1. Introduction

Lenticular screening refers to a print technology where multiple images are spatially multiplexed and then printed onto the flat side of a plastic, lenticular lens array such that only a single image is visible when viewed from any particular incident angle (Fig. 1). Assuming the screen is viewed with the lens axes vertically arranged, the screen forms a stereoscopic array where each eye sees a unique image thus creating the illusion of depth. As such, the component images are typically of a 3-D scene viewed from a sliding camera (Fig. 2), but dissimilar images are sometimes used to create a scene-wipe effect. Now although stereoscopic imaging can be traced back to Sir Charles Wheatstone and the invention of the stereoscope in 1838 and even back to 1692 and the French painter Gois-Clair [1], it wasn’t until the 1940s that lenticular imaging became of commercial interest with the introduction of inexpensive manufacturing processes for plastic, lenticular sheets [2]. Today, lenticular printing is now commonly performed using lithographic presses printing directly to the plastic lens arrays. There is also significant interest in printing directly on sheets with hexagonally packed lens arrays.

For digital halftoning where a continuous-tone original image is converted to a binary representation for printing, lenticular screening creates a unique and challenging set of problems. Traditionally, halftoning converts an image into binary dots arranged either as a regular grid of round dot clusters that vary in size according to tone or as a stochastic arrangement of isolated dots varying in their spacing. In the case of a periodic arrangement, halftoning schemes are generally referred to as amplitude modulated (AM) halftoning where dark shades of gray are represented by large, black dot clusters and light shades by small clusters. In the case of aperiodic patterns, halftoning schemes are generally referred to as frequency modulated (FM) halftoning where dark shades of gray are represented by a tight packing of printed black dots and light shades by a loose packing.

 

Fig. 1. Illustration of the lenticular imaging process (provided by http://www.lenstar.org).

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Frequency modulated techniques are generally considered to be the preferred approach due to the lower visibility of the isolated pixels, as opposed to clusters, and to aperiodic textures, as opposed to regular grids [3]. Because these techniques achieve a higher apparent resolution than AM, FM halftoning is also commonly referred to as high fidelity halftoning. But because dots are isolated, FM halftones are much more susceptible to distortions caused by variations in the size and shape of printed dots [4–6]. As such, only predictable printing processes like ink-jet devices can employ these techniques.

Amplitude modulation, by clustering pixels, creates patterns less susceptible to distortions caused by dot variability as clustering greatly improves the consistency in the size and shape of printed dots [7]. As such, AM techniques are sometimes the only techniques that can be used in unreliable devices such a electrophotographic (laser) printers. AM halftoning is also used in highly reliable devices where the device resolution in terms of dot addressability (i.e. 10,000 dots per inch) is much greater than the resolution in terms of printed dot size (i.e. 10 micron spot size). For these devices, the corresponding AM screens can achieve higher screen frequencies than the apparent resolution of the FM screens because printed dots, placed side-by-side, result in a dot area only slight larger than a single, isolated dot. As such, AM screens can address a multitude more unique gray-levels per unit area than a FM screen.

Now for lenticular imaging, the image to be halftoned is formed by splicing the columns of the component images, and because of the manner of packing multiple views/images into a single print, the apparent resolution is reduced along the horizontal axis by a factor equal to the number of views. As such, there is an even greater need for the screening technique to maximize the apparent resolution of a given device but to do so without introducing spatial correlation between columns from neighboring views, where the printed status of a pixel in one column affects the resulting printed tone of pixels in neighboring columns. AM halftoning would, therefore, seem inappropriate given the lower apparent resolution of AM screens and the manner in which AM halftoning divides the composite image into halftone cells that are then replaced with a cluster representing the average gray-level for the entire block [8], irrespective of the image content within the cell.

 

Fig. 2. The component images of a lenticular image as seen through a lenticular lens array. [Media 1]

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Frequency modulated halftoning, through error-diffusion [9], would seem an appropriate screening technique for lenticular imaging because of its behavior to isolate printed dots, but the manner in which quantization error is diffused into neighboring pixels has unwanted consequences given the fact that neighboring pixels may be from different component images. Pixels in the halftone image will, therefore, be correlated when the corresponding pixels are statistically independent in the continuous-tone, composite image. A demonstration of the effects of error-diffusion halftoning, using the 12-weight Stucki [10] filter and a serpentine raster [11], of the composite images can be seen in Fig. 3 where we show the simulated visual response to a 12-frame, lenticular image sequence with streaking artifacts most visible in the first frame where a white text string is visible across the top of the image. This white logo is the result of dot suppression caused by the black letters from the last frame of the sequence. Streaking artifacts are also visible in the first frame to the left and right of the central dolphin where extreme gray-level pixels, of the neighboring channel, show up as a negative image blended with the current view.

