The following blog is a paper I have written 2010 for the 2010 SMPTE International Conference on Stereoscopic 3D for Media and Entertainment

**Abstract.** For stereoscopic pictures the geometric dimension on set and the geometric situation at the screening interact with each other. The combination of both produces the final depth perception at the observer. The screen size is believed to be a central variable in the process of creating a stereoscopic picture. The information of the screen size must be known during shooting. But different screen sizes at different theaters raise the question: What is the target screen size for stereoscopic feature film? This paper tries to examine this question.

**Keywords.** Screen size, target screen size, disparity, relative parallax, absolute parallax, divergence, binocular rivalry, panum’s fusion area,

Introduction

Stereoscopic feature film entails new challenges to every part of filmmaking. Preproduction, production and postproduction are confronted with new questions. Postproduction facilities, for example, have to implement new processes to modify stereoscopic characteristics, like *»stereo refinement«* and *»depth grading«*.

But the big difference between stereoscopic and traditional »planar« feature film is that the stereoscopic process chain must be regarded as a whole, from the trigonometric dimensions on set all the way to the geometrical situation when screening the final product.

Figure 1. Stereoscopic production chain must be seen as one process.

Most concerns in stereo-photography are made about how to choose the target screen size. Stereographers need this value (target screen size or projection magnification) to feed in their calculators – but why?

Equation 1 shows one fundamental equation for stereoscopic picture (approximation). [KUHN1999]

- s = screen plain
- IA = inter-axial distance
- f = focal length
- p = max. absolute parallax (on the image sensor)

Lets examine p:

The max. absolute parallax **p** is limited by the “prohibition of divergence”. [KUHN1999] The absolute parallax **p** on the screen must not be bigger than inter inter pupillary distance **PD**. Therefore a magnification **m** must be known (See equation 2).

The magnification **m** is the ratio of image sensor size **c** to screen size **C** (see equation 3):

The stereographer has control over all variables (distance to the screen plane, focal length, interaxial distance, image sensor size) except from the screen size.

What happens if the stereoscopic content is shown on different screen sizes?

When a stereoscopic movie is made for a five meter screen for example, the maximum absolute parallax may be equal to the distance of the eyes (≈ 6.3 cm). If the same feature film is projected on a 10 meter screen everything is doubled. So the maximum parallax is 12.6 cm. This cause the eyes to »toe out« or diverge when they try to fixate this point. Too much divergence is bad because it cause eyestrain and headache. If on the other hand the screen size is diminished – the parallax is diminished as well and will maybe cause reduction of the depth effect.

But it is not practical and not possible to shoot a stereoscopic movie to fit for all screen sizes. So what should be the best trade off for a target screen size and are all these concerns necessary?

There are no real statements about the optimal screen size in literature. A well-known consideration is to go for 12 meter screen width. [Lipt2010] But is this practical?

In this chapter the reader will find an analytical approach to find the optimal target screen size for cinema projection. On the way to the answer the reader will find an alternative approach to this subject.

## 1 Preconditions

Some preconditions must be defined to fixate some of the variables.

## 1.1 Viewing distance

Regulation communities like *THX* and *SMPTE* have the same motivation for defining the optimal distance for a theatrical planar screening. The spatial resolution of one eye is crucial. At the optimal distance the eye can just not resolve the resolution or the raster of an image. For a normal person with 100% (20/20 acuity) in a typical screening environment (photopic vision) the resolution of one eye is about one arc minute per cycle. [POYN2007]In context to spatial resolution the viewing distance or angle depends on the screen width and the spatial resolution of the image. The viewing angle in degree is described by equation 4. The resulting distance is described in equation 5. The optimal distance will be called »sweet-spot«.

The nearest distance for this assumption is the first row of a typical cinema. The typical first row is at 0.8 of the screen height for a cinema-scope presentation. [SWAR2005] The distance is computed by the equation 6. Another »bad« seat is the distance where one pixel covers two arc minutes of the observer. This »bad-spot« is closer than the screen width and describes a non-optimal seat in the first half of a theatre. The distance is computed by the equation 7.

