Recognition of Blurred Images by Invariants

Recognition of images degraded by blur is a complex and serious challenge. The blur may arise from wrong focus, diffraction, camera shake, object motion, atmospheric turbulence, and similar factors.

We assume the blur can be modeled by convolution

\[z(x)=(h*u)(x) + n(x),\]

where \(z\) is the acquired image, \(u\) is the ideal image, \(h\) is the point-spread function (PSF) of the system and \(n\) is a random additive noise.
You can see examples of photos blurred with different types of blur below.

There are basically four approaches to handle the blur (see Fig. 2):

Image restoration
Image description by blur invariants
Blur-invariant similarity measure between two images
Brute force (augmentation of the training set by blurred images)

In this long-term project, we have focused on blur invariants (BIs); see Fig. 2 the second column. BIs are image descriptors that are not affected by blur of a certain class \(I(f*h)=I(f)\).

For different blur classes we have different invariants. Examples of blur classes are centrosymmetric blur, radially symmetric blur, N-fold symmetric blur, axially symmetric blur, Gaussian blur, and others.

Generally, the invariants are defined in Fourier domain as a ratio of two spectra
\(I(f)=\frac{F(f)}{F(Pf)}\), where \(P\) is a projection operator that projects the image space onto the assumed blur class. Different blur classes require the use of different projection operators \(P\), which leads to specific invariants for the given blur type. For the sake of efficient and stable computation, these invariants are not used in Fourier domain directly. Instead, we compute their Taylor coefficients, which can be accomplished in the image domain without computing Fourier transforms.

Unlike image restoration, blur invariants do not provide an estimation of the original scene. They are compressive image descriptors only but they are sufficient to resolve numerous practical tasks. A typical example of successful application is matching of a blurred template against a database of clear images.

A blurred portrait (top) is to be matched against a sharp group photograph. This is a typical scenario where blur invariants yield good recognition results and perform much faster than image restoration followed by cross-correlation matching.

Related publications:

FLUSSER, Jan, et al. Projection operators and moment invariants to image blurring. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2014, 37.4: 786-802.
KOSTKOVÁ, Jitka, et al. Handling Gaussian blur without deconvolution. Pattern Recognition, 2020, 103: 107264.
FLUSSER, Jan, et al. Blur invariants for image recognition. International Journal of Computer Vision, 2023, 131.9: 2298-2315.

Contact person

Jan Flusser