available from [29].

A3 – a set of 116 images acquired using an Olympus Stylus XZ2 consumer camera. Because the camera detector uses a Bayerpattern RGGB color filter array, the grayscale Bayerpattern image was converted to a color image by taking for each group of RGGB subpixels the average of two green subpixels as the intensity of the green pixel component and the intensities of the red and blue subpixels as the intensities of the red and blue pixel components, respectively. Thus, the image resolution is as close to the detector resolution as possible without interpolation of all components (i.e., equal in the case of red and blue components, lower for the green component). The total number of pixels in the color image equals ¼ of the total number of subpixels in the camera detector. For extracting Bayerpattern images from RAW camera files, dcraw version 9.24 [25] was used. To allow analyzing how noise affects the RDLSmodified transformation, the A3 set also contains six series of images (Fig. 2 j–o). Each series is taken from the same scene reaching the detector (fixed manual focus and aperture were applied) but captured with different ISO speeds (100 to 12,800 in eight steps) resulting in different amounts of noise. These series are not perfect; due to the wind, the movement of the Sun, etc., there are small differences between the images in a given series. The A3 images are noticeably sharper than the A1 and A2 images. Image sizes are 1992 × 1493, available from [29].

Sets A1red.3, A2red.3, and A3red.3, all of whose images were reduced three times from A1, A2, and A3, respectively; a pixel of a reducedsize image is calculated as an average of nine pixels from a fullsize image.
Free tools used for resizing, color component transformations, prediction, etc. are available from [8]. A free research implementation for examining the effects of RDLSRDgDb and other transformations on the Reader’s images was prepared [10].
4 Results and discussion
4.1 Effects of RDLS on entropy and bitrates of transformed chrominance components
This subsection describes the preliminary research that allowed determination of some useful RDLS properties and finding effective methods for estimating transformation effects. In this part of the research, only the Smoothing filter was used and, because RDgDb and RDLSRDgDb affect only two of the three image components (that are transformed into the chrominance components Dg/dDg and Db/dDb), RDLS effects were analyzed only for these components.
Effects of RDLS on the entropy of Dg and Db components
Set  Dg/dDg  Db/dDb  

component  predicted AVG  predicted MED  component  predicted AVG  predicted MED  
H _{0}  ∆H _{0}  H _{0}  ∆H _{0}  H _{0}  ∆H _{0}  H _{0}  ∆H _{0}  H _{0}  ∆H _{0}  H _{0}  ∆H _{0}  
Waterloo  6.4489  1.15%  3.5465  1.17%  2.7440  2.68%  6.3846  1.25%  3.5554  1.34%  2.7679  2.77% 
Kodak  5.6283  0.01%  2.8993  0.03%  2.7112  0.05%  5.6355  0.01%  2.9571  −0.04%  2.8793  0.01% 
EPFL  6.3105  0.02%  3.3350  −0.23%  3.2630  −0.17%  6.2441  0.00%  3.5963  −0.92%  3.5175  −1.46% 
A1  5.0821  −0.11%  2.2484  −4.49%  2.2080  −5.12%  6.2161  −0.01%  2.6813  −0.16%  2.5769  −0.24% 
A2  6.2611  −1.84%  4.9566  −12.16%  5.1361  −13.01%  6.7713  −0.33%  5.4503  −2.66%  5.6381  −2.84% 
A3  6.0461  −3.97%  5.2656  −14.14%  5.4417  −15.35%  5.9153  −0.75%  5.0694  −2.39%  5.2540  −2.45% 
