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...izes energy in the higher frequencies of the solution, opting for a smooth and, hence, blurry image. To achieve reconstructed images with sharper edges, in the spirit of the total variation criterion =-=[32]-=-, [33] and a related method called the bilateral filter9 [34], [35], a robust regularizer called bilateral-TV (B-TV) was introduced in [3]. The B-TV regularizing function looks like 9 Note that by ado...

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... for a smooth and, hence, blurry image. To achieve reconstructed images with sharper edges, in the spirit of the total variation criterion [32], [33] and a related method called the bilateral filter9 =-=[34]-=-, [35], a robust regularizer called bilateral-TV (B-TV) was introduced in [3]. The B-TV regularizing function looks like 9 Note that by adopting a different realization of the bilateral filter, [27] h...

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...the relative motion between these images followed the translational model, we only needed to estimate the motion between the luminance components of these images [43]. We used the method described in =-=[44]-=- to computed the motion vectors. The shift-and-add result (resolution enhancement factor of 4) is shown in Fig. 8(b) [zoomed in Fig. 8(f)]. In Fig. 8(c) [zoomed in Fig. 8(g)], the method of [3] is use...

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...mit can be achieved by acquiring and fusing several low-resolution (LR) images of the same scene, producing high-resolution (HR) images; this is the basic idea behind super-resolution techniques [1], =-=[2]-=-, [3], [4]. In the last two decades a variety of super-resolution methods have been proposed for estimating the HR image from a set of LR images. Early works on SR showed that the aliasing effects in ...

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...y of motion estimation. Another approach is transforming the problem to a different color space, where chrominance layers are separated from luminance, and SR is applied only to the luminance channel =-=[7]-=-. Both of these methods are suboptimal as they do not fully exploit the correlation across the color bands. In Section VI, we show that ignoring the relation between different color channels will resu...

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...ion of SR in that early work, in general, SR is a computationally complex and numerically ill-behaved problem in many instances [6]. In recent years, more sophisticated SR methods were developed (see =-=[3]-=- and [6]–[10] as representative works). 2 Three CCD cameras which measure each color field independently tend to be relatively more expensive. Note that almost all super-resolution methods to date hav...

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.... Early works on SR showed that the aliasing effects in the LR images enable the recovery of the HR-fused image, provided that a relative subpixel motion exists between the under-sampled input images =-=[5]-=-. However, in contrast to the clean and practically naive frequency domain description of SR in that early work, in general, SR is a computationally complex and numerically ill-behaved problem in many...

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...super-resolution and demosaicing models are special cases of the general model presented above. In particular, in the super-resolution literature the effect of color-filtering is usually ignored [3], =-=[9]-=-, [10] and, therefore, the model is simplified to In this model, the LR images and the HR image are assumed to be monochromatic. On the other hand, in the demosaicing literature only single frame reco...

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... smooth and, hence, blurry image. To achieve reconstructed images with sharper edges, in the spirit of the total variation criterion [32], [33] and a related method called the bilateral filter9 [34], =-=[35]-=-, a robust regularizer called bilateral-TV (B-TV) was introduced in [3]. The B-TV regularizing function looks like 9 Note that by adopting a different realization of the bilateral filter, [27] has pro...

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...inance interpolation is done using edge directions computed in a steerable wavelet pyramidal structure. Other examples of popular demosaicing methods available in published literature are [20], [21], =-=[22]-=-, [23], [24], [25], and [26]. Almost all of the proposed demosaicing methods are based on one or more of these following assumptions: 1) In the constructed image with the mosaicing pattern, there are ...

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...nergy in the higher frequencies of the solution, opting for a smooth and, hence, blurry image. To achieve reconstructed images with sharper edges, in the spirit of the total variation criterion [32], =-=[33]-=- and a related method called the bilateral filter9 [34], [35], a robust regularizer called bilateral-TV (B-TV) was introduced in [3]. The B-TV regularizing function looks like 9 Note that by adopting ...

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...the green channel and the first derivative of the red (or blue) channels are used to estimate the edge direction in the green channel. The smooth hue and gradient-based methods were later combined in =-=[16]-=-. In this iterative method, the smooth hue interpolation is done with respect to the local gradients computed in eight directions about a pixel of interest. A second stage using anisotropic inverse di...

