Keystone Distortion

This paper concentrates on inherent Distortion Values that typically effect the overall shape of the picture area and the relative shapes of all subjects being viewed as well as the audience's perception of imaged picture quality. 

The term Distortion can have several different meanings, depending on the context of its use.  For this paper, Distortion is limited to the following five forms:

• Keystone Distortion: Horizontal and/or Vertical Trapezoidal picture shape.
• Anamorphic Stretch: Horizontal and/or Vertical Image Stretch dependent on the keystone geometry.
• Geometric Distortion: Horizontal distortion dependent on screen curvature.

In all cases, distortion is defined as “the change in shape of the projected image relative to the shape of the projected image on a flat screen with the projector being perpendicular to the screen and centered.”  Any distortions based on viewing angle or viewer position in the theater are not addressed in this paper.

Since distortion changes only the shape of objects in the picture and not how well the viewer can see them, there is little basis for objectively derived design limits. In a theater setting, the eye will normally accommodate small amounts of distortion, which is therefore not perceived. However, when moderate to severe amounts of distortion are present, particularly when images of straight line objects such as flagpoles and the horizon are visible, the effect becomes apparent to the viewer and distracts from the overall presentation of the viewed image. Based on our collective experience, research and testing, we have found that when each of the above listed distortions have a value less then 5% (absolute value) the audience does not perceive any picture degradation due to these effects and will find the picture to be pleasing.

Keystone distortion & anamorphic stretch

Keystone Distortion and Anamorphic Stretch are forms of geometric distortion that bring about a trapezoidal image of a nominally rectangular picture.  They are usually produced when a picture is projected from a position such that the line of sight or optical axis of the projector is not precisely orthogonal to the screen. In this situation, Keystone Distortion and Anamorphic Stretch occur because the film plane and image plane are no longer parallel to each other. Figure 1, below, shows an illustration of an extreme example of vertical keystone distortion.
 

^{Figure 1: Vertical Keystone Distortion}

^{Figure 2: Shows an illustration of an extreme example of simultaneous horizontal and vertical keystone distortion.}

The most typical example of these distortions is found when a projector is placed high up in a theater and tilted downward toward the screen (Refer to Figure 3 below).  The vanishing point, V, is defined such that a line through the projection lens parallel to the film would intersect the plane of the screen at a point high up above the roof of the theater.  The undistorted image is represented by AA and the screen image by BB. The screen image will be keystoned, and anamorphically stretched because BB is longer then AA in the vertical direction. It should be noted that this also applies in the horizontal direction if the projector is placed left or right of the screen centerline.  

In short, keystone distortion and anamorphic stretch each have both a vertical and horizontal component. As an example, the following is a calculation of Anamorphic Stretch in the vertical direction. All values in the following refer to defined reference lengths and angles illustrated in Figure 3. For example KB  refers to the distance defined from the point marked K to the point marked B.

^{Figure 3: Projector / Screen Geometry}

^{Figure 4: Projector / Screen Geometry for Keystone Calculation}

Throw  = PK = 100 feet  = 1200"

Downward tilt of the projector = q = 15E             

Lens focal length = fN= 30mm = 1.181"

Film Format = 35mm 1.85 (0.825" x 0.446")

Note: In the following calculations all distance results are rounded to the nearest inch.

A) The 1/2 projected angle, a, in the vertical direction is calculated by the following formula

a = Tan^-1 [ 2 YN ) fN]  

            where YN = film format size (please note the film orientation)

            thus

a = Tan^-1 [ 2( 0.446 ) 1.181]  = 10.69E                    

B) Then we see that

            KB = PK ( Tan (q - a)    

            KB = 100' ( Tan (15E - 10.69E) = 7' 6"                  

            and

            KBN = PK ( Tan (q + a)

            KBN = 100'  ( Tan (15E + 10.69E) = 48' 1"

     KBN is the projector height above the bottom of the image on the screen.

C) The vertical height of the screen image is

            BBN= (KBN - KB)

            BBN= (48' 1"- 7' 6") = 40' 7"

D) AAN, the height of the undistorted image is given by

            AAN = 2 ( PC ( Tan (a)

            where

            PC = PK ( Sec (q)

            so

            AAN = 2 ( PK ( Sec (q) ( Tan (a)

            AAN = 2 ( 100'  ( Sec (15E) ( Tan (10.69E) = 39' 1"

E) The Anamorphic Distortion (Stretch) in the vertical direction is defined by

            % Anamorphic Distortion = [(BBN ) AAN) - 1] ( 100

            % Anamorphic Distortion = [(40' 7" ) 39' 1") - 1] ( 100 = 3.8%

Note:  The above calculations can also be applied to the horizontal direction, if your projector is displaced horizontally at an acute angle relative to the screen.

Curved screen distortion

Today many theater screens are curved in an attempt to produce more even illumination and to add apparent depth to the picture. The overall goal is to make the picture more pleasant to the viewer. However, when a screen is curved, the actual viewed picture contains a form of geometric distortion. In all cases, the screen is curved in one orientation (horizontal) and concave to the viewers (with screen edges closest to viewers).  Since the shape of the curve is a simple segment of a cylinder, the distortion that it yields is only in one orientation (horizontal).

As shown in Figure 5:

CA    = 1/2 picture width (Flat Screen) No distortion
CBN    = FB ) 2 = 1/2 picture width (Curved Screen)  
CD     = Projection Distance from projector to screen center
 

^{Figure 5: Projector / Curved Screen Geometry. In the above diagram, G = location of the center of curvature of the screen where} 

Distortion tolerance

Based on our collective experience, research and testing, we have found that when each of the above listed distortions have a value less then 5% (absolute value), the audience does not perceive any picture degradation due to these effects and will find the picture to be pleasing. This applies to real life imagery and not necessarily to a rectilinear test grid image. When viewing a motion picture in a theater, the viewed image is not simultaneously referenced with a “non-distorted” grid, so the observer cannot perceive the suggested tolerance values.

Conclusion

In this paper we have shown how to calculate five forms of distortion: vertical and horizontal Keystone Distortion, vertical and horizontal Anamorphic Stretch and Geometric Distortion due to screen curvature. Several examples were given to illustrate the geometry involved and the order of magnitude of the results that will be seen based on typical geometries. Design limits of 5% maximum for each of the types of distortions defined are proposed based on experience, research and previous testing. No distortion effects based on viewing angle were considered since these are dependent on the location of the observer in the theater. The distortions defined herein are only dependent on the physical geometry of the screen in reference to the projector position and orientation as well as the lens focal length.