Depending on the image shooting and display conditions, the relative size of the reproduced image in the depth direction may become smaller. In such cases, the apparent thickness of the reproduced object is distorted in the depth direction, causing a poor sense of depth. This phenomenon is sometimes called the cardboard effect  because it makes three-dimensional objects look like cardboard cut-outs.
To quantitatively ascertain the thickness of a stereoscopic image based on the camera-to-object distance, we introduced the thickness term Ec expressed by the following formula:
Here, Ms is the magnification of the reproduced image, and Lb and Ld are the depth distances of the object in real space (where the object image is actually shot) and stereoscopic image display space, respectively. Ec expresses the degree of local changes in the depth direction.
Table 5 shows the results calculated by applying this value to the parameters of parallel and toed-in configurations shown in Table 2.
Analysis of depth distortion
In the parallel configuration, Ec depends on a2, which is simply the ratio of the angles of view when the image is shot and when the image is displayed. The situation is more complex in the toed-in configuration, where several factors are involved. However, when considering an object close to the depth position of the display screen, it can be seen that it is related to the camera separation (a1), the distance to the intersection of the optical axes when capturing the image (Lc), and the viewing distance (Ls). It is suggested that these parameters should be considered in order to avoid the cardboard effect.
The cardboard effect is a phenomenon whereby individual objects in a scene are perceived as having no depth, although it is generally still possible to ascertain the positional relationships among groups of objects in the depth direction. Although an analysis of the occurrence of this phenomenon is inevitably complex, here, we attempt to predict its occurrence in cases where the analysis is restricted to binocular parallax.
In the discussion of the previous section, we showed that the thickness Ec of an object’s image is affected by various parameters. When the thickness Ec is small, it is predicted that unnatural phenomena such as the cardboard effect can occur.
We tried to apply the geometrical discussion of the previous section to the results of subjective evaluation tests. In the subjective evaluation tests, stereoscopic images were reproduced under a number of different imaging and display conditions, and the test subjects evaluated the subjective thickness of the object .
For the evaluated images, we produced images of the object shown in Fig. 9 under the nine sets of conditions shown in Table 6.
The test subjects were asked to evaluate the subjective thickness of the reproduced stereoscopic image on the following five-grade scale:
5: Very thick
3: Somewhat thick
2: Not very thick
1: Not at all thick
Object used in the subjective evaluation tests
of the cardboard effect
Shooting and display conditions used in subjective
evaluation tests of the cardboard effect
Figure 10 shows the correspondence between the thickness Ec and the subjective evaluation scores. The vertical axis shows the results of normalizing the 5 evaluation categories in psychological space by using the method of successive categories. The calculation and subjective evaluations are clearly correlated.
The cardboard effect is a subjective effect that might be affected by other cues such as motion parallax and shading information. In this Report, we have at least shown the geometrical criteria that cause this effect.
Correspondence with subjective evaluation scores