Every distance greater than the sweet-spot is not important for this scope, because increasing the distance will diminish errors. The »vivid plastic impression« or »roundness« is enhanced or exaggerated if the audience have greater distance to the screen than the theoretical sweet-spot.

In figure 2 all defined spots are sketched in a typical theater room for 2048 pixel spatial horizontal image resolution. Notice, that the theoretical sweet-spot is almost at the last row. This fact pose the question if 2048 pixel image resolution if enough for digital cinema (planar and stereo). At CinePostproduction a more practical sweet-spot was defined which elicit a theoretical resolution of 2600 pixel.

Figure 2. The predefined spots sketched in a typical cinema. [SWAR2005 p.230]

## 1.2 Screen range

A limiting factor for stereoscopic projection is luminance. Half of the light output goes to the left eye and half goes to the right eye. After that 70% of the remaining light is consumed by the optical elements like modulator (3d system) and analyzer (glasses). Another portion is lost due to time multiplex, which is used for a one-projector setup.

The result is, that 15% to 30% of the light output arrives at the observer. This fact limits the screen size. The current practice at cinemas is a peak white luminance is in the range from 3.5 to 5.5 ft-L at 25 meter maximum screen width.

The lower white point luminance around 5 ft-L compare to 14,8 ft-L for planar cinema, influence the human color perception. It is important to color grade the feature under these circumstances and to perform a match grade for planar, and therefore brighter releases.

For this scope the range of screen widths will be 4 meters for very small theaters to 25 meters for big screens.

## 1.3 Divergence

»The amount of divergence that a person can tolerate will depend on each individual, but a normal value for maximum divergence, when viewing a distant object like a movie screen, is about 4 degrees. Since this is a maximum value, it might fatigue the eyes, so a more comfortable value would be about half of this amount, or about 2 degrees.« [SALM2010]

To provide additional headroom for smaller inter pupillary distances the value is reduced to 1° in some calculation.

## 1.4 Parallax and disparity

To take the result from visual science into account it is necessary to define the relation between parallax on the screen and the disparity on the retina. Disparity in degree is defined as the angle difference between two points in space (∡L and ∡R). The same disparity can be measured at the difference between angle α and β – see figure 3. [SALM2010]

Figure 3. The relation disparity in the eye and relative parallax on screen is a triangular relationship

To compute alpha or any other angle of a given point via its projections **Al** and **Ar**, the triangle ∆ **El-Al-Al’** can be used. The distance **El-Al’** is the sum of **PD/2 **and the screen-parallax **(Al-Ar)/2**. (PD is the inter pupillary distance) The distance **Al-Al’** is **d** = distance to the screen. Equation 8 and 9 shows the final equation to calculate one angle.

- apix = screen parallax in pixel
- a = screen parallax
- r = spatial horizontal image resolution
- w = screen width
- d = distance to the screen
- pd = interpupillary distance
- alpha = angle of one point

The minus operant in (**pd/2-a/2**) is chosen instead of the plus operant, because a crossed disparity (negative parallax) will generate negative parallax values by definition.

The disparity **η**** **is the difference between two angles. Equation 10 and 11 show the final Equations to calculate **η**.

- bpix = screen parallax of the second point in pixel
- b = screen parallax of the second point in meter or centimeter
- η = the disparity between point A and B in degree

So disparity is the difference between two point-pairs on the screen. It is the difference between two parallaxes or the relative parallax.

Disparity refers to the eye (intrinsic)

Relative parallax refers to the screen (extrinsic)

An example:

A foreground object appears in front of the screen and has the parallax -10 px (scrub in figure 4). A background object far behind in space has the parallax +24 px (tree in figure 4). So the resulting relative parallax for the image is 34 px or ≈ 34 arc minute disparity for the sweet-spot position – see figure 4.

Figure 4. A captured scene with perceived objects and their projections on the screen.

Note, that the ratio relative parallax to disparity is screen size independent, because the distance to the screen is constrained by the screen width.