In addressing these halftoning issues, the best source of information regarding lenticular halftoning is, perhaps, the patent literature where just some of the many works include that of Goggins [12–14], Pilu [15], Yano [16], and of Iwano et al [17]. A work of particular importance is that of Goggins who describes a lenticular printing system where the component images are reduced in size (number of pixels) along the horizontal axis and then halftoned. The columns of each component halftoned image are then spatially multiplexed together such that the n^th column of each component image falls under the n^th lens. This technique is demonstrated in Fig. 4 where the composite images, generated by means of error diffusion prior to being spatially multiplexed together, show an almost perfect rendition of the original sequence of Fig. 2.

 

Fig. 3. The component images of a lenticular image as seen through a lenticular lens array where error diffusion has been applied after spatial multiplexing. [Media 2]

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Fig. 4. The component images of a lenticular image as seen through a lenticular lens array where error diffusion has been applied prior to spatial multiplexing. [Media 3]

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Fig. 5. The component images of a lenticular image as seen through a lens array, using a printed dot model, where error diffusion has been applied prior to spatial multiplexing with tone correction. [Media 4]

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Now unlike the works of Goggins that recognize the need to maintain statistical independence between pixels of separate component images, we also recognize that printer distortions, such as dot-overlap and dot-loss, will create correlations/interactions between consecutive pixels regardless of the underlying image content. An illustration of this correlation can be seen in Fig. 5 where we show the images produced by error-diffusion using a printed-dot model [18,19] and where tone correction [11] has been applied prior to halftoning. From visual inspection, the overlap of printed dots into neighboring channels creates a ghost image most visible above and below the text string. So from this, we realize that traditional tone correction is incapable of accounting for printer variations in lenticular prints because these techniques rely upon an assumed distribution of dots at each gray-level. As such given a particular pixel’s gray-level in a traditional application, we can easily predict what the average tone level will be around that pixel in the final output from prior measurements and, hence, we will alter that pixel’s gray-level through a reverse table look-up procedure such that we get our desired tone in the printed output. By halftoning each component image independently, we cannot reliably predict how much ink will overlap from the neighboring pixel columns and, as such, don’t know how to alter the current pixel’s gray-level such that the total ink coverage within the pixel column will be equal to the input gray level.

In light of the artifacts created by non-ideal printing devices, this paper describes our work with lenticular halftoning that delineates itself on this later premise of printed-dot interactions. In particular, this paper focuses on our use of model-based error-diffusion as a means of maximizing the apparent resolution within each component image by creating an aperiodic pattern of isolated dots, taking into account the size and shape of printed dots (even isolated ones) when diffusing quantization error between pixels. Model-based error-diffusion was first introduced by Pappas and Neuhoff [19] who used the hard, circular-dot model of Roetling and Holladay [18] to predict the resulting gray-level of each halftone pixel after printing. This predicted gray-level was then used to determine the quantization error to be diffused into soon-to-be-processed pixels as opposed to assuming binary black or white levels.

 

Fig. 6. Model-based error-diffusion.

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In order to minimize the correlation between consecutive columns of lenticular images, we will make use of specially modified error filters such that the quantization error for any particular pixel is only diffused to neighboring input pixels of that same component image. But this diffusion technique creates instabilities in areas where component images differ widely in their gray-levels. As such, we will describe a novel technique for clipping the quantization error such that error does not build up over many pixels. The excess error, beyond the clipping threshold, will not be discarded as proposed by Kim [20] but, instead, will be diffused into the input gray-levels of the nearest neighboring pixels regardless of which component image they belong. The effect of this diffusion is to lighten the gray-levels of the neighboring pixels, and by brightening these neighboring pixels who are responsible for limiting the gamut of the currently processed pixel, we hope to reduce the gamut reduction. This, of course, introduces reverse ghosting, but the assumption here is that we need to eliminate the instability in error diffusion and that introducing mild to moderate reverse ghosting in dark image regions is better than severe ghosting in bright regions.

2. Model-based error-diffusion

Introduced in 1976, Floyd and Steinberg’s error-diffusion [9] is a neighborhood filtering process that converts a continuous-tone image pixel x[n] into a binary pixel of halftone image y[n] such that:

where x_e [n] is the diffused quantization error accumulated during previous iterations as:

with y_e [n] = y[n] - (x[n] + x_e [n]). The diffusion coefficients b_i , which regulate the proportions to which the quantization error at pixel n transfers or diffuses into neighboring pixels, are such that ${\mathrm{\Sigma }}_{i=1}^M$ b_i = 1. Computationally, error-diffusion can be done in-place with the output pixels, y[n], residing in the memory locations of the input pixels, x[n].

