Image Obscuration within Moire PatternsStanford EE 368, Digital Image Processing, Course Report

Instructor: Gordon Wetzstein

Elliott SpelmanStanford University

Department of Art and Art Historyespelman@stanford.edu

Keenan MolnerStanford University

Department of Electrical Engineeringkmolner@stanford.edu

Abstract

The pointwise multiplication of banding images - alter-nating black and white stripes, arranged in linear or sinu-soidal patterns - often leads to interesting and unique visualeffects. One family of combined banding images, referredto as Moire fringe patterns, is the result of interference be-tween periodic bands. Moire images have two components:the base band image and the revealing bands. As the reveal-ing mask is translated across a processed image, differentamounts of shift will cause obvious visual effects, particu-larly beating animation. In this project, we investigate dif-ferent geometric transforms and image modification tech-niques to obscure one or multiple images within a singleMoire band pattern, both in black and white and in color.

1. IntroductionMoire fringe patterns result from the interference be-

tween two different periodic banding images[1]. These pat-terns can be observed in nature, like through window blindsor in between fence posts. Part of their allure is their incor-poration of small periodic patterns to reveal macroscopicshapes. Additionally, the translation of one pattern acrossanother can result in a beating animation, adding a per-ceived motion to an otherwise still scene. Moire patternsare used commercially in counterfeit protection systems[2],as it’s very difficult to replicate an exact fringe pattern.

2. MotivationWe chose to investigate Moire beating shapes primarily

for aesthetic reasons. Just as ”Magic Eye” books present afun, creative, and captivating way to obscure image infor-mation in plain sight, we too wanted to explore these ideas.In addition to spatial encoding, the animation that one ob-serves when translating the revealing mask over the baselayers is mesmerizing. This project makes extensive use of

the Matlab image processing toolbox, with emphasis placedon image capture, morphological image operators, and dis-play methods. Material covered in the EE368 class preparedus well for this creative project.

3. Related WorkChosson et al. generalizes Moire fringe line patterns to

be described by the following equation, representing theinterference lines, indexed by k, between two images [3]:Φ(x,y)Treveal

− Ψ(x,y)Tbaseband

= k. Amidror [4] defines G(x, y) to bethe elevation function that shifts a periodic base banding,later to be obscured with a revealing mask with the sameperiod and orientation. G(x, y) can be discontinuous andcomplicated, and only needs to be represented as a depthimage. Q(x, y) is a geometric transform that maps pointsin the image space shifted by G(x, y) back to the originalperiodic gratings. By substituting Φ(x, y) = Q(x, y) andΨ(x, y) = Q(x, y) − G(x, y) we can recover the Moirerepresentation of the depth profile.

Q(x, y)

T− Q(x, y)−G(x, y)

T= k ⇒ G(x, y)

T= K

Furthermore, Chossen proves that all geometric transformsapplied to the base band and the revealing mask will sim-plify to meet the conditions laid out for G(x,y)

T = k, al-lowing us to see our original depth image, no matter thetransformation. transformation[3].

4. Band GenerationBand generation begins with tuning parameters for the

desired revealing mask. We further explore manipulatingthese parameters in Section 6. Once band period, size, andorientation have been determined, we generate an alternat-ing black and white banding of equal thickness for the re-vealing mask. We then generate a second copy of this band-ing pattern. This secondary copy will be offset in further

steps to create our underlying image. The first copy of thebanding image is our revealing mask. Alternatively, ratherthan generating a black and white banding with equal spac-ing, we can also generate single pixel bands of colors, re-peating with the same period as the black and white band-ing.

5. Image Embedding MethodsGrayscale images representing a depth map are embed-

ded in the base banding patterns, to be revealed later withthe revealing mask. To embed the depth image within thebase banding, we shift position of each banding pixel alongthe vertical direction by the scaled depth value. The bright-est depth map pixels are shifted the greatest distance. Wemake sure to constrain this shift amount to the maximumperiod of the banding in the original band image. This en-sures that each pixel intensity will have a unique transla-tion amount. The amount each pixel in the base banding isshifted by is described by the equation[4]:

ynew = y + round(pixelIntensity × bandPeriod)

Figure 1 shows a grayscale image and it’s embeddingwithin a horizontal, black and white base band layer.

