Microscope Image Processing

subspace for an image at resolution j is denoted V

2

j

, which is in L

2

(R

2

),

and its orthogonal complement subspace is denoted W

2

j

. As with one-dimensional

multiresolution spaces, the sum of the subspaces V

2

j

and W

2

j

forms the

finer approximation space V

2

jþ1

;thatis,

V

2

jþ1

¼ V

2

j

 W

2

j

(7:69)

To construct two-dimensional wavelet bases, we first consider the separable

basis, which is obtained by applying the same filters developed for one-

dimensional signals to each dimension separately. In vector spaces, the separable

two-dimensional multiresolution space is defined by the tensor product of

one-dimensional multiresolution spaces,

V

2

j

¼ V

j

 V

j

(7:70)

Thus the separable two-dimensional scaling function can be written as

f(x, y) ¼ f(x)f(y)(7:71)

which means that one can ...