Instructions for rendering

Rendering 3D scenes for lenticular 3D print (simplified) July 12, 2018 Š Yitzhak Weissman Ver. 3.1

1 THE SHOOTING SCENARIO The art submission for a 3D lenticular print consists of a sequence of images of the object taken from different points in space. There are two photography approaches: linear and angular. Here, we will adopt the angular approach, since it is simpler to explain and to apply. Both approaches, if applied correctly, yield identical results. The angular photography approach is shown schematically in Figure 1.

Figure 1: The angular photography scenario In the angular photography scenario, the shooting points are distributed evenly on an arc. The scenario is defined by four parameters: arc center, shooting distance, shooting angle, and the number of shooting points. The reference plane contains the arc center, and the picture plane will coincide with the reference plane.

The arc center is normally chosen to be close to the object’s center, and the shooting distance can be chosen at convenience. The determination of the shooting angle requires the introduction of an additional concept: the parallax.

2 THE PARALLAX In any two given images, the representations of a given object point appear displaced horizontally. The parallax is the distance between the locations of these points.

The parallax increases with the distance of the object point from the reference plane (Figure 1). The extreme parallaxes of the sequence will be those corresponding to the object points with the largest distances from the 2

reference plane. The parallax corresponding to the object point closest to the shooting points is called “front parallax”. Likewise, the parallax corresponding to the furthest object point is called “back parallax”. Typically, the reference plane will be located between these points, as shown in Figure 1.

The parallax in a given pair of images also increases with the angle between their shooting points. The parallax between two adjacent images in the sequence is called “incremental parallax.” The parallax between the first and the last image is called “total parallax.” These parallaxes are related by

Total parallax =

( N − 1) ⋅ ( Incremental parallax ) ,

where N is the number of the shooting points.

An example of two images from a sequence is shown in Figure 2.

Figure 2: Two images from a sequence

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Figure 3: Measurement of maximal parallaxes for a pair of images The parallax between such images can be measured by superimposing the two images one on top of the other and setting the transparency of the top image to 50%. An example of such measurement is shown in Figure 3. The background in this scene is a two-dimensional picture deposited on a vertical plane so that the background parallax can be measured from the shift of the fence poles. The front object is the hydrangea flowers. In this example the front and back parallaxes are equal, which indicates that the reference plane is centered with respect to the front and the back-scene points.

This measurement is done on digital images, and the result is in pixels. This can be converted to a physical distance by assigning a physical size to the picture width. For example, assume that one wants to print the image in Figure 3 as a picture of width 30cm, and that the parallax and the picture width in pixels are 50 and 1000 respectively. In this case, the measured parallax will be

P ( cm ) =

50 â‹… 30cm =1.5cm 1000

3 THE LENTICULAR SHEET AND THE IMAGED DISTANCE The lenticular sheet is an array of identical cylindrical lenses called lenticules. Optically, it is characterized by three parameters: thickness t, pitch p, and index of refraction n. A cross-section of a lenticular sheet is shown in Figure 4.

p

t

n

Figure 4: Cross-section of a lenticular sheet Lenticular sheets are normally characterized by the lenticules density, rather than by the pitch. These quantities are related by

p=

1 . lenticules density

Equation 1

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The common unit for lenticules density is 1/inch and is called “lpi” (standing for “lenticules per inch”). Thus, a 40lpi sheet has a pitch of 25 mils, or 0.635mm.

The parallax of a given object point determines its imaged distance from the picture plane. It can be shown that

Imaged distance =

Total parallax N − 1 = ⋅ ( Incremental parallax ) , Rp Rp

Equation 2

where Rp is a characteristic of the lenticular lens called “relative aperture,” and is given by

Rp =

np . t

Its value in lenses designed for 3D imaging is typically ~0.5. The parallaxes in Equation 2 correspond to physical distances in the printed picture and must be expressed in units of length like millimeters. It follows from Equation 2 that the imaged distance is proportional to both the incremental parallax and the number of shooting points. The total parallax is proportional to the shooting angle; therefore, the imaged distance is proportional to the shooting angle too.

4 DETERMINATION OF THE SHOOTING ANGLE For best visual quality, the angular increment between adjacent shooting points should be adjusted so that the incremental parallax for all object points is smaller than the lenticular pitch p.

Artists normally wish to display in the picture the largest depth possible. Therefore, the shooting scenario is designed so that the maximal incremental parallax is equal to the lenticular pitch. This requirement determines the angular increment.

One way to determine this angular increment is to perform a calibration, which consists of taking two photographs of the object with a certain angular separation α, as shown in Figure 5, and to measure the maximal parallax P. Denoting the image width by W, the angular increment β is determined by the following equation:

P β 1 ⋅ = , W α Nl

Equation 3

where Nl is the number of lenticules in the picture:

Nl

( picture width ) ⋅ ( lenticules density )

Equation 3 can be rewritten as

= β

Equation 4

W ⋅α .