Noting that the printed dots of an ink-jet or similar printer can be accurately modeled as a binary, round, circular-dot such that an isolated black pixel is completely covered with ink and with portions of neighboring white pixels partially covered by ink, it should be obvious that the binary halftones printed by error diffusion will always print darker than their ratio of black to white pixels. As such, images will typically be tone-corrected prior to halftoning in order to compensate for this ink overlap. For an alternative approach, Pappas and Neuhoff [19] proposed model-based error-diffusion as a means of accounting for dot overlap in the halftoning process where, as depicted in Fig. 6, a model of the printed dot is used to predict the ink coverage within each halftone pixel after printing and then using this modeled ink coverage in the calculation of the corresponding quantization error.

 

Fig. 7. Calculation of the error term in model-based error-diffusion.

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In model-based error-diffusion, the output pixel, y[n], is still determined as defined in eqn. (1), but in this case, the error terms, y_e [n - i] for i = 1,2,… ,M, are calculated at each iteration and cannot be stored in an error image buffer. That is, assuming an ideal printer means that the quantization error, y_e [n], can be diffused and stored in an error buffer, e[n], such that:

Using a dot model such as the hard circular-dot, which affects neighboring pixels, means that the final value of y[n] is not determined until all its neighbors have been quantized. So in model-based error-diffusion, the error term, e[n], has to be calculated at each iteration prior to quantizing x[n] + e[n] as:

where ỹ[n - i] is the modeled tone for output pixel y[n - i] assuming y[n + i] for i = 1,2,… are not printed (Fig. 7). From eqns. (3) and (6), model-based error-diffusion can be summarized as:

Now in order to predict the resulting gray-levels that will be produced by the printer for a given dot pattern, we need an accurate model of the printing device. Such a model can be specified by a formula, as performed by Pappas and Neuhoff [19] using the hard-circular dot model depicted in Fig. 7, or by table look-up where the table is generated by analysis of printed test patterns from the target device. Such an analysis was performed by Baqai and Allebach [21] who used a test pattern where a 145×145 binary image is divided into blocks of size 6×6. Inside each 6×6 is a 3×3 sub-block that, over the entire pattern, represents every possible combination of 9 bits (512 patterns).

 

Fig. 8. The (top) traditional and (bottom) lenticular Stucki error-diffusion filters for a four component, lenticular image where halftoning can now be done after the spatially multiplexing of images but with the same results as if done prior to. Color coding has been used to indicate which component image each pixel belongs while the arrows indicate the raster scanning direction.

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By repeatedly printing and scanning this pattern and then performing an automated search for a specific bit-pattern, Baqai and Allebach aligned the printed 6×6 blocks from multiple scans according to the fudiciary pixels surrounding the block at its 4 corners. Once aligned, the 6×6 blocks would be averaged together and then the average gray-level corresponding to the center pixel of the 3×3 sub-block would be entered into the look-up table (LUT) for that particular 9-bit combination. Noting that many printing devices produce radially symmetric dots (on average), the total size of the LUT can be reduced from 512 entries down to 102 given that many of the 9-bit combinations, when printed as 3×3 blocks, are permutations (flips, rotations, etc.) of other combinations. Wang et al [22] even took the unique approach of modeling the average gray-level of the region between the centers of the printed pixels forming a 2×2 cluster, reducing the LUT to 16 entries.

3. Lenticular screening

In extending model-based error diffusion to lenticular printing, we note that, as a neighborhood process, error-diffusion assumes correlation between neighboring pixels [3], and in the case of a step-edge, error-diffusion will diffuse error across the edge to create a blurred edge in the halftone [20], and supposing that the image is composed of multiple images spatially multiplexed together, this blurring will result in columns of image slice A bleeding into slice B. The overall impact of this bleeding depends on how well correlated the two slices are at this point but ghosting artifacts are the result as was demonstrated in Fig. 3. So in order to eliminate or minimize the effects of bleeding quantization error between component images, we propose using error filters that restrict the quantization error from pixels of slice A to only those neighboring pixels also from slice A. Under the assumption that there are an equal number of image columns under each and every lenticular lens such that corresponding columns of the same component image are equally spaced apart, we need only splice an equal number of zero columns between the columns of a traditional error filter as illustrated in Fig. 8 using Stucki’s 12-weight error filter [10].

For situations where the number of columns under each lens is not equal for all lenses such as when printing 1,000 dpi halftones onto 80 lens-per-inch material with even numbered lenses having 12 pixel columns and odd numbered lenses having 13 columns, the distribution of error will not be restricted to the appropriate channel by simply splicing zeros into the error filter. As such for each lens whose number of underlying columns is less than the maximum number of columns under any one particular lens (the even lenses with 12 columns), we can insert temporary spacer columns into the original continuous-tone image either by duplication of an existing column, by inserting a pre-specified constant value, or arbitrary combination of existing columns. After halftoning this up-sampled, continuous-tone image using the proposed zero-spliced error filter, we then delete the spacer columns from the halftone image, thereby, generating an appropriately sized, binary image with, in our example, even numbered columns having 12 pixel columns and odd numbered columns having 13.

