hadamard matrix is matrix of order n i.e nxn matrix with elements -1 or +1.

orthogonal 4x4 matrix

1 1 1 1
1 -1 1 -1
1 1 -1 -1
1 -1 -1 1

normalised 4x4 matrix

       | 1 1 1 1   |
 1/2   | 1 -1 1 -1 | 
       | 1 1 -1 -1 |
       | 1 -1 -1 1 |

a) Hit-and miss helps to derive information on how objects in binary images are related to their background.Uses two structuring elements, one for the object of a given image and other for background. Hit-and-miss for image A with element E(E1,E2)

AxE=(A.E1) subtract (A+E2)

b)Opening and closing are based on dilation and erosion. Opening of image A by structuring element E is

A.E=(A - E) + E

Closing of image A by structuring element E is

A.(blod)E=(A + E) - E

where + means Dilation operation and - means Erosion operation

Both this operations are used to smooth contours of objects.