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...erpolation step, while adaptive median filtering is used as the enhancement step. A different iterative implementation of the median filters is used as the enhancement step of the method described in =-=[18]-=- that take advantage of a homogeneity assumption in the neighboring pixels. Iterative MAP methods form another important category of demosaicing methods. A MAP algorithm with a smooth chrominance prio...

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...ly where the scalar is the step size. (17) V. RELATED METHODS As mentioned earlier, there has been very little work on the problem we have posed here. One related paper is the work of Zomet and Peleg =-=[39]-=-, who have recently proposed a novel method for combining the information from multiple sensors, which can also be used for demosaicing purposes. Although their method has produced successful results ...

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...those measured images looks like, one of these Bayer filtered LR images is reconstructed by the method in [14] and shown in Fig. 4(b). The above method is implemented on Kodak DCS-200 digital cameras =-=[41]-=-, so each LR image may be thought of as one picture taken with this 12 SNR is defined as IH ��� @' a' A, where ' , ' are variance of a clean frame and noise, respectively.s152 IEEE TRANSACTIONS ON IMA...

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... luminance component of the image than the details in chrominance component [20]. 6) The human eye is more sensitive to chromatic changes in the low spatial frequency region than the luminance change =-=[24]-=-. 7) Interpolation should be preformed along and not across the edges. 8) Different color bands are correlated with each other. 9) Edges should align between color channels. Note that even the most po...

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...nsition assumption is logical for smooth regions of the reconstructed image, it is not successful in the high-frequency (edge) areas. Considering this fact, gradient-based methods, first addressed in =-=[14]-=-, do not preform interpolation across the edges of an image. This non-iterative method uses the second derivative of the Red and Blue channels to estimate the edge direction in the Green channel. Late...

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... common cost functions to measure the closeness of the final solution to the measured data is the least-squares (LS) cost function, which minimizes the norm of the residual vector (see [9], [10], and =-=[36]-=- as representative works). For the case where the noise is additive white, zero mean Gaussian, this approach has the interpretation of providing the maximum likelihood (ML) estimate of [9]. However, a...

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...is limit can be achieved by acquiring and fusing several low-resolution (LR) images of the same scene, producing high-resolution (HR) images; this is the basic idea behind super-resolution techniques =-=[1]-=-, [2], [3], [4]. In the last two decades a variety of super-resolution methods have been proposed for estimating the HR image from a set of LR images. Early works on SR showed that the aliasing effect...

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...pic inverse diffusion will further enhance the quality of the reconstructed image. This two step approach of interpolation followed by an enhancement step has been used in many other publications. In =-=[16]-=-, spatial and spectral correlations among neighboring pixels are exploited to define the interpolation step, while adaptive median filtering is used as the enhancement step. A different iterative impl...

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...on is done using edge directions computed in a steerable wavelet pyramidal structure. Other examples of popular demosaicing methods available in published literature are [20], [21], [22], [23], [24], =-=[25]-=-, and [26]. Almost all of the proposed demosaicing methods are based on one or more of these following assumptions: 1) In the constructed image with the mosaicing pattern, there are more green sensors...

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...that early work, in general SR is a computationally complex and numerically ill-behaved problem in many instances [6]. In recent years more sophisticated SR methods were developed (See [3], [6], [7], =-=[8]-=-, [9], [10] as representative works). Note that almost all super-resolution methods to date have been designed to increase the resolution of a single channel (monochromatic) image. A related problem, ...

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...inst the second term (regularization cost). Generally speaking, choosing could be either done manually, using visual inspection, or automatically using methods like generalized cross-validation [29], =-=[30]-=-, L-curve [31], or other techniques. How to choose such regularization parameters is in itself a vast topic, which we will not treat in the present paper. C. Monochromatic Spatial Regularization Tikho...

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...resolution. To date, there is very little work addressing the problem of color SR. The typical solution involves applying monochromatic SR algorithms to each of the color channels independently [11], =-=[12]-=-, while using the color information to improve the accuracy of motion estimation. Another approach is transforming the problem to a different color space, where chrominance layers are separated from l...

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...The luminance interpolation is done using edge directions computed in a steerable wavelet pyramidal structure. Other examples of popular demosaicing methods available in published literature are [21]–=-=[27]-=-. Almost all of the proposed demosaicing methods are based on one or more of these following assumptions. 1) In the constructed image with the mosaicing pattern, there are more green sensors with regu...