 

Fig. 9. The component images of a lenticular sequence as seen through a lens array, using a printed-dot model, where the proposed model-based error diffusion technique has been applied after spatial multiplexing. [Media 5]

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In the case of non-lenticular lens arrays where the spatial multiplexing of pixels from multiple component images leads to an arbitrary distribution, we need a means by which we can properly distribute error from previously processed pixels. To do so, we create a so-called “map” image as a way of searching the post-multiplexed image for nearest neighboring pixels from the same component image as the pixel currently being quantized. Specifically, each component image is assigned a unique identification number/tag such that the pixel, map[n], of the map image is set equal to the identification number/tag associated with the component image corresponding to x[n]. So in the process of quantizing x[n], we search a fixed-sized neighbor of the already processed pixels looking for those pixels whose corresponding map image has the same identification number/tag. Of those pixels corresponding to the same component image, we define a set of filter weights {b_i :i = 1,2,…}, which sum to one, and calculate the resulting error term e[n].

For a demonstration of the halftones produced by the proposed MBED technique, Fig. 9 shows the resulting lenticular image sequence that, in comparison with Fig. 5, shows greatly reduced ghosting artifacts in the regions of the traveling text string. But these artifacts are not wholly eliminated as is evident in the light gray regions directly neighboring black pixels in the frames directly before or after the subject frame (i.e. top of first frame) - limiting the maximum intensity value to 0.67 (0=black, 1=white). These artifacts are a direct consequence of the gamut reduction produced by dot overlap where the printed dots, from dark-gray slices, overlap into the pixels of neighboring light-gray slices creating the impression that three frames are visible instead of just the one. This gamut reduction has the potential to create an instability [24–27] where quantization error builds up uncontrollably, suppressing dots in later regions where we may, otherwise, want dots to be printed.

In order to eliminate this unregulated build-up of error, we could use a technique similar to that of Kim et al [20] who clipped the quantization error across step-edges in order to reduce the amount of bleeding of quantization error across discontuities in gray-level. Specifically, Kim et al would compare the accumulated quantization error with the current input pixel, which after crossing a step-edge would jump in value. This jump in value would, likewise, create a jump in the distance of the accumulated error value, x_e [n], with the input pixel, x[n], and, hence, trigger a clipping operation defined by some threshold, T.

In our case, we could simply clip the error, regardless of the current input gray-level, beyond a similar threshold (T = 0.80), but while this approach will eliminate dot suppression, it does not address the problem of ghosting where the dark regions of one component image show up in neighboring component images. So instead of simply discarding the excess error, we will diffuse it into the dark-gray pixels of the nearest, neighboring, component images. By doing so, we increase the gray-levels of these neighboring dark regions to, thereby, reduce the number of printed dots responsible for limiting the light-gray output gamut. In this manner, we distribute the effects of dot-overlap symmetrically to both the extreme light and dark gray-levels. In the case of Fig. 9, the maximum intensity level increases from 0.67 to 0.82 with an increase in intensity for the black text from 0.07 to 0.19. For comparison, the same ghosting artifacts in Fig. 5 have a gray level of 0.77.

4. Conclusions

Digital halftoning is the process of converting continuous-tone images to binary for display by digital printers, and as printers are achieving ever higher resolutions, the variation in the size and shape of printed dots are having an increasingly negative impact on the overall quality of the resulting prints. As such, modern halftoning algorithms need to take into account the behavior of the device to print isolated dots in order to counteract the effects of printer distortion. For lenticular screening, the problem of accounting for printer distortion is especially difficult because of the need to maintain statistical independence between component images. In this paper, we have described a series of modifications to Pappas and Neuhoff’s model-based error-diffusion algorithm where our approach is to use specially designed error filters that restrict quantization error to pixels of the same slice. And in order to maintain stability, we employ a novel clip-and-diffuse process of the quantization error at each pixel. Now while ghosting artifacts were not completely eliminated, the proposed technique of model-based halftoning did greatly reduce the overall visibility of artifacts where only in cases of bright gray-levels above 0.82 did artifacts occur. Regardless, this issue of dot overlap and it’s corresponding gamut reduction remains a challenge for future works in this area as our a long list of additional concerns. These other concerns include issues related to the alignment of the lens array with the halftone sampling grid during printing, and it these other issues that need to be addressed in future works.