A1red.3x  5.0475  0.00%  2.7309  −1.84%  2.7930  −2.49%  6.1965  −0.02%  3.3058  −0.14%  3.3418  −0.11% 
A2red.3x  6.0348  −0.65%  4.4527  −4.43%  4.5144  −5.24%  6.5321  −0.06%  4.9038  −0.85%  4.9917  −1.05% 
A3red.3x  5.5786  −1.22%  4.3292  −5.93%  4.4119  −6.83%  5.5091  −0.38%  4.1134  −2.23%  4.1936  −2.09% 
Effects of RDLS on JPEGLS bitrates of Dg and Db components
Set  Dg/dDg  Db/dDb  

r  ∆r _{bitrate}  w _{set}  ∆r _{set}  r  ∆r _{bitrate}  w _{set}  ∆r _{set}  
Waterloo  2.7110  2.62%  1024  3.02%  2.7807  2.94%  1024  3.14% 
Kodak  2.5276  0.06%  1024  0.06%  2.6981  0.03%  1024  0.03% 
EPFL  3.0266  −0.14%  1024  0.03%  3.2570  −1.80%  1024  0.03% 
A1  2.0804  −5.03%  4  −4.94%  2.4208  −0.51%  8  −0.44% 
A2  4.9253  −12.83%  1  −12.83%  5.4428  −2.94%  1  −2.93% 
A3  5.2158  −15.28%  2  −14.95%  5.0314  −2.25%  8  −1.96% 
A1red.3x  2.6001  −2.32%  16  −2.05%  3.1462  −0.10%  64  −0.07% 
A2red.3x  4.3451  −5.23%  16  −4.53%  4.8241  −0.99%  32  −0.40% 
A3red.3x  4.1827  −6.93%  16  −3.88%  3.9884  −2.00%  64  −1.02% 
Effects of RDLS on JPEG 2000 bitrates of Dg and Db components
Set  Dg/dDg  Db/dDb  

r  ∆r _{bitrate}  w _{set}  ∆r _{set}  r  ∆r _{bitrate}  w _{set}  ∆r _{set}  
Waterloo  3.4148  0.39%  1024  0.64%  3.4718  0.30%  1024  0.68% 
Kodak  2.4668  0.06%  1024  0.06%  2.5583  0.04%  1024  0.04% 
EPFL  3.1521  −0.18%  512  −0.01%  3.3800  −1.98%  1024  −0.01% 
A1  2.1119  −4.09%  4  −3.98%  2.3961  −0.71%  4  −0.65% 
A2  4.9789  −12.69%  1  −12.69%  5.5086  −2.70%  1  −2.68% 
A3  5.2546  −14.67%  2  −14.36%  5.0651  −2.29%  8  −2.00% 
A1red.3x  2.6654  −1.96%  16  −1.73%  3.1760  −0.10%  64  −0.07% 
A2red.3x  4.4528  −5.22%  16  −4.55%  4.9384  −1.01%  32  −0.39% 
A3red.3x  4.2694  −6.59%  16  −3.71%  4.0820  −2.24%  64  −1.21% 
Effects of RDLS on JPEG XR bitrates of Dg and Db components
Set  Dg/dDg  Db/dDb  

r  ∆r _{bitrate}  w _{set}  ∆r _{set}  r  ∆r _{bitrate}  w _{set}  ∆r _{set}  
Waterloo  4.1733  0.26%  1024  0.59%  4.2199  0.00%  1024  0.66% 
Kodak  3.0526  0.02%  1024  0.03%  3.1215  0.00%  1024  0.01% 
EPFL  3.4972  −0.15%  512  −0.02%  3.7092  −1.17%  512  −0.03% 
A1  2.6630  −2.77%  4  −2.72%  2.9376  −0.32%  8  −0.27% 
A2  5.1057  −12.04%  1  −12.04%  5.6047  −2.62%  1  −2.60% 
A3  5.4194  −13.87%  2  −13.58%  5.2392  −2.15%  8  −1.89% 
A1red.3x  3.0258  −1.46%  16  −1.31%  3.4711  −0.09%  64  −0.06% 
A2red.3x  4.6751  −4.65%  16  −4.09%  5.1376  −0.90%  32  −0.28% 
A3red.3x  4.5679  −5.93%  16  −3.26%  4.3779  −1.93%  64  −1.04% 
A noticeable bitrate increase due to RDLS is observed for Waterloo images and the JPEGLS algorithm—over 2.5% for both chrominance components. Actually, this change is a result of the much greater bitrate deterioration in the case of the two computergenerated images containing many sharp edges. The bitrates of “frymire” (Fig. 2b) and “serrano” (Fig. 2c) were worsened by over 22% and 10%, respectively. Predictive JPEGLS outperforms other algorithms for artificial images. The Smoothing filter changes the characteristics of such images. Histograms of these images may be globally or locally highly sparse [23, 33] and application of the Smoothing filter results in an increase of the number of active levels. For other algorithms, the Waterloo bitrates are worsened by no more than 0.4%. Kodak bitrates are worsened by no more than 0.06%, and for all the other sets the RDLS application results in bitrate improvement.
