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# Two variants of creating pairs of triangles via four pixels

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## Contexts in source publication

Context 1
... nm is index of vertex, in front of base, to mark triangle plane. As show in Fig. 2, there are two different ways to draw pair of triangles and calculate pixels intensity. When magnification S is integer, x and y can get discrete integer values from 0 to S. To calculate pixels, with coordinates (x, y), intensity are used formulas (4, 5). When triangles are situated as shown in Fig. ...
Context 2
... decide what is a better way to situate triangles planes, was used intensity vector direction that direction is from pixel with lowermost intensity to pixel with topmost intensity of four pixels (M 00 , M 10 , M 01 and M 11 ). Vector can takes eight discrete directions and was marked with number that explained in Figure 3. Vector mark is main characteristic that allow solving what variant, of two shown in Fig. 2, is better. For example, when vector value is 2, the best way to situate triangles is shown in Fig. 2a, in other hand, when vector value is 4, the best way to situate triangles is shown in Fig. 2b. In comparison of triangles in both of Fig. 2 pictures, is evident that in Fig. 2b, sharpen of triangle plane ( M 00 , M 10 , M 01 ) will increased and plane ( M 11 , M 10 , M 01 ) became flatten, so magnified image sharpness will slightly increase. This image magnification method was implemented as C++ function in conjunction with free image processing toolkit CImg. For compressed image processing was used Image Magic package. Scaling down and magnify factors were selected integer numbers. For example scaling factor 3 means that both width and height of image will be divided or multiplied by 3. Naturally magnified image has some distortions. To measure distortions was calculated root-mean-square-error (RMSE) between original and magnified images. First of all original image was scaled down with scale factor, then image was magnified with the same scale factor, and after that calculated RMSE between original and magnified images for each pixel on each colour layer R, G and B. For comparison, magnification was executed with three well known magnification methods: block, linear, bicubic interpolation and introduced triangle based magnification that will be call “Triangle”. Two different types of compressed digital image formats JPG and PNG were used. JPG is lossy, but very popular, image format and PNG is lossless format. For there first test series were selected 9 JPG and 9 PNG images with different size and significant difference in number of small and large objects. Test series with scaling factor 2, 3, 4, 5, 6, 7, 8 was produced with each image. The second test was the same but with 5000 images with various size and format types for statistic analysis. First test series of image magnification RMSE is shown in Fig. 4 for JPG images and Fig. 5 for PNG ...
Context 3
... decide what is a better way to situate triangles planes, was used intensity vector direction that direction is from pixel with lowermost intensity to pixel with topmost intensity of four pixels (M 00 , M 10 , M 01 and M 11 ). Vector can takes eight discrete directions and was marked with number that explained in Figure 3. Vector mark is main characteristic that allow solving what variant, of two shown in Fig. 2, is better. For example, when vector value is 2, the best way to situate triangles is shown in Fig. 2a, in other hand, when vector value is 4, the best way to situate triangles is shown in Fig. 2b. In comparison of triangles in both of Fig. 2 pictures, is evident that in Fig. 2b, sharpen of triangle plane ( M 00 , M 10 , M 01 ) will increased and plane ( M 11 , M 10 , M 01 ) became flatten, so magnified image sharpness will slightly increase. This image magnification method was implemented as C++ function in conjunction with free image processing toolkit CImg. For compressed image processing was used Image Magic package. Scaling down and magnify factors were selected integer numbers. For example scaling factor 3 means that both width and height of image will be divided or multiplied by 3. Naturally magnified image has some distortions. To measure distortions was calculated root-mean-square-error (RMSE) between original and magnified images. First of all original image was scaled down with scale factor, then image was magnified with the same scale factor, and after that calculated RMSE between original and magnified images for each pixel on each colour layer R, G and B. For comparison, magnification was executed with three well known magnification methods: block, linear, bicubic interpolation and introduced triangle based magnification that will be call “Triangle”. Two different types of compressed digital image formats JPG and PNG were used. JPG is lossy, but very popular, image format and PNG is lossless format. For there first test series were selected 9 JPG and 9 PNG images with different size and significant difference in number of small and large objects. Test series with scaling factor 2, 3, 4, 5, 6, 7, 8 was produced with each image. The second test was the same but with 5000 images with various size and format types for statistic analysis. First test series of image magnification RMSE is shown in Fig. 4 for JPG images and Fig. 5 for PNG ...