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...atial resolution. To date, there is very little work addressing the problem of color SR. The typical solution involves applying monochromatic SR algorithms to each of the color channels independently =-=[11]-=-, [12], while using the color information to improve the accuracy of motion estimation. Another approach is transforming the problem to a different color space, where chrominance layers are separated ...

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...R in that early work, in general SR is a computationally complex and numerically ill-behaved problem in many instances [6]. In recent years more sophisticated SR methods were developed (See [3], [6], =-=[7]-=-, [8], [9], [10] as representative works). Note that almost all super-resolution methods to date have been designed to increase the resolution of a single channel (monochromatic) image. A related prob...

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...h standard deviation equal to one. As the relative motion between these images followed the translational model, we only needed to estimate the motion between the luminance components of these images =-=[42]-=-. We used the method described in [43] to computed the motion vectors. The Shift-And-Add result (resolution enhancement factor of 4) is shown in Figure 8.b (zoomed in Figure 8.f). In Figure 8.c (zoome...

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... values in the Red and Blue channels. 3 This is the scenario for the more expensive 3-CCD cameras. 4 This is the scenario for cheaper 1-CCD cameras. 5sA variation of this method was later proposed in =-=[15]-=-, where the second derivative of the Green channel and the first derivative of the Red (or Blue) channels are used to estimate the edge direction in the Green channel. The smooth hue and gradient base...

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...is limit can be achieved by acquiring and fusing several low-resolution (LR) images of the same scene, producing high-resolution (HR) images; this is the basic idea behind super-resolution techniques =-=[1]-=-–[4]. In the last two decades, a variety of super-resolution methods have been proposed for estimating the HR image from a set of LR images. Early works on SR showed that the aliasing effects in the L...

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...ing. The luminance interpolation is done using edge directions computed in a steerable wavelet pyramidal structure. Other examples of popular demosaicing methods available in published literature are =-=[21]-=-–[27]. Almost all of the proposed demosaicing methods are based on one or more of these following assumptions. 1) In the constructed image with the mosaicing pattern, there are more green sensors with...

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... that early work, in general, SR is a computationally complex and numerically ill-behaved problem in many instances [6]. In recent years, more sophisticated SR methods were developed (see [3] and [6]–=-=[10]-=- as representative works). 2 Three CCD cameras which measure each color field independently tend to be relatively more expensive. Note that almost all super-resolution methods to date have been design...

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... multiframe image fusion techniques, suggests that heavytailed noise distributions such as Laplacian are more appropriate models (especially in the presence of the inevitable motion estimation error) =-=[37]-=-. In [3], an alternate data fidelity term based on the norm is recently used, which has been shown to be very robust to data outliers Note that the norm is the ML estimate of data in the presence of L...

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...in contrast to the clean and practically naive frequency domain description of SR in that early work, in general, SR is a computationally complex and numerically ill-behaved problem in many instances =-=[6]-=-. In recent years, more sophisticated SR methods were developed (see [3] and [6]–[10] as representative works). 2 Three CCD cameras which measure each color field independently tend to be relatively m...

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...Y ˆA aIH��� PSS 2 R� w �ˆ 0 ˆ� 18 The S-CIELAB measure is a perceptual color fidelity measure that measures how accurate the reproduction of a color is to the original when viewed by a human observer =-=[42]-=-. In our experiments, we used the code with default parameters used in the implementation of this measure available at http://white.stanford.edu/~brian/scielab/scielab.html. XsFARSIU et al.: MULTIFRAM...

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...polation is done using edge directions computed in a steerable wavelet pyramidal structure. Other examples of popular demosaicing methods available in published literature are [20], [21], [22], [23], =-=[24]-=-, [25], and [26]. Almost all of the proposed demosaicing methods are based on one or more of these following assumptions: 1) In the constructed image with the mosaicing pattern, there are more green s...

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...ge of a homogeneity assumption in the neighboring pixels. Iterative MAP methods form another important category of demosaicing methods. A MAP algorithm with a smooth chrominance prior is discussed in =-=[19]-=-. The smooth chrominance prior is also used in [20], where the original image is transformed to YIQ representation. The chrominance interpolation is preformed using isotropic smoothing. The luminance ...

