Fourier descriptors & moments

1. FOURIER DESCRIPTORS &
MOMENTS
Submitted by,
D.Shanmugapriya
II-M.sc (IT)
Department of CS & IT
Nadar saraswathi college of arts &
science
Theni.

2. INTRODUCTION
 Fourier descriptors a method used in object
recognition and image processing to represent the
boundary shape of a segment in an image.
 The first few terms in a fourier series provide the
basis of a descriptor.
 This type of object descriptor is useful for recognition
tasks because it can be designed to be independent of
scaling , translation , or rotation.

3. PROPERTIES OF FOURIER
DESCRIPTORS
Scaling :
 It means multiplying x(k) and y(k) by some constant.
 Hence , fourier descriptors are scaled by the same
constant.
Starting point:
 Changing starting point is equivalent to translation of
the one-dimensional signal s(k) along the k
dimension.
 Hence , translation in the spatial domain is a phase –
shift in the transform.

4. COMPLEX FOURIER
DESCRIPTORS
 The shape is now described by a set of N vertices
{z(i):i=1,…..N}
 Corresponding to N points of the outline.
 The fourier descriptors {c(k):-N/2} are the
coefficients of the fourier transform of Z.
 The descriptors C(K) describe the frequency contents
of the curve.
 For K=0 , C(K) represents the position of the center
of gravity of the shape.

5. APPLICATIONS OF FOURIER
DESCRIPTORS
 Such a technique is commonly used for pattern
recognition like chromosome classification ,
identification of aircrafts or identification particules.
 A big issue about fourier descriptors is how many
terms should be kept from the fourier descriptors is
how many terms should be kept from the fourier
transform so the description is efficient.

6. ADVANTAGES OF FOURIER
DESCRIPTORS
The advantage is that it is
possible to capture coarse shape
properties with only a few
numeric values , and the level of
detail can be increased (or
decreased) by adding (or
removing) descriptors elements.

7. MOMENTS
 In image processing , computer vision and related
fields .
 An image moment is a certain particular weighted
average (moment) of the image pixel’s intensities , or
a function of such moments , usually chosen to have
some attractive property or interpretation.
 Image moments are useful to describe objects after
segmentation.

8. RAW MOMENTS
 For a 2D continuous function f(x , y) the
moment(sometimes called a “raw is moment”)of
order(p + q) is defined as
Mpq = ∫ ∫ x (p) y(q) f(x , y) dx dy
for p , q= 0 , 1 , 2…… adapting this to scalar image
with pixel intensities I(x , y) raw image moments Mij
are calculated by ,
MIJ=∑ ∑ x(i)y(j) I(x , y)
x y

9. CENTRAL MOMENTS
 Central moments are defined as
µ pq = ∫ (x-x) (y-y) f (x , y)d
 The eigenvectors of this matrix correspond to the
major and minor axes of the image intensity , so the
orientation can thus be extracted from the angle of
the eigen value towards the axis closest to this
eigenvector.

10. VELOCITY MOMENTS
 In the field of computer vision , velocity moments are
weighted averages of the intensities of pixels in a
sequence of images , similar to image moments but in
addition to describing an object’s shape also describe
its motion through the sequence of images.
 These are automated identification of a shape in an
image when information about the motion is
significant in its description.
 There are currently two established versions of
velocity moments.

11. MOMENT INVARIANTS
 Moments are well-known for their application in
image analysis , since they can be used to derive
invariants with respect to specific transformation
classes.
 The term invariant moments is often abused in this
context.
 While moment invariants are invariants that are
formed from moments , the only moments that are
invariants themselves are the central moments.

12. APPLICATIONS
 Applied Hu moment invariants to solve the
pathological brain detection problem.
 Doerr and florence used information of the object
orientation related to the second order central
moments to effectively extract translation- and
rotation –invariant object cross sections from micro-
x-ray tomography image data.

13. THANK YOU