The ratio relative parallax (in pixel) to disparity (in minute of arc) is:

- 1:1 for sweet-spot
- 1:2 for bad-spot
- 1:3,3 first row
- Viewing distance is a function of screen width
- Screen width range is 4-25 meters
- The allowed divergence amount is 1° to 2°
- The relation between relative parallax and disparity is angular and stays constant for every screen width

## Summary of pre-conditions

- Viewing distance is a function of screen width
- Screen width range is 4-25 meters
- The allowed divergence amount is 1° to 2°
- The relation between relative parallax and disparity is angular and stays constant for every screen widt

## 2 Calculations

Some single aspects about stereoscopic projection will be contemplated. At the end al respective conclusions will be summarized and a final statement may be formulate. All calculations refer to 2048 pixels spatial horizontal image resolution.

## 2.1 Maximum parallax without divergence

Plot 1 shows the screen width (in centimeter) vs. the interocular parallax (in pixel). For this parallax the eyes look parallel, greater parallax cause divergence.

Plot 1. Shows the relation between screen size and interocular parallax.

The bigger the screen the less pixel are available until the interocular-parallax is reached, because the pixel dimension on the screen gets bigger.

Note, that the ratio »distance – screen width« cause the ratio »relative parallax – disparity« to stay constant for every screen width. In other words: one pixel disparity is the same angular disparity for every screen width, for a constant viewing distance relations.

Some statements can be justified:

- The bigger the screen, the smaller the maximal parallax and therefore the smaller the relative parallax.
- Smaller screen sizes cannot be easily adopted to bigger screens (compare 40 px at 3 meters and 6 px at 20meters)
- Reviewing screens smaller than 5 meters are inappropriate. After 7 meters the function might be smooth enough for practical purpose.

On the one side a bigger maximal parallax may give more flexibility in shooting a stereoscopic feature, but on the other side it is less compatible for changes in screen size. To determine a good trade off for parallax budget and compatibility more aspects have to be taken into account.

## 2.2 Parallax and divergence

The x-axis of plot 2 is absolute parallax in pixel. The y-axis is distance to the screen. The straight colored lines describe the relation between parallax and the distance for a given screen width, where divergence is exactly 1 degree. The three curves shows the three predefined spots in the cinema (first row, bad-spot, sweet-spot).

Plot 2. Shows the relation between distance to the screen and absolute parallax, where divergence is 1°.

The three curves aspire a fixed value for infinite screen width:

- first row: 11.97 px
- bad-spot: 26.36 px
- sweet-spot: 58.22 px

Some statements can be justified:

- There is a broad tolerance between interocular-parallax and comfortable divergence.
- A parallax of 12 pixel will never cause any uncomfortable divergence for any screen size
- A parallax of 17 pixel will not cause uncomfortable divergence for todays screening situation.
- A parallax of 31 pixel will not cause uncomfortable divergence for the most seats in a cinema.

Plot 3 is analog to plot 2. The plot shows the distance to the screen for 2° divergence.

Plot 3. Shows the relation between distance to the screen and absolute parallax, where divergence is 2°.

The three distance curves again aspire a fixed value for infinite screen width:

- first row: 23.93 px
- bad-spot: 52.75 px
- sweet-spot: 116.47 px

Some statements can be justified:

- A parallax of 24 pixel will never cause a divergence greater than 2° for any screen size.
- A parallax of 29 pixel will not cause a divergence greater than 2° for today’s screening situation.
- A parallax of 52 pixel will not cause a divergence greater than 2° for the most seats in a cinema.

Divergence only occurs when a point with a parallax greater than the inter pupillary distance is actually fixated. So a stereoscopic sequence may have greater parallax values in a frame, if the attention (and therefore fixation) of the observer is on a closer object, like a foreground object. So divergence might not be a limiting factor for stereoscopic cinema at all?

A limiting factor that is more severe is binocular rivalry or the boundaries of patent stereopsis (comfortable and quantitative stereopsis). Stereopsis is the sense of depth perception. Is there a need for a depth budget of 50 px ≈ 50 min of arc disparity and more?

## 2.3 What disparity is appropriate

There are some values given by physiologic science that can be adopted for stereoscopic cinema.

