Patent Application: US-28843402-A

Abstract:
image interpolation is becoming an increasingly important topic in digital image processing , especially as consumer digital photography is becoming ever more popular . from enlarging consumer images to creating large artistic prints , interpolation is at the heart of it all . this invention present a new method for image interpolation , which produces high quality results , yet it &# 39 ; s speed is comparable to polynomial interpolation .

Description:
this invention presents a method for enlarging an image by factor k in the horizontal and vertical directions . before we explain the general procedure , for enlargement by any factor k , we focus our attention to the case when the enlargement factor k is an even integer . in order to better understand the procedure we will further use a working example where the interpolation factor is a factor of k = 6 . later , we will show how to generalize the procedure to any real k . in the drawings of fig3 fig4 fig5 and fig6 we use the following convention to label the different classes of pixels . for the interpolated pixels ( 10 ) we use a black square , for the pixels that are being processed ( 11 ) we use a gray square , for the unprocessed pixels ( 12 ) we use a white square , and for the original pixels ( 13 ) we use a black square with white diagonal crosses . further , it is our initial assumption that we are working with a gray scale image . at the end , we will extend the interpolation procedure to color images as well . here are the interpolation steps : 1 . in the original image , each pixel has one diagonal and one nondiagonal direction . the diagonal directions can be : 45 degrees ( also named diagonal - 1 ), 135 degrees ( also named diagonal - 2 ), or diagonal - neutral . the nondiagonal directions can be zero degrees ( also named horizontal ), 90 degrees ( also named vertical ), or nondiagonal - neutral . in fig1 - b , pixels p 1 , p 2 , p 3 , p 4 , p 5 , p 6 , p 7 , p 8 , and p 9 are the original image pixels . given a positive threshold value th , which can be determined adaptively or preselected , the diagonal and nondiagonal directions of pixel p 5 are determined as follows : d 1 = abs   ( 1 2  ( p 4 - p 6 ) - p 5 ) , d 2 = abs   ( 1 2  ( p 3 - p 7 ) - p 5 ) , d 3 = abs   ( 1 2  ( p 2 - p 8 ) - p 5 ) , d 4 = abs   ( 1 2  ( p 1 - p 9 ) - p 5 ) . ( b ) if abs ( d 1 − d 3 )& lt ; th the direction is nondiagonal - neutral , else if d 1 & lt ; d 3 the direction is horizontal , ( c ) if abs ( d 2 − d 4 )& lt ; th the direction is diagonal - neutral , else if d 2 & lt ; d 4 the direction is diagonal - 1 , this means that from the original image we can form two new images : the first image , which we call diagonal direction label image ( 4 ), corresponds to the diagonal direction ( in which each pixels will have one of the three possible labels , namely diagonal - 1 , diagonal - 2 , or diagonal - neutral ) and the second , which we call nondiagonal direction label image ( 5 ), corresponds to the nondiagonal direction ( in which each pixels will have one of the three possible labels , namely horizontal , vertical , or nondiagonal - neutral ). if we focus our attention on image ( 4 ), it is unlikely that in any region of the image we will find an isolated pixel labeled diagonal - 1 and being surrounded by diagonal - 2 pixels . the reason for this is that in most images , edge pixels are clustered together . if we do find a diagonal - 1 pixel surrounded by diagonal - 2 pixels , then most likely the labeling of the diagonal - 1 pixel was a mistake and the actual label should have been a diagonal - 2 label . to further improve the robustness of the labeling algorithm , images ( 4 ) and ( 5 ) are median filtered and these now form the new labels for the diagonal and nondiagonal directions . next , here is the step - by - step procedure for interpolating an image by an even factor , namely k = 6 : 2 . upsample the image as in fig3 to obtain image ( 7 ). in this case , the upsampling factor is k = 6 . 