Image Downscaling Based on Neugebauer Model

We propose a method for image downscaling based on Neugebauer model, which expresses an arbitrary color as a weighted combination of primary colors called the Neugebauer primaries, which are the vertex colors in RGB color cube. The primary colors are highly saturated and can produce higher contrast image contents than the other colors inside the color cube. We define a similarity measure for evaluating the closeness between an arbitrary color and the Neugebauer primaries, and utilize it as a weight value for the weighted averaging pixel fusion. Experimental results show that the proposed method can produce downscaled images having the similar colors to the Neugebauer primaries.


Introduction
Image downscaling is a kind of image resizing or scaling techniques, which include image enlargement (1) , image interpolation (2) , image super resolution (3), (4) , contentaware image resizing (5) , making low resolution pixel art from pictures of standard sizes or vector images (6) , and enlargement of pixel art (7) .Downscaling is necessary to view a high-resolution image on digital cameras, mobile devices and social networks.
Conventional methods such as nearest neighbor, bilinear and bicubic interpolations are content-invariant, and therefore, some details in original images may be smoothed out by downscaling procedures.To overcome this problem, Kopf et al. proposed a content-aware image downscaling method that adapts the shape of downscaling kernel, yielding sharper and more detailed downscaled results (8) .Oztireli and Gross (9) proposed a method which optimizes the perceptual quality of the downscaled results with SSIM (10) .
In this paper, we propose an image downscaling method which focuses on the Neugebauer primaries that have high saturation and can produce high contrast images.
Compared to Kopf's method (8) and Oztireli's method (9) , the proposed method is computationally efficient because the proposed method simply computes a weighted average without optimization procedures such as the expectationmaximization algorithm used by Kopf et al.Experimental results show that the proposed method produces downscaled images with similar colors to the Neugebauer primaries.

𝑋𝑋∈𝑆𝑆
. Therefore, it can be seen that each   indicates the similarity between an arbitrary color  and a primary color  ∈ .Based on this observation, we define a similarity measure between a color  and the set of the Neugebauer primaries  as follows: This similarity (, ) satisfies the following inequalities: (, ) ≤ 1. ( The equality in ( 4) is achieved when  = [ ] as for all  ∈ , and the equality in ( 5) is achieved when  is a primary color as follows.For example, if  = , then, from (2), we have = 0 for  ≠ , (8) and hence (, ) =   = 1.The similar equation holds for every primary color.for  = 1, … ,  � <  and  = 1, … ,  � <  denotes the RGB color vector at the pixel position (, ) in  .We suppose that  � and  � are given by a user as well as .

Integer Scaling Factors
Suppose that  =  �  + , 0 ≤ ̂<  � , where  and ̂ denote the quotient and remainder, respectively.If ̂= 0, then the scaling factor in the vertical direction is just : If ̂> 0, then we upscale the vertical direction of  with the scaling factor  � ( + 1)/ by Lanczos interpolation with size parameter 2, and downscale the upscaled image with the scaling factor ( + 1): The horizontal direction is also scaled in the similar manner to the vertical direction described above.To preserve the aspect ratio of a given image, we will use a common  to both the vertical and horizontal directions.

Image Downscaling by Weighted Averaging
In this subsection, we suppose that  =  �  and  = where we used the abbreviation and  denotes a parameter for controlling the effect of the weights; for  > 1, the similarity is enhanced, and for  < 1 , it is suppressed, especially for  = 0 , (11) outputs a uniform average color in the block.

Experimental Results
We used the images in Fig. 1 for image downscaling experiments (These images are available on the Web: https://graphics.ethz.ch/~cengizo/imageDownscaling.htm).Fig. 2 shows the image downscaling results of Fig. 1(a) by the proposed method with  = 0.1, 1, 2, 10 from top to bottom, and the image sizes are 200 × 150, 100 × 75, 40 × 30, 16 × 12 pixels from left to right.Fig. 3 shows grayscale images in which each pixel value denotes the similarity value of the corresponding pixel in Fig. 2.This figure shows that the downscaled images with larger value of  have larger similarity values in each resolution, that is also confirmed in the next figure, where we calculate the mean similarity value defined by and show them in Fig. 4, where the vertical axis denotes ̅ , and the horizontal axis denotes .The red, green, blue and cyan lines denote  = 2, 4, 10, 25 , respectively.For all tested values of , ̅ increases monotonically with .subsumpling, bicubic, box, Lanczos methods, respectively.Figs.5(e)-(h) show the results with bilateral kernel (12) , generalized sampling (13) , content-adaptive downscaling (8) and perceptual downscaling (9) , respectively.Fig. 6 shows the similarity images calculated from the downscaled images in Fig. 5. Fig. 7 shows the mean similarity values ̅ of 128 × 96 downscaled images by the compared methods.In Fig. 7, the vertical axis denotes ̅ , and the horizontal axis denotes the names of the compared methods: subsampling, bicubic, box, Lanczos, bilateral, generalized, content, perceptual and the proposed ( = 2) methods from the left to right.Among the compared methods, the subsampling method obtained the largest value of ̅ , and the proposed method is better than the subsampling one.

Conclusions
We proposed a method for downscaling images based on the Neugebauer model, where we defined a similarity measure between a color and the Neugebauer primaries.
Experimental results showed that the downscaled images by the proposed method had higher similarity values to the Neugebauer primaries than the compared methods.

Fig. 5 Fig. 5 .
Fig. 5.The image downscaling results by conventional methods.The size of each image is 128 × 96.

Fig. 8
shows the downscaled images of the images in Figs.1(b)-(d) from top to bottom, where the four columns from the left to right correspond to the generalized, contentaware, perceptual and proposed methods, respectively.