Panum`s fusion area is the disparity range a human perceives an object as a single sharp object. Its range is about 6 arc minute around the fixation point. [JULE2006] For stereoscopic screening under the pre-conditions mentioned above this is a relative parallax of 6 px at sweet-spot and 3px at bad-spot and 1.8px and first row.

A form of binocular rivalry (eye suppression) is the phenomenon between foreground and background where eyestrain occurs. Binocular rivalry is the antithesis to fusion. [HERS2000] It happens, when the difference (disparity) between the two images is too big. It starts more or less at 25 arc minute disparity. That is 25 Px relative parallax for sweet-spot, 12,5Px for bad-spot and 7,6 Px for first row. After this value the perceived depth perception decrees and start to suppress.

Plot 4 shows the relation binocular disparity in minutes of arc to the perceived depth. The blue and green areas are the comfortable ones (patent stereopsis). But only the green area can be used in actual cinema screening at a resolution of 2048 pixel at sweet-spot or greater distance.

Plot 4. Shows the relation between binocular disparity and perceived depth. [HERS2000 p.56]

An image can have greater values of disparity and still looking ok, but as a base these clues can be consulted. Characteristics that alter the start of binocular rivalry are for example image size, temporal and spatial frequencies, eccentricity, illumination, vergence mechanism and practice. Remember, the mentioned disparity is the difference between two parallax values.

- Relative parallax is more crucial than absolute parallax.
- Is the classical cinema architectural situation suitable for stereoscopic movies? (Compare first row with sweet-spot in terms of disparity)
- Stereoscopic cinema needs more horizontal spatial resolution (blue part of plot 4)

## 3 Conclusion for target screen size

First all the statements are summarized

- For every screen the relation relative parallax (in pixel) to disparity (in min of arc) is 1:1 for »sweet-spot«, 1:2 for »bad-spot« and 1:3.3 for »first row« due to the pre-condition of distance.
- The bigger the screen, the smaller the maximal parallax and therefore the smaller the relative parallax when composing for no divergence.
- Smaller screen sizes can not be easily adopted to bigger screens (compare 40 px at 3 meters and 6 px at 20meters)
- Reviewing screens smaller than 5 meters are inappropriate
- There is a broad tolerance between interocular-parallax and comfortable divergence.
- A parallax of 12 pixel will never cause any uncomfortable divergence for any screen size
- A parallax of 17 pixel will not cause uncomfortable divergence for today’s screening situation.
- A parallax of 31 pixel will not cause uncomfortable divergence for the most seats in a cinema.
- A parallax of 24 pixel will never cause a divergence greater than 2° for any screen size.
- A parallax of 29 pixel will not cause a divergence greater than 2° for today’s screening situation.
- A parallax of 52 pixel will not cause a divergence greater than 2° for the most seats in a cinema.
- Panum`s fusion area is 6 min of arc around the fixation point. For stereoscopic screening under the pre-conditions mentioned above this is a disparity of 6px at sweet-spot.
- Binocular rivalry more or less starts at 25 arc minute.
- Relative parallax is more crucial than absolute parallax.
- Is the classical cinema architectural situation suitable for stereoscopic movies? (Compare first row with sweet-spot in terms of disparity)
- Stereoscopic cinema needs more spatial horizontal resolution (blue are of plot 4)

## 3.1 Consideration for absolute parallax

The maximal absolute parallax can be enlarged by divergence considerations. For more flexibility the range of interocular-parallax can be expand to some extend (see plot 2 and 3). Note, that the most stereoscopic calculators calculate parallax for no divergence. So a stereographer should discern between mathematical target screen size (producing no divergence), desired absolute parallax and review screen size. For judging depth perception a screen equal or larger then 7 meters is practical (smooth region of plot 1).