PNl 5

It is common to choose the value of α between 4º to 10º. Since P is proportional to α, the value of β is independent from the choice of α. Accordingly, the shooting angle is (N – 1)β, where N is the number of shooting points.

Example: Suppose that one wishes to make a 10” wide picture from a 40lpi sheet. The number of lenticules in the picture will be 10x40 = 400. An angle of 5º was chosen for the calibration, and the measured maximal parallax was 50 pixels. The image width was 1000 pixels. Using Equation 4 one gets

= β

1000 = ⋅ 5 0.25 50 ⋅ 400

Figure 5: Calibration measurement

5 CHOICE OF THE SEQUENCE LENGTH According to Equation 2, the imaged distance from the picture plane increases linearly with the sequence length N. Therefore, artists are motivated to use large sequence lengths.

The sequence length is limited by the available resources. In real photography, longer sequences require longer shooting time or more expensive equipment. In rendering, longer sequences require longer computing time.

However, there is a more fundamental limitation on the sequence length: a lenticular picture can resolve only a certain limited number of images Nr. Increasing the sequence length beyond Nr will cause blurring which increases with the distance from the picture plane. This degrades the visual quality of the picture.

It is common to use sequences with lengths that exceed Nr to extend the displayed depth, even at the cost of blurring certain regions in the picture. Pop3DArt accepts sequences with lengths up to 5Nr, thus allowing to extend the displayed depth by a factor of 5.

The value of Nr depends on the lenticular pitch and the printer resolution. It is a primary figure of merit of the lenticular picture. The values of Nr achievable with the Pop3DArt printers are given in Table 1.

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Table 1:

Characteristics of lenticular sheets used in Pop3DArt for 3D pictures (n = 1.56) lpi (1/inch)

thickness pitch (mm) (mm)

Nr

Maximal distance from picture plane for a sequence of length Nr (mm)

70

1

0.363

6

4

60

1.2

0.423

8

6

40

2

0.635

12

15

20

4

1.27

24

62

Artists interested in displaying maximal depth may be tempted to choose the 20lpi sheet for all pictures. However, the effective resolution of a lenticular picture is the same as the lenticules density. Thus, a picture made form a 20lpi sheet will appear as if it is printed at a pixel density of 20dpi. Such pictures appear “dull” if the number of lenticules is too small. As a rule of thumb, it is recommended to have at least 500 lenticules in the picture. This recommendation implies a minimal picture width as a function of the lpi value.

Table 2:

Recommended minimal widths of pictures lpi (1/inch)

minimal width (cm)

70

18

60

21

40

32

20

64

6 CHOICE OF REFERENCE (PICTURE) PLANE The arc center of the camera trajectory lies on the reference plane (Figure 1), which becomes the picture plane in the lenticular picture. The choice of this plane affects the depth which the picture can display.

Figure 6 illustrates how the picture plane position is chosen to optimize the depth display. The closest object point to the viewer is denoted by F, and the furthest (visible) object point by B. The object displayed depth is b + f. Since the imaging limitation is on the maximal distance from the picture plane, it follows that the maximal depth is achieved when both b and f are equal to the maximal imaging distance (Table 1). In this optimal choice, b = f, and, therefore, this choice is called “balanced”.

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Figure 6: Choosing the position of the picture plane. The viewer is on the right.

In Figure 6 we have also shown schematically the two planes which limit the unblurred display volume (Table 1). The choice of the picture plane in Figure 6 (a) is unbalanced (f > b), and therefore non-optimal. The object point F lies outside the allowed imaging volume, and therefore will appear blurred.

In choice (b), the corresponding distances b’ and f’ are equal, so it is balanced and will show maximal depth. In this choice both points F and B lie on the boundary of the slowed imaging volume, so they will be displayed sharp.

Note that the choice of the picture plane does not affect the displayed object depth: b + f = b’ + f’.

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7 THE RENDERED SEQUENCE The sequence should be rendered with an angular increment of β, and the same shooting distance that was used for the calibration.

Verify that the parallax between successive views is as required. To improve the precision of this verification, we suggest that you measure the total relative parallax between the first and the last images of the sequence and divide the result by the number of views minus one.

We prefer jpg format, though any other standard format is acceptable. The images should be named "1.jpg", “2.jpg”, …, with “1.jpg” corresponding to the extreme right point, as it appears for a viewer standing behind the camera and looking at the scene.

8 CONCLUDING REMARKS Readers interested to learn more about the subject should consult my book “Lenticular Imaging,” (www.pop3dart.com/lenticular-imaging) where it is treated rigorously and more generally. Those who do that may encounter slight discrepancies between the book and this article. These discrepancies are caused by some simplifications used here and are negligible for practical purposes.

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