Question

In object recognition, translating an image by a few pixels in some direction should not affect the category recognized. Suppose that we consider images with an object in the foreground on top of a uniform background. Suppose also that the objects of interest are always at least 10 pixels away from the borders of the image. Are the following neural networks invariant to translations of at most 10 pixels in some direction?

Here the translation is applied only to the foreground object while keeping the background fixed. If your answer is yes, show that the neural network will necessarily produce the same output for two images where the foreground object is translated by at most 10 pixels. If your answer is no, provide a counter example by describing a situation where the output of the neural network is different for two images where the foreground object is translated by at most 10 pixels.

(a) Neural network with one hidden layer consisting of convolutions (5x5 patches with a stride of 1 in each direction) and a softmax output layer.

(b) Neural network with two hidden layers consisting of convolutions (5x5 patches with a stride of 1 in each direction) followed by max pooling (4x4 patches with a stride of 4 in each direction) and a softmax output layer.

Answer 1

Welcome to Chegg!

The neural network will be able to detect the object as it is. If we normalize the image then it makes no difference to the overall image, since the overall image (area of interest) remains the same.

Extract features of the image (object or object characteristics) which remains same with respect to translation of the image and create a classifier based on these features.

The normalization can be done if we consider geometric moments as follows:

Total mass and position of the center of mass can be found by respectively.

So, to make translation invariant image, change the image to

These equations are true for a single hidden layer as well as a multi hidden layer. So, apply these equations two times from in part 'b' above.

Softmax output can be found by applying the following equation (once in both the parts):

Images for part 'a' and 'b' are given below for representation purpose (Downloaded from the internet).

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