Recognition of object by finding correspondences between features of a model and an image. Alignment repeatedly hypothesize correspondences between minimal set of features of a model and an image and then tries to find model poses. For computing poses a model of projection must be selected. A minimal number of points needed to compute a model pose is three. Alignment

General idea: 1.Given an input image and a candidate model, establish correspondence between them. 2.Determine transformation from the model to the image 3.Apply the recovered transformation to the model 4.Compare the transformed model with the viewed object 5.Based on this comparison choose the best model Alignment cont.

General steps before alignment : 1.Selection of object of interest in the picture. 2.Segmentation – delineation of a sub-part of the image to which subsequent recognition process will be applied. 3.Image description – extraction of information which will be used for matching the viewed object with stored object models 4.Extracting an alignment key. Alignment key is an information used to bring the viewed object and models into alignment. Before, During and After Alignment

Alignment 1.Viewed object is brought into correspondence with a large number of models stored in the memory. 2.Individual alignments General steps after alignment: 1.Indexing (classification)– use some criteria to “filter out” unlikely models. Matching Before, During and After Alignment

We consider a work “3D pose from 3 corresponding points under weak perspective projection” of T.D.Alter The problem is to determine the pose of 3 points in space given 3 corresponding points in image. It gives direct expressions for 3 matched model points in image coordinates and an expression of a position in the image of any additional, unmatched model point. 3D pose from 3 corresponding points

Hypothesize a correspondence between three model points and three image points. Compute the 3D pose of the model from three-point correspondence. Predict the image positions of the remaining model points and extended features using the 3D pose. Verify whether the hypothesis is correct by looking in the image near the predicted positions of the model features for corresponding image features. Alignment Algorithm

Fig.1 Model points undergoing perspective projection to produce image points The perspective solution

Let image points be extended as follows: Then The problem is: given find a,b, and c. From the law of cosines: Given a, b, and c, we can compute the 3D locations of the model points: The perspective solution cont.

Approximate perspective projection closely in many cases. Less complicated. Conceptually simpler. We do not need to know the camera focal length and the central point. Fewer solutions (four for perspective an two for weak perspective). Justification of the weak perspective approximation

Weak-Perspective Solution Fig.2 Model points undergoing orthographic projection plus Scale to produce image points

To recover the 3D pose of the model we should know the distances between the model points and distances between the image points The parameters of the geometry in Fig. 2 are (will be proved later ): See eq. (7)-(13). Weak-Perspective Solution cont.

Fig. 3 Small solid representing orthographic projection plus scale of three model points into an image. Computing the Weak-Perspective Solution From Fig. 3 we have three constraints:

Computing the Weak-Perspective Solution cont. Multiplying (3) by –1 and adding all three gives Squaring (4) and using (1) and (2) to eliminate and which leads to biquadratic in s :

where The positive solutions of biquadratic are Computing the Weak-Perspective Solution cont.

From (1),(2) and (4) Computing the Weak-Perspective Solution cont.

The solution fails when the model triangle degenerates to a line, at which case a=0. Computing the Weak-Perspective Solution cont.

Let the image points be Given we can invert the projection to get the tree model points: where unknown w can not be recovered. Image location of a fourth model point.

Denote the model points in arbitrary model coordinate frame. Using solve the following vector equation for the “extended affine coordinates”, of Let Using the three model points with Image location of a fourth model point.

Substituting (17)-(19) into (15) we’ll get Image location of a fourth model point.

To project, first apply the scale factor s : Then project orthographically to get the image location of the fourth point: Image location of a fourth model point.

3D pose from 3 corresponding points under weak perspective projection” T.D.Alter., MIT A.I.Memo No. 1378, D Pose from 3 Points Using Weak Perspective T.D.Alter, IEEE Transactions on Pattern Analysis and Machine Intelligence,v.16 No.8,1994 References