For the A2 and A3 sets, RDLS resulted in an improvement of the Dg component bitrate for all algorithms of over 12%. Results for the former set were obtained using the strongest denoising tested (w = 1) for each individual image, suggesting that even greater improvements could be possible if even stronger denoising was used. For the Db component, the bitrate improvement is smaller (2.1–2.9%), but for most A2 images, the strongest denoising was the best. In general, RDLS effects on bitrates are similar to the observed effects on residual image entropy, especially to those obtained using MED. For A1, the improvements are smaller (2.7–5.0% for Dg, below 1% for Db), yet still practically useful. Compared with fullsize images, bitrates of reducedsize images are less improved by RDLS, e.g., Dg bitrates of A2red.3x and A3red.3x are improved by 4.6–6.9%.
Looking at the results of individual images (not shown in the tables), it may be noticed that for each image in the acquisition device optical resolution, as well as for all reducedsize A1 images, RDLS improved the bitrates of both chrominance components with all the compression algorithms examined. For reducedsize A2 and A3 images, the bitrates are improved on average, but not for each individual image. The opposite may be said about Waterloo images, whose bitrates were noticeably increased on average but not for each individual image, e.g., an improvement was consistently observed for the wellknown “lena” image (Fig. 2a).
In both figures, we see that bitrates of the G component are significantly lower than bitrates of the remaining primary color components. The acquisition detector noise is reduced in the case of the G component during the conversion from the Bayerpattern color filter array detector image to the color image. In the A3 set, the G pixel component was calculated as the average of two green subpixels from the camera detector, while the remaining color components were taken directly from single detector subpixels. Lower noise levels may also be a result of the different characteristics of the bandpass green filter and other filters constituting the color filter array. A3red3x images are much less noisy; here also G contains the least noise because it is actually an average of 18 green subpixels, whereas other components are averages of 9 subpixels each.
As already noted, RDLS improves bitrates of chrominance components of each A3 image as well as for almost every image of the A3red.3x set. Figures 3 and 4 show how this improvement depends on the image noise level and component. When the level of noise is high (for A3 and noisier A3red.3x images), the bitrate of the chrominance component (i.e., Dg or Db) is close to the bitrate of the primary color component used to calculate the chrominance, whose bitrate is worse (to R and B, respectively). For the highest noise levels (noisier A3 images), the chrominance bitrate is even worse than the inferior bitrates of the primary color components. Clearly, RDgDb is not the best color space transformation for these images. When we apply RDLS, the bitrate of the RDLSRDgDb chrominance component (dDg or dDb) is close to the bitrate of the untransformed primary color component that becomes this chrominance component during the transformation (G or B, respectively). Therefore, the RDLS bitrate improvement is high if the component being modified during transformation is compressed better than the other component used to calculate the chrominance (calculation of Dg in this case). For the A3 set series, the bitrate decreased from 14.1% to 15.9%. If the component being modified is compressed worse (calculation of Db), then both unaltered and RDLSmodified chrominance bitrates are close to the bitrate of the component being modified. We get a smaller but still practically useful bitrate decrease due to RDLS (2.1–2.3% for A3 series), which should probably be attributed to transferring less noise during transformation while still removing the correlation.
When the level of noise is low (less noisy A3red.3x images), the effect of removing the correlation has a greater impact on the resulting bitrate than the effect of transferring noise to the chrominance component from color components, and the bitrate of chrominance is significantly lower than bitrates of both color components. An interesting field of future research is the theoretical analysis of the aforementioned RDLS behavior.
