ElectRoidWarE: Non-rectangular Sampling Grids

So far our discussion has been on image sampling on a rectangular grid. This is the most commonly used sampling grid structure because many display systemssuch as TV use raster scanning, that is, scanning from left to right and top to bottom. Also, most digital cameras have their sensors built in rectangular grid arrays.However, it is possible to use nonrectangular grids, such as hexagonal sampling grid, to acquire a digital image. An advantage of using hexagonal sampling grid is that the acquired image has 13.4% less data than that acquired using the rectangular sampling grid [2]. It has also been found that edge detection is more efficient with hexagonally sampled images. Hexagonal sampling is used widely in machine vision and biomedicalimaging.

Although the compression standards such as Moving Picture Experts Group (MPEG) use rectangular grid structure for coding still and moving imagesespecially for motion estimation and compensation, it may be more efficient to employ hexagonal grids for such purposes for better accuracy in motion estimation and higher compression ratio.

Figure 1Effect of nonideal sampling: (a) a 64 × 64 BW image, (b) rectangular sampling

image of size 32 × 32 pixels, (c) sampled image, (d) 2D Fourier transform of (a), (e) 2D Fourier transform of (b), (f) 2D Fourier transform of the sampled image in (b), (g) 2D Fourier transform of an ideal impulse, (h) 2D Fourier transform of (a) sampled by impulse, and (i) image in (c) obtained by filtering by an ideal lowpass filter.

The Barbara image from the originalrectangular sampling to hexagonal sampling grid using equation and displaythem. Figure 3.a is the cropped image of Barbara using rectangular sampling grid. The corresponding hexagonally sampled image is shown in Figure 3.b. If M and N are the number of rows and columns, respectively, of an image, then the size of the hexagonally sampled image will be M + N − 1 × M + N. In the MATLAB code listed below, conversion from hexagonal grid back to rectangular grid is also included. The code reads the specified image, converts to hexagonal grid, displays the respective images, converts hexagonal to rectangular grid, and displays the valid region.

Figure 2 Converting from rectangular to hexagonal sampling grids: (a) pixels on a rectangular grid with unit spacing in both horizontal and vertical directions, (b) pixels on hexagonal grids after applying the transformation in equation, (c) zoomed version of (a), and (d) zoomed version of (b). In both (c) and (d), the zoom factor is 4.