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Effect of Non-Integer Scaling on the Recovery of Data-bearing Marks

The spatial quantization imparted by printed pixels becomes significant when printing finely detailed bitonal images such as data-bearing halftones. This paper explores the consequences non-integer scaling has on data recovery error from such data-bearing marks. We verify that different printer manufacturers use nearest-neighbor scaling, and conducted hundreds of print and mobile camera capture measurements to quantify data recovery errors as a function of printed pixel replication factor. We tested the effects of multiple payloads represented by several data-bearing images, printed on a variety of printers, and captured at different camera distances. The analog print–and-capture experiments are compared with digital simulations, using several error measures. The results support a surprising conclusion: there is no significant advantage in forcing printed pixel replication to be an integer. Introduction Embedding data in and recovering data from hardcopy are key steps in a variety of information-rich applications. As the resolution and close-focus ability of mobile cameras increase, the marks for embedding data in hard copy continue to decrease in size. Embedding schemes such as QR or data matrix codes involve represented symbols with relatively large binary squares, and are robust to scaling effects. Higher density solutions such as data-bearing halftones [1][2] can require more careful registration for accurate decoding. In the case of stegatones [3], the focus of this paper, data is represented by single-pixel shifts of halftone clustered dots. A convenient assumption that helps promote robust recovery of these designs is that a physical rendering from input pixels to device pixels must be scaled by an integer, such that single pixel perturbations translate cleanly into the rendered image space. There are, however, situations where this integer relationship cannot be guaranteed. An important case is that when a document is prepared with a fixed size mark, but the printer is not yet known. 400 dpi data-bearing halftones have yielded a good blend of payload density and recoverability. While a document with such a mark will render beautifully on a 1200dpi printer (with a replication factor of 3X), a potential problem arises with a 600dpi printer. The printed pixel replication factor will be 1.5, meaning some input pixels are represented by 2x2 device pixels and some are represented by a single (1x1) pixel. Changing the image size will also introduce non-integer replication factors. For fine details, the effects of non-integer scaling can be significant, and in fact, a halftone strategy that attempts to create imagery suitable for a set of different scales has been proposed [4]. Consider the example of scaling a one-pixel wide line as shown in Figure 1(a) by a factor of 1.6. Nearest-neighbor scaling will render the line as shown in (b). The uniform input line is now composed of segments with 4 different shapes. Bilinear scaling produces the gray scale version in (c) which is visually more uniform but cannot be rendered on most printers which are inherently bitonal. Figure 1. Scaling a one-pixel wide line by 1.6x. (a) Input, (b) Nearestneighbor scaling, (c) Bilinear scaling. The generation of a stegatone is illustrated in Figure 2. A continuous-tone input image (a) is halftoned (b) using a classical 45-degree clustered-dot screen. A data payload in embedded by single-pixel shifting the halftone clusters to produce the stegatone in (c). For this particular 200x200 pixel image the payload capacity is 1488 raw bits, with an error-corrected capacity of 376 bits. Nearest-neighbor scaling the stegatone by a factor of 1.6 produces the binary image in (d). A small area from the left side of the stegatone is enlarged in Figure 3. A region form the input image is shown in (a) with uniform 2x2 pixel clusters. The 1.6x scaled version of the same area in (b) reveals clusters of size 3x3, 4x4, 3x4, and 4x3 pixels. Figure 2. Stegatone generation and printing. (a) Gray scale input image, (b) Clustered-dot halftone, (c) Stegatone, and (d) Scaled by 1.6x when printed. (c) (b) (a)