>> Invariants

LEADERS IN THE FIELD

We are not just inventors of complex moments

During last fifteen years we contributed significantly to the theory of moments invariants. This includes derivation of complete systems of invariants with respect to affine transformation, with respect to blurring by kernels with various types of symmetries and also invariants to combined degradations.

Rotation moment invariants

Translation, rotation, and scaling (TRS) are the simplest transformations of spatial coordinates. TRS, sometimes called similarity transform can be described as

...

u = a₀ + a₁x - b₁y

Affine moment invariants

The affine transform is general linear transformation of space coordinates of the image:

u = a₀ + a₁x + a₂y

v = b₀ + b₁x + b₂y.

The affine moment invariants are features for pattern recognition computed ...

3D Rotation Moment Invariants

The geometric moments in three dimensions (3D) are defined:

Tensor method

The moment tensor is defined:

where x¹=x, x²=y and x³=z. If p indices equal 1, q indices equal 2 and r indices equal 3, then

Rotation Invariants of 2D Vector Fields

Vector fields are a special kind of multidimensional data. In each pixel, the field is assigned to a vector that shows the direction and the magnitude of the quantity that has been measured.

Affine Moment Invariants of Vector Fields

Vector fields are a special kind of multidimensional data. In each pixel, the field is assigned to a vector that shows the direction and the magnitude of the quantity, which has been measured.

Affine Moment Invariants of Tensor Fields

Tensor fields are a special kind of multidimensional data. In each pixel or voxel, the field is assigned to a tensor.

Moment invariants to convolution

Moment Invariants to Convolution are mathematical properties or features of shapes that remain constant when the shape is convolved with a kernel, making them useful for analyzing patterns and texture...

Point-Based Projective Invariants

Point-Based Projective Invariants are mathematical properties of a set of points that remain unchanged under projective transformations, such as translation, rotation, scaling, and perspective distort...