Figure 1. Embedding an image into the base bands

By using a grayscale depth map, rather than a black andwhite image, we achieve multiple perceived intensities withthe superposition of the revealing mask. Additionally, theshapes at different depths will beat through different col-ors or intensities as the revealing mask is translated acrossthe image. For simplicity, we’ve decided that all revealingmasks will be translated only in the vertical direction.

5.1. Black and White morphological image modifi-cation

By turning a simple black and white image into agrayscale image, we achieve more visually stimulating ef-fects as the revealing mask is translated over the the basebandings. In effect, we are simulating depth images with-out actually having depth data. We use morphological im-age operators such as dilation, erosion, and skeletonization

to separate the outlines, background and foreground of theimage. We then take the Euclidean distance transform be-tween the skeleton of the foreground and the shape outline,as well as between the skeleton of the background and theshape outline[3].

To present a sharp contrast between the foreground andthe background, we remap distance transform values to liebetween 1 to 0.5 for the foreground and 0.0 to 0.5 for thebackground. Edges between the original foreground andbackground shapes are presented in these depth maps as astep between 0.5 and 0.0, a large step when the revealingmask added. The original image and the resulting imagefrom our morphological image processing are show in Fig-ure 2.

Figure 2. Before and after mophological image processing

5.2. Depth Image embedding

In addition to accepting simulated depth images, ourbanding offset algorithm obviously works with true spatialdepth information as well. We’ve enabled our project tocapture depth maps with a Kinect camera and embed thisimage into the base bands. The depth resolution is limitedto the period size of the revealing mask to prevent aliasingin the final superimposed image. Figure 3 shows the orig-inal depth map and the recovered image with a revealingmask.

Figure 3. Hidden and revealed image

5.3. Multi-image embedding

Our image processing pipeline is not limited to the ob-scuration of single images. The same process also allowsfor multiple depth maps to be obscured within the same

image, each of which can be revealed with a correspond-ing revealing mask. To do so, we generate two embeddedimage patterns and their revealing masks independently ofeach other [5]. Then, we combine the two embedded imageswith a pointwise OR operation for black and white images,or a pointwise multiplication for color images.

Figure 4. Multiple images within the same base banding

The results shown in Figure 4 show a combined base-band image and the two revealed images when using differ-ent revealing masks. More than two images could be com-bined with this same method, but artifacts between imagesbegin to show up if parameters in the revealing masks aretoo similar.

5.4. Beating Shapes

Shifting the revealing layer in the vertical direction byone pixel at a time shifts all unoccluded pixels in the basebanding by one grayscale level. This creates a beating ef-fect. The number of steps in the beating effect is determinedby the period of the base banding. Figure 1 has a period ofT = 4 and shows four frames of the successive image se-quence.

Figure 5. Beating Moire fringe patterns

6. Band Manipulation MethodsWith these aforementioned banded image generation

processes, we explore modifying our baseline parametersand the resulting images. We look at changes to bandspacing, band angle, the introduction of color banding, andlarger geometric transforms that can be applied to the entireimage.

6.1. Band Spacing

When generating the bandings for the base layer andthe revealing mask, we choose a period in pixels for howlarge we desire our bands to be. Larger periods give rise to

wider spacing, but also present the opportunity to achievemore discrete levels of perceived grayscale. This is a resultfrom the way we form our shifted baseband images, mov-ing each pixel in the y direction by the value of the depthmap, proportional to the period of the banding: ynew =y + round(pixelIntensity × bandPeriod).

While this constraint ensures that our maximum shift isno larger than the period of the banding to prevent aliasing,the larger band period also allows for more discrete per-ceived grayscale levels when interfered with the revealingmask. For example, a banding with a period of T = 2 wouldonly allow for the revealing mask to either be fully trans-missive, or fully occluding at each translation in the y di-rection, resulting in only two different perceived grayscalelevels. A banding with a period of T = 4 Allows for fullocclusion, three-quarters occlusion, half transmission, andquarter transmission, resulting in four perceived grayscalelevels. The results show in Figure 6 show the differencebetween band periods of T = 2 and T = 4 pixels.