## 3.2 Consideration for relative parallax

For depth perception only relative parallax is relevant. With the 1:1 relation for relative parallax and disparity it is feasible to stay within the ranges known from physiological science. The 1:1 relation is only true for the sweet-spot. For seats with greater distance the angles become smaller and won’t cause any error. For seats closer to the sweet-spot the angles and so the disparity gets greater. It is a trade off between compatibility and depth budget. The relative parallax is the limiting factor in practical approach. The first third of a theater is a problem zone because the disparity difference between first row and sweet-spot is hugh (compare 7.6 Px to 25 Px). Unfortunately a content creator (stereographer, Dop, Director) must pick a position in the theater to master for. Seats closer to the screen will have less depth and more disparity. To master for the first row is unpractical, because a lot of depth budget is lost. To master for the sweat-spot is unpractical as well, because most of the seats in a theater will have a smaller distance to the screen.

The last and most important guideline is human cognition. It is the job of the stereographer and the stereo-grader to judge every shot in a real time big screen environment and correct both the absolute and relative parallax if it is needed. This is not done, by simply shifting the images horizontal. A horizontal Image translation will not change the relative parallax or disparity!

## Final Conclusion

The master-screen size question is not answered with a single value, but a range is provided, giving freedom for creative approach. There are two target screen sizes:

- A theoretical screen size that is used to calculate the desired disparity (shot dependent, can vary from shot to shot)
- A practical screen size to judge depth perception of stereoscopic content ( >7 meter).

What happen when a stereoscopic content for cinema is watched on small screen sizes like Consumer Displays or Laptops? If the choice of distance is calculate the same, the angular proportion stay constant (1 px relative parallax ≈ 1 arc minute disparity). Normally the ratio distance to the display is greater compare to cinema. Additionally in such small distances (within grasping reach) other physiological effects come into account, like change in »size constancy«. Stereoscopic feature film may be reviewed and »HIT- modified« for small screen deliveries in future, as it is modified nowadays in terms of color for HDTV Releases.

Stereoscopic film has no problem with different screen sizes but with different viewing angles.

## Books

[HERS2000] Hershenson, Maurice: Visual Space perception, A Primer. MIT Press. Cambridge 2000.

[JULE2006] Jules, Bela: Foundation of Cyclopean Perception. MIT Press. London 2006.

[KUHN1999] Kuhn, Gerhard: Stereofotografie und Raumbildprojektion. Vfv Verlag. Gilching 1999.

[POYN2007] Poynton, Charles: Digital Video and HDTV, Algorithms and Interfaces. Morgen Kaufmann Publishers, San Francisco 2007.

[SWAR2005] Swartz, S., Charles: Understanding Digital Cinema. Focal Press. Oxford 2005.

## Internet

[Lipt2010] Lipton, Lenny: What to do about the big screen <http://lennylipton.wordpress.com /2008/04/10/what-to-do-about-the-big-screen/> (16.02.2010)

## Lecture Notes

[SALM2010] Salmon, Thomas O. OD, PhD, FAAO: Vision Science 3 -Binocular Vision Module, Lecture – Stereopsis. Lecture Notes.

Hy Danilele

this is very helpful for my studies, thanks!

When you say that the ratio between pixel disparity and arc minute is 1:1 for the sweet pot, can you explain me better that point because if i make the math this doesn’t match, or maybe you can link the reference where you found that!

ps: are you from Siracusa?? 🙂

Hi Earl,

sorry for the delayed response, wasn`t on wordpress for a long time.

I believe the answer to your question can be found in the definition of the sweetspot:

For a normal person with 100% (20/20 acuity) in a typical screening environment (photopic vision) the resolution of one eye is about one arc minute per cycle. [POYN2007]In context to spatial resolution the viewing distance or angle depends on the screen width and the spatial resolution of the image. The viewing angle in degree is described by equation 4. The resulting distance is described in equation 5. The optimal distance will be called »sweet-spot«.

Where do you have problems with the math?

all the best

Daniele

Thank you for answering!

I explain better my problem.

I have made some stereo images (just 3 squares in a uniforme backgroung) and i have put 1 pixel of disparity on the screen(negative parallax), and now i would like to know this disparity in arc minute.

Let’s say distance: 2m, screen with: 43 cm and a horizontal resolution of 1600. We will find a disparity of 0,46 arc min.

Is this right?

Earl