THE SURFACE CONSTRAINT:
AN APPROACH TO SOLVE THE STEREO DEPTH PROBLEM
((János Geier)) Department of General Psychology, Eötvös Loránd University, Budapest, Hungary.
Purpose A global computational model and computer simulation for the stereopsys is presented. A new constraint has been developed to avoid false targets and to solve the problem related to half-occluded regions (i.e. problem of “no-man’s land” ).
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Supported by MAKA, (US-Hungarian Joint Fund) J.F. No. 360
What is stereopsys?
It is a kind of perception.
What is perception?
It is problem-solving.
(see Gregory’s ‘object-hypothesis’).
The main question:
What is the problem in stereopsys that we need to solve?
1. The classical answer
In accordance with the classical approach (Julesz, Marr), the problem of stereopsys is equivalent to the stereo correspondence problem, i.e.: what is the best matching between the left and right images?
To solve this, some constraints are employed to avoid false targets: ordering, smoothness, uniqueness, primitive similarity, etc.
All of the well-known stereopsys models are based on this classical approach. The computation of the depth of points that are placed in the half occluded areas is very difficult. Among the classical stereopsys models we can’t find a really effective algorithm that can solve the problem of half occluded regions.
2. My answer
The real problem of stereopsys is:
which 3D surface is the one that is in the best ‘harmony’ with the given stereo image pair?
Present computational theory is based on the following definition of surface constraint:
Take a stereo image pair with known camera-parameters, and project the left and right images back onto the original 3D object in its original position, then the projected images will perfectly overlap on the surface of the object, except in the half-occluded regions.
3. The computational theory based on the surface constraint
(i) what is the computational goal?
(ii) what is the algorithm for solving?
(i) The computational goal is to find a representation of the 3D surface that provides the best fulfillment of the surface constraint.
How can we measure the best overlap (i.e. the best fulfilling)?
Suppose the 3D surface is represented in the computer program by a
grid with size nx*ny, where for all (x,y) gridpoints the z value of depth is given:grid={ xij,yij,zij; i=1..nx, j=1..ny}
The measure for the overlap on a given 3D surface is defined by the following value:
where
L(.,.) and R(.,.) is the intensity distribution on the left and right image respectively, and
point to the left retina, and
point to the right retina.
The computational goal in computer terms is to find the
grid that results in the minimal Q(grid) value.Or, in other words: during the computation of
Q measure, the hypothetical 3D surface plays a mediating role between the two images. The 3D surface that gives the minimal Q value gives the solution of the stereo depth problem; this will be the best “object-hypothesis” for the 3D surface.(ii) The algorithm for finding the minimal
Q value.Minimizing the
Q value is a nx*ny dimensional optimum problem. We can find the optimum by using a multidimensional random searching algorithm.Results
The computer algorithm has been tested on real life images and computer generated random dot stereograms. The matching is correct at nearly all 3D points.
Conclusions