RDLS may certainly be seen as a method of improving the bitrate of chrominance components of certain types of images (e.g., images in acquisition device optical resolution that were not subject to further processing and, to a lesser extent, those processed only by size reduction), both in cases where unmodified color space transformation improves or worsens chrominance bitrates. However, for other images (Waterloo), a noticeable bitrate increase due to the RDLS application was observed. For the noisiest A3 images (see Fig. 3), although RDLS significantly improves the bitrate of the RDgDb transformed image, it is even better to compress the untransformed RGB image. The above results were obtained by RDLS with only the Smoothing filter. Using a more diverse filter set should avoid the above bitrate increase, e.g., using the Null filter that causes the RDLSmodified transformation effectively to be skipped.
4.2 RDLS effects from a practical standpoint
To evaluate the effects of RDLS, the entropy and bitrates of chrominance components were analyzed in the previous section, but in practice, we are more interested in the overall bitrate improvement achievable for an entire threecomponent image at an acceptable cost. Therefore, hereafter let us focus on bitrate improvements of threecomponent images due to RDLS. Based on observations from Section 4.1, selecting the special filter cases None and Null should also be allowed, besides selecting Smoothing filters. Performing actual compression of transformed components (obtained using RDLSs with different denoising filters) to determine the best filter for a specific image component may be unacceptable from a practical standpoint because it increases the compression time too much. Therefore, a practical method that allows for quickly finding the denoising filters for an image without employing an actual compression algorithm is needed.
RDLS effects for threecomponent images and efficiency of entropybased estimation for RDLS filter selection
Set  JPEGLS  JPEG 2000  JPEG XR  

r _{RDgDb}  ∆r _{bitrate}  ∆r_{H0_pMED}  r _{RDgDb}  ∆r _{bitrate}  ∆r_{H0_pMED}  r _{RDgDb}  ∆r _{bitrate}  ∆r_{H0_pMED}  
Waterloo  8.8653  −0.68%  −0.65%  11.1428  −0.82%  −0.81%  13.2978  −0.36%  −0.30% 
Kodak  9.5673  0.00%  0.00%  9.4754  0.00%  0.00%  10.8725  0.00%  0.00% 
EPFL  10.3383  −2.19%  −2.19%  10.7240  −1.44%  −1.44%  11.6723  −0.81%  −0.80% 
A1  6.3869  −3.25%  −3.20%  6.3920  −3.77%  −3.44%  8.2259  −1.03%  −0.85% 
A2  14.8272  −5.39%  −5.37%  15.0354  −5.22%  −5.22%  15.4139  −4.95%  −4.95% 
A3  15.4922  −6.01%  −5.96%  15.6774  −5.81%  −5.74%  16.1822  −5.45%  −5.39% 
A1red.3  8.3402  −0.76%  −0.76%  8.5181  −0.65%  −0.65%  9.6403  −0.49%  −0.49% 
A2red.3  13.6966  −2.01%  −1.99%  14.1401  −2.00%  −1.98%  14.8451  −1.78%  −1.77% 
A3red.3  13.0793  −2.88%  −2.83%  13.4509  −2.81%  −2.79%  14.3621  −2.49%  −2.48% 
Looking at the bitrate improvements obtained without using the entropy estimation, it is clear that the greatest improvements were obtained for A2 and A3 sets; depending on the algorithm and the set, the bitrate improvement is from 5.0% to 6.0%, which is significant for the lossless image compression and useful from a practical standpoint. Sets A2 and A3 contain unprocessed images in the optical resolution of acquisition devices, the A1 set resembles them in this respect. In addition, for the A1 set, the improvements are better than in the case of standard image sets and quite good in the case of JPEGLS and JPEG 2000 (3.1% and 3.8%, respectively). All in all, the RDLS with the simple Smoothing filter, and even simpler Null and None filters, appears an effective method for improving bitrates of images in the optical resolution of acquisition devices. Bitrates of reducedsize sets A1red.3, A2red.3, and A3red.3 were improved to a smaller extent than their fullsize counterparts, however, the improvement for A2red.3 and A3red.3 (from 1.8% to 2.8%) is still useful. Improvements for the standard sets (Waterloo, Kodak, and EPFL) do not exceed 1.0%, except for the EPFL set in the case of JPEGLS and JPEG 2000. By using more advanced denoising filters, we would probably obtain better results for these images (as well as for images for which simple filters already result in significant bitrate improvements).