3 . our next step is to process the diagonal pixels , which are the gray pixels ( 11 ) in fig4 - a . in order to do this , we segment the image into regions of interest that are ( k + 1 )×( k + 1 ) ( in our case the regions are 7 × 7 ) and focus our attention on one region of interest at a time . in fig4 - a one such region of interest is ( 14 ), which is depicted more clearly in fig4 - b . furthermore , fig4 - a has four such 7 × 7 regions of interest . we label this region of interest ( 14 ) as being diagonal - 1 , diagonal - 2 or diagonal - neutral based on the majority of the diagonal labeling of the nearby original pixels ( 13 ). for example , if three of the four nearby pixels are labeled as diagonal - 1 , then the region will be labeled diagonal - 1 . the nearby region is a predefined , a priory fixed size . the smallest nearby region would include at least the original pixels ( 13 ) in the region of interest ( 14 ). if the nearby region would be increased by 6 pixels , this would include all the original pixels ( 13 ) shown in fig4 - a plus all the other original pixels on the left and at the top of ( 14 )— not shown in the figure . once the region of interest ( 14 ) is labeled , we use 1 - dimensional interpolation as follows : ( a ) if the region of interest is diagonal - 1 , do 1 - dimensional interpolation in the diagonal - 1 direction . else if the region of interest is diagonal - 2 , do 1 - dimensional interpolation in the diagonal - 2 direction . else use the interpolated pixel values obtained from bi - cubic interpolation . in our example , fig5 - a depicts the completion of this step ( 8 ). here , from left - to - right , top - to - bottom , the first region of interest was labeled diagonal - 2 , the second was labeled diagonal - 1 , the third and fourth were labeled diagonal - 2 . ( b ) next , go back to each region of interest and use 1 - dimensional interpolation to interpolate in the other directions using the known interpolated values . for the other directions the sampling rate is k / 2 ( this is why we need k to be an even integer .) in fig5 - a , the other directions are from left - to - right , top - to - bottom : diagonal - 1 , diagonal - 2 , diagonal - 1 and diagonal - 2 . after this step is complete we have a diagonal grid ( 9 ) in which the interpolated diagonal lines cross themselves in original pixels as in fig5 - b . notice that the 1 - dimensional interpolation does not have to be polynomial interpolation , although this is the most common and we used cubic interpolation . 4 . next , we process of horizontal pixels , which are the gray pixels in fig6 - a . to do this , segment the image into ( k + 1 )×( k + 1 ) regions of interest . in fig6 - a there are four such regions of interest . this time , two of the regions of interest are ( 15 ) ( i . e . known pixels on top and bottom ) and two are ( 16 ) ( i . e . known pixels to the left and to the right ). however , this does not affect the processing of each region of interest . similarly to the previous step , label the regions of interest as horizontal , vertical or nondiagonal - neutral based on the majority of the nondiagonal labeling of the nearby original pixels ( 13 ). once the region of interest ( 15 ) and ( 16 ) is labeled , use 1 - dimensional interpolation as follows : ( a ) if the region of interest is horizontal , do 1 - dimensional interpolation in the horizontal direction for each row . else if the region of interest is vertical , do 1 - dimensional interpolation in the vertical direction for each column . notice again that the 1 - dimensional interpolation does not have to be polynomial interpolation , although this is the most common and for this step we use linear interpolation . if non - linear interpolation is to be used then the sampling grid is no longer uniformly sampled and more care needs to be taken in finding the cubic coefficients . for completeness we provide one such way here . in fig2 the pixel values at x 1 , x 2 , x 4 , and x 5 are known , and we want to find the pixel value at x 3 ( notice that the sampling grid is non - uniform ). further , we want to fit a cubic polynomial through the known samples . this means that : with a , b , c , d unknown . the four unknowns can be found by setting up a 4 × 4 linear system using y i = ax i 3 + bx i 2 + cx i + d for i = 1 , 2 , 4 , 5 . this system is invertible as long as x 1 , x 2 , x 4 , and x 5 are distinct and y 3 is determined uniquely . 