Kumar Srijan (200602015)Syed Ahsan(200601096)

Problem Statement

To create a Neural Networks based multiclass object classifier which can do rotation, scale and translation invariant object recognition.

 

Translation Invariance

Scale Invariance

Rotation Invariance

Translation, Rotation and Scale Invariance

SolutionNormalize the image so that scaled and

translated images look the same.Extract features from the images which are

invariant to rotation.Create a classifier based on these features.

All these images are same if we consider scale and translation invariance and are equal to:

This is accomplished by normalization with respect to first two orders of geometric moments.

Geometric Moment for an image is :

The zeroth order moment, M00, represents the total mass of the image.

The two first order moments, (M10, M01), provide the position of the center of mass.

Translation InvarianceTranslation invariance is achieved by

transforming the image into a new one whose first order moments, M10 and M01, are both equal to zero.

So, we transform the original image (f(x,y)) into: f(x + x’, y + y’)

where x’= M10/M00

and, y’= M01/M00

Scale InvarianceEnlarge or reduce the object such that its zeroth

order moment, M00 , is set equal to a predetermined value β.

This is done after making the image translation invariant.

Done by changing the image to a new function:f(x/a,y/a)where,a=sqrt(β/M00)

Rotation InvarianceZernike moments for an image is defined as:

Here, n represents the order and m represents the repetition.

For n=5, the valid values of m are : -5,-3,-1,1,3 and 5

Rotation InvarianceNow suppose the image is rotated by an angle

φ so,

Thus, |Znm| can be taken as rotational invariant feature of underlying image function.

Feature ExtractionBinarize the image first according to some

thresholdNormalize it to make translation invariant and

scale invariant Calculate Zernike moments of g(x,y) from 2nd

order to nth order (since, the 0th order moment is = β/π and 1st the first order moments are = 0 for all images after making them scale and translation invariant ).

ClassificationIt is done using a multi layer neural network.

In this case, we used only one hidden layer.

Back Propagation of error is used for learning.

Classifier DetailsThe activation function used at the output layer

and the hidden layer is Symmetric Sigmoid Function.

Sigmoid Function

Classifier DetailsThe symmetric sigmoid is simply the sigmoid that is

stretched so that the y range is 2 and then shifted down by 1 so that it ranges between -1 and 1.

If f(x) is the standard sigmoid then the symmetric sigmoid is g(x) = 2*f(x) – 1. So, this becomes symmetric about the origin.

Classifier Details26 nodes in the output layer.Number of hidden layer nodes can be varied.Number of input layer nodes is equal to the

length of feature vector.

Training of the ClassifierWe have used the Back Propagation Algorithm.Initialize all Wij’s to small random values.Present an input from class m and specify the

desired output. The desired output is -1 for all the output nodes except the mth node which is 1.

Calculate actual outputs of all the nodes using the present value of Wij’s.

This is done by mapping the total input at the node according to the symmetric sigmoid function.

Training of the Classifier

Training of the ClassifierNow, we adjust weights by:

where (n+1), (n), and (n-1) index next, present, and previous respectively. α is a learning rate similar to step size in gradient search algorithms.

ζ is a constant between 0 and 1 which determines the effect of past weight changes on the current direction of movement in weight space.

All the training inputs are presented cyclically until weights stabilize.

ExperimentationWe trained the classifier using 4 images of each of the

letters of the alphabet. A sample of training data:

Zernike moments from 2nd to nth order were calculated for all images and treated as feature vectors.

TestingSimilar kinds of images with translation,

scale and rotation imbalances were used for testing.

Some images used in testing:

A total of 104 images are 4 for each alphabet with various scale, translation and rotation imbalances were used as testing data.

ResultsEXPERIMENT – IKeeping number of features = 47 (2nd – 12th

order). Results for varying the number of hidden layer nodes.

Results

ResultsEXPERIMENT – IIKeeping number of hidden nodes = 50 and

varying the length of feature vector.

Results

InferencesGood classification can be achieved even by the

use of one hidden layer.Use of very few hidden layer nodes can may lead

to a very bad classifier.After a limit, there is a saturation in the amount

of performance one can get by increasing the number of hidden layer nodes.

For very good classification, one must have sufficient number of features.

Use of too many features does not guarantee better classifier.