Using the entropy of residual component image obtained using the MED predictor as an estimator of the component compression effects results in overall color image bitrate improvements which are very close to those obtained when the denoising filter was selected based on the actual bitrate of the compression algorithm. They are worse by up to 0.07 percentage points for all the sets and compression algorithms except for the A1 set in the case of JPEG 2000 and JPEG XR (where the estimation effects are worse by up to 0.31 percentage points). The other estimators were tested as well: the entropy of component and entropy of residual component image obtained using the AVG predictor (not reported in Table 5), however, they were significantly worse. In addition, the correlation between the actual color image bitrate and bitrate estimated with the entropy of residual component images obtained using the MED predictor, measured with the Pearson correlation coefficient, is very high. It is at least 0.94 for all the sets and compression algorithms except for the Waterloo set in the case of JPEG 2000 and JPEG XR. This estimator is good enough to be used in practice.
The obtained results suggest that the application of RDLS to the RDgDb color space transformation with simple denoising filters is especially effective for images in the native optical resolution of acquisition devices both when the actual bitrate and its estimation using MED are employed for selecting denoising filters. Let us test the following statistical hypothesis H_{1}: for images in the native optical resolution of acquisition devices, RDLS results in mean bitrate improvement greater than for other images by at least 3 percentage points. Applying the onetailed independent samples t test (considering samples from two populations: 136 bitrate improvements due to RDLS obtained for images from sets A1, A2, and A3 and 177 improvements obtained for all the other images) at a confidence level 99.5% (α = 0.005), we find that H_{1} is valid for the JPEGLS, JPEG 2000, and JPEG XR algorithms, and for both methods of selecting denoising filters.
The entropybased estimator, although not involving actual compression and being significantly simpler than actual compression, is quite complex compared with the transformation itself. For each denoising filter, we have to denoise the color component and compute the chrominance component. For all these chrominance components, we have to compute the residual image and calculate its entropy. Overall, the denoising of components is the most timeconsuming part of the entropyestimationbased filter search process. Fortunately, it can easily be parallelized because this may be done independently for image components, separate image regions, and different filters. Another possibility is to exploit similarities among filters. Computing the Smoothing filter with 3 × 3 pixel window for a specific component of a specific pixel requires 10 integer arithmetic operations. Computing it when the center point weight is 1 requires nine operations or only five if the filter is applied to all pixels inside a contiguous area. Having computed it for a given filter window center point weight (weights of other points are 1 s), computing the filter for another center point weight can be done in just three such operations. The estimation method used in this study may be seen as an adaptation of an automatic method of selecting a color space out of a large set of color spaces presented by Strutz [39], where component transformation was selected based on the memoryless entropy of the residual component image obtained using the MED predictor. Strutz also found that computing the residual image entropy from only a small subset of image pixels (up to 10,000 pixels) is sufficient for close to optimum transformation selection; an application of a similar approach to RDLS is investigated in [36].