5 . our final concern are the white pixels in fig6 - a . for an image , the border pixels will remain unprocessed . one way to handle this is very common in the image processing literature and that is to mirror the image across the border in order to process the border pixels . mirror extension is more desired since it tends to introduce the least amount of high frequencies and intuitively , this reduces ripples around the border pixels ( i . e . minimizes the gibbs phenomenon ). now that we have a clear understanding of the interpolation process for k - even , let &# 39 ; s extend the interpolation procedure to any k : 1 . for interpolation when k is and odd integer proceed in two steps . first interpolate for 2k ( notice that 2k is even ) and then downsample by two . for example , to interpolate by 3 , first interpolate by 6 and then downsample by 2 . 2 . for k a non - integer factor , write k as a fraction of two integers ( say k = m / n ). then interpolate by m and downsample by n . in this patent , down - sampling by n means taking every other n th pixel , starting with the first one . downsarnpling by 6 the image in fig3 one obtains the original image . until now most of the discussion was concerning gray scale images . some caution must be given when converting the algorithm to a color image . in particular , the caution must be taken in the labeling step ( step 1 ) of the algorithm . we suggest two possible approaches . in practice , both methods yield similar results , with a slight preference towards the second approach : 1 . the first approach is to use only the green channel in the labeling step to obtain images ( 4 ) and ( 5 ). then use these labels to do the directional 1 - dimensional interpolation for each color channel separately ( steps 2 - 5 ). 2 . a better option is to first convert from rgb to cielab and then use the distance defined in the cielab space to define d 1 through d 4 , namely : d 1 2 = [ 1 2  ( l 4 - l 6 ) - l 5 ] 2 + [ 1 2  ( a 4 - a 6 ) - a 5 ] 2 + [ 1 2  ( b 4 - b 6 ) - b 5 ] 2 d 2 2 = [ 1 2  ( l 3 - l 7 ) - l 5 ] 2 + [ 1 2  ( a 3 - a 7 ) - a 5 ] 2 + [ 1 2  ( b 3 - b 7 ) - b 5 ] 2 d 3 2 = [ 1 2  ( l 2 - l 8 ) - l 5 ] 2 + [ 1 2  ( a 2 - a 8 ) - a 5 ] 2 + [ 1 2  ( b 2 - b 8 ) - b 5 ] 2 d 4 2 = [ 1 2  ( l 1 - l 9 ) - l 5 ] 2 + [ 1 2  ( a 1 - a 9 ) - a 5 ] 2 + [ 1 2  ( b 1 - b 9 ) - b 5 ] 2 and use these distances to obtain the label images ( 4 ) and ( 5 ). then use these labels to do the directional 1 - dimensional interpolation for each color channel separately ( steps 2 - 5 ). of course , other distance metrics can also be tried for the definitions of d 1 , d 2 , d 3 , d 4 . one comment is in order for the computational cost . if one uses polynomial interpolation for the 1 - dimensional interpolation steps , after the classification of the pixel directions ( step one ) is complete , the number of multiplications and additions is the same as doing standard bi - polynomial interpolation . this means that the algorithm very fast and ready for today &# 39 ; s real - time graphical applications . we conclude this patent with two examples of the image interpolation algorithm applied to two difference images . the first image is a 512 × 512 rings image down - sampled by two , without pre - filtering , and then interpolated back to its original size . the downsampling process introduces some aliasing that manifests itself as extra rings going from the top of the image to the lower - right hand corner . cubic interpolation ( fig7 - a ) maintains the aliasing introduced by the downsampling process , while our interpolation ( fig7 - b ) removes it very nicely ( the alias is also more noticeable when the image is kept at arm &# 39 ; s length ). notice that the image in fig7 - b is much sharper . the second example is a leaf image interpolated by a factor of 4 . in fig8 - a we have the cubic interpolation and in fig8 - b we have the results of the interpolation presented in this patent . notice how much sharper and less jagged our interpolation is . 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