The Quotient Image:

Class Based Recognition and Synthesis Under Varying Illumination Conditions

Our approach is based on a new result showing that a set of all images generated by varying lightening
conditions on a collection of objects all having the same shape but differ in their surface texture (albedo) can be characterized analytically. The characterization involves a single signature image per object and
three model images (taken from distinct lightening conditions) of any one of those objects. The product between the signature image of an object and the linear subspace determined by the three
(fixed) model images generates the image space of that particular object. The second result is on how to obtain the signature image from a database of example images of several objects while proving that for arbitrary texture variation the signature image obtained is invariant to illumination conditions. The work is demonstrated by varied animations .

 Previous Work
Consider a space of all images (image space) of an object under varying illuminations. Each n-pixels image can be represented as an n-dimensional point in the image space. When the surface of the object is matte, one can easily show that the image space lives in a 3 dimensional subspace (Shashua 91,97)

Linear combinations of 3 differently illuminated images


of the same objects yield novel images of the object under new illuminations. 

Synthesis Task

Given a new image, b


and a database of other images of the same class,

generate new images from b, simulating the change of illumination.

 Recognition Tasks

Given a database of images of N objects each under 3 different illuminations and an image of an object
taken under novel lightening condition, Identify the object.

The Quotient Image
 We have developed a method to find a signature image which can be used as an object's
 representative under given view point.


 This signature image is actually the quotient between the novel image b and a linear combination of 3 model images.
We name this image the 'Quotient Image'.
The quotient image should be invariant to varying illumination conditions
of the novel image b.



   Here, each of the images was formed from a different set (in terms of illumination) of 3 model images.


We have conducted a wide range of experimentation on the algorithm presented above. In the first ones to be demonstrate here we used a high quality database prepared by Thomas Vetter and his associates.

Example 1


                                 Compare to ...          Compare to ... 

This animation was generated                                      This animation was generated             This animation was generated from
from linear combination of the above                            from 10 objects database                  one object database(see above)
3 original images                                                          using q_image method                         using q_image method

The same algorithm can be applied with color images converting the original RGB matrix to HSV and and Using the V matrix.
The new V matrices formed in the process are then recombined again to H and V.

                                                              Bill and Monica in a New Light

                                   Original image                   Q image                                                   original image                  Q image


                                               New generated images                                                                     New generated images

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