Figure 6. Banding with Period of T = 2 and T = 4

6.2. Band Angle

Keeping the band period the same, we can rotate the an-gle of both the base banding and the revealing mask by thesame angle, θ, and achieve another family of moire patterns,still fulfilling the G(x,y)

T = k requirements [4]. While weare still able to recover the embedded image from the occlu-sion at different angles, the perceived grayscale value mayappear different, due to the interpolation use when rotatingthe base banding. Different rotations have different interpo-lations, resulting in different overlap at the same position onthe image. Figure 7 shows the resulting interference after a20◦ and 40◦ rotation to the bandings, with period kept thesame.

Figure 7. Banding with rotation of 20◦ and 40◦

Additionally, band rotation obscures the desired embed-ded image in the original base bands better than in tradi-tional horizontal bandings. This is advantageous for hidingvisually sensitive images or making bandings more difficultto copy. Figure 8 shows the a horizontal and rotated basebanding without a revealing mask, both with the same im-age and same band period. It is more difficult to spot theoriginal image in the rotated bands.

Figure 8. Unmasked banding with rotation of 0◦ and 40◦

6.3. Band Color

As previously mentioned, Moire fringe patterns can beproduced in color if the color banding matches the same pe-riod, angle, and transformation as the revealing mask. Sim-ilarly to how changing the band period increases the num-ber of perceived grayscale levels, Larger periods result in agreater number of colors. This is demonstrated in Figure 9with four and six different band colors.

Figure 9. Four and six color banding

6.4. Band Translation

To make banding patterns more complex, we introducea geometric shift at each pixel in both the banding and therevealing mask. Like the banding rotation, this fulfills allmathematical requirements for Moire patterns. For the pur-poses of our demonstration, we defined a cosine shift, mov-ing each pixel along the horizontal direction by a specificamplitude. The amplitude of the shift changes dependingon the cosine of the position. Both the period of the waveand the amplitude of the offset are adjustable. This workswith both black and white and color banding images. Twodifferent cosine waves are demonstrated in Figure 10.

Figure 10. Banding images with cosine wave transformations

Wave translations can be thought of as position depen-dent angular rotations. Like the increased visual complexityof the obscured image when band angles are rotated, the ob-scured image banding becomes almost completely impossi-ble to understand without the revealing mask. This is showin Figure 11

Figure 11. Banding images with cosine wave transformations

7. Future Work

We’d like to investigate other transformations that couldbe applied that satisfy the equation for G(x,y)

T = k, likerotational Moire pattern revelation. We believe that differ-ent revealing masks with horizontal, vertical, and rotationalpreference could all be introduced into the same basebandimage, each revealing a different depth map when translatedwith the correct revealing mask with the same orientationpreference. One could potentially create 3D patterns that

give the appearance of beating fringe patterns dependingon the angle of observation, like the projection of a fencedown onto the ground when viewed at an angle. Addition-ally, we suspect there to be an audio analogy to these imagetransformation processes, with the revealing pattern repre-senting the waveform amplitude and the orientation repre-senting the phase. These ideas could also be combined withphone cameras, where the shifted band image is passed be-tween two individuals, to be revealed later using an app withknowledge of the encoding.

Another avenue of research would be to quantify howwell an image can be obscured - how much of the originalimage remains to the naked eye. This is a difficult thing toautomate as it involves input from human observers. Eachobscured image would need a collection of guesses on whatthe original image might contain. This type of aggregate re-search would be fun to run as a web game - perhaps, similarto the Magic Eye game mentioned earlier in this paper, peo-ple would enjoy attempting to determine what the obscuredimage truly represented.

8. AcknowledgmentsWe thank Prof. Gordon Wetzstein for teaching and men-

torship.

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[3] Sylvain M Chosson and Roger D Hersch. Beating shapes re-lying on moire level lines. ACM Transactions on Graphics(TOG), 34(1):9, 2014.

[4] Isaac Amidror, Itzhak Amidror, Itzhak Amidror, and ItzhakAmidror. The theory of the moire phenomenon. Number LSP-BOOK-2000-001. Springer, 2000.

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