4.3 Denoising filter selection based on properties of the acquisition process
The model of the acquisition process for determining the RDLSRDgDb filters, trained using all A3 and A3red.3 images for JPEG 2000 lossless compression
ISO speed  Size: full  Size: reduced threefold  

dDg  dDb  dDg  dDb  
100  Smoothing, w = 4  Smoothing, w = 32  Smoothing, w = 16  Smoothing, w = 64 
200  Smoothing, w = 2  Smoothing, w = 16  Smoothing, w = 8  Smoothing, w = 32 
400  Smoothing, w = 2  Smoothing, w = 16  Smoothing, w = 8  Smoothing, w = 32 
800  Smoothing, w = 2  Smoothing, w = 16  Smoothing, w = 4  Smoothing, w = 32 
1600  Smoothing, w = 2  Smoothing, w = 8  Smoothing, w = 4  Smoothing, w = 32 
3200  Null  Smoothing, w = 4  Smoothing, w = 2  Smoothing, w = 16 
6400  Null  Smoothing, w = 4  Smoothing, w = 2  Smoothing, w = 16 
12,800  Null  Smoothing, w = 2  Smoothing, w = 1  Smoothing, w = 8 
Efficiency of the model of the acquisition process as an estimator of transformation effects for lossless component compression
Set  JPEGLS  JPEG 2000  JPEG XR  

r _{RDgDb}  ∆r_{bitrate}  ∆r_{DPC}  r _{RDgDb}  ∆r_{bitrate}  ∆r_{DPC}  r _{RDgDb}  ∆r_{bitrate}  ∆r_{DPC}  
A3  15.4922  −6.01%  −5.86%  15.6774  −5.81%  −5.65%  16.1822  −5.45%  −5.33% 
A3red.3  13.0793  −2.88%  −2.26%  13.4509  −2.81%  −2.22%  14.3621  −2.49%  −2.02% 
5 Conclusions and further work
An unwanted side effect of a reversible color space transformation that consists of LSs is that while removing correlation it contaminates transformed components with noise from other components. To avoid increasing noise while preserving other transformation properties, by integrating denoising into LSs a reversible image component transformation was obtained that is a sequence of RDLSs. The method was evaluated for several testsets of images using JPEGLS, JPEG 2000, and JPEG XR compression algorithms in lossless mode, RDgDb color space transformation, simple linear denoising filters (11 Smoothing filters), and special RDLS filter cases: Null and None.
RDLS was found effective for all algorithms for all tested unprocessed images in the optical resolution of acquisition devices. RDLSRDgDb resulted in improving the bitrates of all such images in cases when unmodified color space transformation either improves or worsens bitrates compared with the untransformed image. Overall, the average improvement of the threecomponent color image bitrate was 5.0–6.0% for two out of the three sets of such images (A2 and A3). A smaller improvement was observed for the third set (1.2–3.8%) and for images of reduced sizes (0.5–2.8%). For two out of the three standard test image sets, among others containing computergenerated and sharpened images, the improvements were from 0.4% to 2.2%. The RDLS effects were found to be similar for various compression algorithms and the memoryless entropy of the residual component image obtained using the MED predictor proved to be an efficient estimator of transformation effects for the selection of denoising filters. This estimator may be applied before performing an actual compression and independently of the compression algorithm used. For images whose acquisition parameters were available (A3 and A3red.3), a model inspired by the DPC approach was proposed that allowed for a virtually costless selection of denoising filters and was especially effective for images in the optical resolution of acquisition devices (A3).
The method presented in this paper is of a general nature and is applicable to other transformations that are sequences of LSs. RDLS has already been found effective for color space transformations that are more complex than RDgDb (RCT, YCoCgR, and LDgEb) [36]. Significant bitrate improvements were obtained for grayscale nonphotographic images by the application of RDLS to a twodimensional DWT in JPEG 2000 lossless compression [35, 37]. In the ongoing research, RDLS is applied to the threedimensional DWT in JPEG 2000 standard part 10 (JP3D, [3, 13]) lossless compression of volumetric medical images. It is also employed to combine predictive and transform (DWT) coding in lossless grayscale image compression. Further research plans include joining RDLSmodified color space transformations and the RDLSmodified DWT to obtain an image compression algorithm effective for a broad range of image types. Other fields of future research are the theoretical analysis of RDLS effects on noisy image bitrates and finding better denoising filters.
Notes
Acknowledgments
The author thanks Tytus Bernaś, Piotr Fabian, and the anonymous reviewers for their constructive comments. This work was supported by the 02/020/BK_19/0171 grant from the Institute of Informatics, Silesian University of Technology.
Compliance with ethical standards
Conflict of interests
Patent PL 229208 was obtained for the transformation presented in this paper.
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