Context 4
... magnification is a process of obtaining an image at resolution higher than taken from image sensor. Image magnification synonyms are interpolation, enlargement, zooming, etc. To create higher resolution image, previous image must be complemented with new pixels and their intensity must be calculated. Commonly, image magnification is accomplished through convolution of the image samples with a single kernel – typically the nearest neighbour [1, 2] bilinear, bicubic [3] or cubic B-spline kernel [4] interpolation and training-based algorithms [5, 6]. Intensity surface can be explained as landscape with hills and hollows, rides and valleys. Any 3 dimensional (3D) surfaces, with some approximation, can be created from triangles. Look at the image intensity as 3D surface build from triangles where vertexes are image pixels with intensity as z axis (Fig. 1). To magnify image, new image must be complemented with new pixels, added necessary columns and rows of pixels, and calculated new pixels intensity. In this work is made assumption, that new pixels intensity is situated on suitable triangle planes (Figure 1). During image magnification contours are blurred because of slope sharpness reduction. This is the main problem for all known magnification methods. Look at small part of intensity surfaces in Figure 2. It is evident, that, in this situation, sharpness of slope is better when triangles are created like in Figure 2b than in Figure 2a. Accordingly the base of both triangles must be diagonal with less intensity gradient. That partially reduce magnification blur. So must be located distinctive points on intensity surface and different magnification algorithm applied. There are two things that must be solved: choose the best arrangement of triangles and calculate new pixels intensity. To explain triangle based magnification method, get four neighbouring pixels from original image and create two triangles planes throws these pixels as vertexes (black dots in Figure 2). These pixels 3D coordinates is known, because they are taken from original image. White dots are new pixels that were inserted to magnify image. Their intensity must be calculated. Early was decided, that new pixels will situated on two triangles planes. Triangle plane can be formulated as equation of three coplanar vectors over spatial points M 00 ( 0 , 0 , Z 00 ) , M 10 ( S , 0 , Z 10 ) , M 01 ( 0 , S , Z 01 ) where x and y coordinates are related with magnification S ...
Context 5
... magnification is a process of obtaining an image at resolution higher than taken from image sensor. Image magnification synonyms are interpolation, enlargement, zooming, etc. To create higher resolution image, previous image must be complemented with new pixels and their intensity must be calculated. Commonly, image magnification is accomplished through convolution of the image samples with a single kernel – typically the nearest neighbour [1, 2] bilinear, bicubic [3] or cubic B-spline kernel [4] interpolation and training-based algorithms [5, 6]. Intensity surface can be explained as landscape with hills and hollows, rides and valleys. Any 3 dimensional (3D) surfaces, with some approximation, can be created from triangles. Look at the image intensity as 3D surface build from triangles where vertexes are image pixels with intensity as z axis (Fig. 1). To magnify image, new image must be complemented with new pixels, added necessary columns and rows of pixels, and calculated new pixels intensity. In this work is made assumption, that new pixels intensity is situated on suitable triangle planes (Figure 1). During image magnification contours are blurred because of slope sharpness reduction. This is the main problem for all known magnification methods. Look at small part of intensity surfaces in Figure 2. It is evident, that, in this situation, sharpness of slope is better when triangles are created like in Figure 2b than in Figure 2a. Accordingly the base of both triangles must be diagonal with less intensity gradient. That partially reduce magnification blur. So must be located distinctive points on intensity surface and different magnification algorithm applied. There are two things that must be solved: choose the best arrangement of triangles and calculate new pixels intensity. To explain triangle based magnification method, get four neighbouring pixels from original image and create two triangles planes throws these pixels as vertexes (black dots in Figure 2). These pixels 3D coordinates is known, because they are taken from original image. White dots are new pixels that were inserted to magnify image. Their intensity must be calculated. Early was decided, that new pixels will situated on two triangles planes. Triangle plane can be formulated as equation of three coplanar vectors over spatial points M 00 ( 0 , 0 , Z 00 ) , M 10 ( S , 0 , Z 10 ) , M 01 ( 0 , S , Z 01 ) where x and y coordinates are related with magnification S ...

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