AI-图像不变矩-图像识别的数学基础

AI基础-图像不变矩







In image processing, computer
vision and related fields, an image moment is a certain particular weighted average (moment)
of the image pixels' 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. Simple
properties of the image which are found via image moments include area (or total intensity), its centroid,
and information about its orientation.

Raw moments

For a 2D continuous function f(x,y)
the moment (sometimes
called "raw moment") of order (p + q)
is defined as

Mpq=∫−∞∞∫−∞∞xpyqf(x,y)dxdy


for p,q = 0,1,2,... Adapting this to scalar (greyscale) image with pixel intensities I(x,y), raw image moments Mij are calculated by
Mij=∑x∑yxiyjI(x,y)


In some cases, this may be calculated by considering the image as a probability
density function, i.e., by dividing the above by
∑x∑yI(x,y)


A uniqueness theorem (Hu [1962]) states that if f(x,y) is piecewise continuous and has nonzero values only in a finite part of the xy plane, moments of all orders exist, and the moment sequence
(Mpq) is uniquely determined by f(x,y). Conversely, (Mpq) uniquely determines f(x,y). In practice, the image is summarized
with functions of a few lower order moments.

Examples

Simple image properties derived via raw moments include:

Area (for binary images) or sum of grey level (for greytone images): M00
Centroid: { x, y } = {M10/M00, M01/M00 }

Central moments

Central moments are
defined as

μpq=∫−∞∞∫−∞∞(x−x¯)p(y−y¯)qf(x,y)dxdy


where x¯=M10M00

 and y¯=M01M00

 are
the components of the centroid.

If ƒ(x, y) is a digital image, then the previous equation becomes
μpq=∑x∑y(x−x¯)p(y−y¯)qf(x,y)


The central moments of order up to 3 are:
μ00=M00,

μ01=0,

μ10=0,

μ11=M11−x¯M01=M11−y¯M10,

μ20=M20−x¯M10,

μ02=M02−y¯M01,

μ21=M21−2x¯M11−y¯M20+2x¯2M01,

μ12=M12−2y¯M11−x¯M02+2y¯2M10,

μ30=M30−3x¯M20+2x¯2M10,

μ03=M03−3y¯M02+2y¯2M01.


It can be shown that:
μpq=∑mp∑nq(pm)(qn)(−x¯)(p−m)(−y¯)(q−n)Mmn


Central moments are translational invariant.

Examples

Information about image orientation can be derived by first using the second order central moments to construct a covariance
matrix.

μ20′=μ20/μ00=M20/M00−x¯2

μ02′=μ02/μ00=M02/M00−y¯2

μ11′=μ11/μ00=M11/M00−x¯y¯


The covariance matrix of the image I(x,y)

 is
now
cov⁡[I(x,y)]=[μ20′μ11′μ11′μ02′]

.
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 eigenvector associated with the largest eigenvalue towards the axis closest to this eigenvector. It can be shown that this angle Θ is given by the
following formula:
Θ=12arctan⁡(2μ11′μ20′−μ02′)


The above formula holds as long as:
μ20′−μ02′≠0


The eigenvalues of the covariance matrix can easily be shown to be
λi=μ20′+μ02′2±4μ′112+(μ′20−μ′02)22,


and are proportional to the squared length of the eigenvector axes. The relative difference in magnitude of the eigenvalues are thus an indication of the eccentricity of the image, or how elongated it is. The eccentricity is
1−λ2λ1.

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. However, while moment invariants are invariants that are formed from moments, the only moments that are invariants themselves are the central moments.

Note that the invariants detailed below are exactly invariant only in the continuous domain. In a discrete domain, neither scaling nor rotation are well defined: a discrete image transformed in such a way is generally an approximation,
and the transformation is not reversible. These invariants therefore are only approximately invariant when describing a shape in a discrete image.

Translation invariants

The central moments μi j of any order are, by construction, invariant with respect to translations.

Scale invariants

Invariants ηi j with respect to both translation and scale can
be constructed from central moments by dividing through a properly scaled zero-th central moment:

ηij=μijμ00(1+i+j2)


where i + j ≥ 2. Note that translational invariance directly follows by only using central moments.

Rotation invariants

As shown in the work of Hu et al., invariants with respect to translation, scale,
and rotation can be constructed:

I1=η20+η02


I2=(η20−η02)2+4η112


I3=(η30−3η12)2+(3η21−η03)2


I4=(η30+η12)2+(η21+η03)2


I5=(η30−3η12)(η30+η12)[(η30+η12)2−3(η21+η03)2]+(3η21−η03)(η21+η03)[3(η30+η12)2−(η21+η03)2]


I6=(η20−η02)[(η30+η12)2−(η21+η03)2]+4η11(η30+η12)(η21+η03)


I7=(3η21−η03)(η30+η12)[(η30+η12)2−3(η21+η03)2]−(η30−3η12)(η21+η03)[3(η30+η12)2−(η21+η03)2].


These are well-known as Hu moment invariants.

The first one, I1, is analogous to the moment
of inertia around the image's centroid, where the pixels' intensities are analogous to physical density. The last one, I7, is skew invariant, which enables it to distinguish mirror images of otherwise identical
images.

A general theory on deriving complete and independent sets of rotation moment invariants was proposed by J. Flusser. He showed that the traditional set of Hu moment invariants is not independent nor complete. I3 is
not very useful as it is dependent on the others. In the original Hu's set there is a missing third order independent moment invariant:
I8=η11[(η30+η12)2−(η03+η21)2]−(η20−η02)(η30+η12)(η03+η21)


Later, J. Flusser and T. Suk specialized the theory for N-rotationally symmetric shapes case.

Hu moment invariants to solve the Pathological Brain Detection (PBD) problem.