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...ixels. Iterative MAP methods form another important category of demosaicing methods. A MAP algorithm with a smooth chrominance prior is discussed in [19]. The smooth chrominance prior is also used in =-=[20]-=-, where the original image is transformed to YIQ representation. The chrominance interpolation is preformed using isotropic smoothing. The luminance interpolation is done using edge directions compute...

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...ds. This property, combined with the assumption that the red/green and blue/green ratios are similar for the neighboring pixels, make the basics of the smooth hue transition method first discussed in =-=[13]-=-. Note that there is a negligible correlation between the values of neighboring pixels located on the different sides of an edge. Therefore, although the smooth hue transition assumption is logical fo...

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...he luminance interpolation is done using edge directions computed in a steerable wavelet pyramidal structure. Other examples of popular demosaicing methods available in published literature are [20], =-=[21]-=-, [22], [23], [24], [25], and [26]. Almost all of the proposed demosaicing methods are based on one or more of these following assumptions: 1) In the constructed image with the mosaicing pattern, ther...

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...also exploits the difference in sensitivity of the human eye to the frequency content and outliers in the luminance and chrominance components of the image. In parallel to our work, Gotoh and Okotumi =-=[40]-=- are proposing another MAP estimation method for solving the same joint demosaicing/super-resolution problem. While their algorithm and ours share much in common, there are fundamental differences bet...

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...ge of a homogeneity assumption in the neighboring pixels. Iterative MAP methods form another important category of demosaicing methods. A MAP algorithm with a smooth chrominance prior is discussed in =-=[18]-=-. The smooth chrominance prior is also used in [19], where the original image is transformed to YIQ representation. The chrominance interpolation is preformed using isotropic smoothing. The luminance ...

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... has been a two-step process. The first step 7 It is convenient to think of h @�A ae @�Ah@�A, where h@�A models the incoherent down-sampling effect of the CCD and e @�A models the color-filter effect =-=[28]-=-. 8 Note that color super resolution by itself is a special case of this model, where vectors † @�A and ‰ @�A are of size ‘Rw 2 I“ and matrices „ @�A and h @�A are of size ‘Rw 2 R� w “ for any color b...

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...t) against the second term (regularization cost). Generally speaking, choosing could be either done manually, using visual inspection, or automatically using methods like generalized cross-validation =-=[29]-=-, [30], L-curve [31], or other techniques. How to choose such regularization parameters is in itself a vast topic, which we will not treat in the present paper. C. Monochromatic Spatial Regularization...

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...ensive 3-CCD cameras. 4 This is the scenario for cheaper 1-CCD cameras.sFARSIU et al.: MULTIFRAME DEMOSAICING AND SUPER-RESOLUTION OF COLOR IMAGES 143 fact, gradient-based methods, first addressed in =-=[14]-=-, do not perform interpolation across the edges of an image. This noniterative method uses the second derivative of the red and blue channels to estimate the edge direction in the green channel. Later...

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...ng B-TV regularization to the luminance component will result in this desired property [3]. The luminance image can be calculated as the weighted sum X L =0.299X R +0.597X G +0.114X B as explained in =-=[37]-=-. The luminance regularization term is then defined as: J1(X) = C. Spatial Chrominance Penalty Term P� P� l=−P m=−P α |m|+|l| �X L − S l x Sm y X L�1. (12) Spatial regularization is required also for ...

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...term (regularization cost). Generally speaking, choosing λ could be either done manually, using visual inspection, or automatically using methods like Generalized Cross-Validation [28], [29], L-curve =-=[30]-=-, or other techniques. How to choose such regularization parameters is in itself a vast topic, which we will not treat in the present paper. 11sC. Monochromatic Spatial Regularization Tikhonov regular...

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...h standard deviation equal to one. As the relative motion between these images followed the translational model, we only needed to estimate the motion between the luminance components of these images =-=[43]-=-. We used the method described in [44] to computed the motion vectors. The shift-and-add result (resolution enhancement factor of 4) is shown in Fig. 8(b) [zoomed in Fig. 8(f)]. In Fig. 8(c) [zoomed i...

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...d term (regularization cost). Generally speaking, choosing could be either done manually, using visual inspection, or automatically using methods like generalized cross-validation [29], [30], L-curve =-=[31]-=-, or other techniques. How to choose such regularization parameters is in itself a vast topic, which we will not treat in the present paper. C. Monochromatic Spatial Regularization Tikhonov regulariza...