A method for correction of nonhomogenous illumination based on optimization of parameters of B-spline shading model with respect to Shannon's entropy is presented. The evaluation of Shannon's entropy is based on Parzen windowing method (Mangin, 2000) with the spline-based shading model. This allows us to express the derivatives of the entropy criterion analytically, which enables efficient use of gradient-based optimization algorithms. Seven different gradient- and nongradient-based optimization algorithms were initially tested on a set of 40 simulated retinal images, generated by a model of the respective image acquisition system. Among the tested optimizers, the gradient-based optimizer with varying step has shown to have the fastest convergence while providing the best precision. The final algorithm proved to be able of suppressing approximately 70% of the artificially introduced non-homogenous illumination. To assess the practical utility of the method, it was qualitatively tested on a set of 336 real retinal images; it proved the ability of eliminating the illumination inhomogeneity substantially in most of cases. The application field of this method is especially in preprocessing of retinal images, as preparation for reliable segmentation or registration.

Improper scene illumination as well as nonideal acquisition conditions due to for example, misadjusted imaging system can introduce severe distortions into the resulting image. These distortions are usually perceived as smooth intensity variations across the image. According to the terminology commonly used in processing of magnetic resonance (MR) images, we call these systematic intensity level inhomogeneities as the

Most existing bias correction methods assume that the bias field is multiplicative, slowly varying, and tissue independent. Many techniques ignore the noise and apply a log transform to make the bias field additive. The known illumination correction methods can be categorized in the following groups: filtering, segmentation based, surface fitting, and other methods.

Illumination inhomogeneities are generated during acquisition process in systems with different modalities. Here, the proposed illumination correction method will be applied on retinal images from confocal scanning laser ophthalmoscope (CSLO). This correction is an important pre-processing task in image segmentation and/or multimodal registration [

The earliest bias correction techniques were based on phantoms [

Linear filtering methods [

Some simple methods, as [

The expectation maximization (EM) algorithm is proposed in [

Segmentation-based approaches raise the problem of selecting the number of classes, which have to be explicitly modeled. Furthermore, these algorithms unfortunately tend to converge to a local nonoptimal minimum for some bias configurations, especially when more than two tissue classes are modeled [

To overcome the segmentation problems, bias correction methods not requiring the segmentation were designed based on a chosen image quality criterion. In [

Specific methods of illumination correction were proposed in the frame of retinal image processing and analysis. Simple and fast methods using large-kernel median filter to obtain a low-pass correction coefficients were used for CSLO image preprocessing in [

Other approaches exist, for example, in applications including illumination correction for face recognition [

In this paper, we focus our attention on methods estimating the parametric illumination field using the quality criterion derived from the information theory. We use a multiplicative model of nonuniform illumination and parametric local bias model for formulation of criterion function and its derivatives (Section

For the purpose of illumination correction process, we need a model of image creation. We assume, that each tissue class (vessels, optic disc, retinal surface) has a different mean value

Model of image acquisition. Ideal image is corrupted by noise, multiplicative illumination, and finite resolution of the imaging device (characterized by Gaussian point spread function).

On the right-hand side of (

Thus, under the assumption that the bias involved in image creation process is multiplicative, and that it is the only important disturbance, we can reconstruct the original signal as

(a): Example of a line of input signal:

In this section, we extend the Likar’s algorithm [

Generally, in accordance with [

In order to regularize the problem of finding the bias field that would optimally correct the illumination inhomogeneity, we used the bases in the following form:_{i}

We have found that the illumination distortion of retinal images shows a significant spatial variance, while both illumination models mentioned in [

Hence, we decided to use the locally defined mean-corrected and normalized reciprocal bias model based on B-splines, formalized as follows:

For the case of two dimensions, _{x1}_{x2}_{1}_{2}_{k}_{i}_{i }^{m}c_{i}_{i}

Formation of grid of control points for B-spline intensity transform.

Similarly, we can derive the normalization condition from (

Because this transformation may generally produce out-of-range intensity values, we use intermediate image representation with extended intensity range. The resultant image is finally computed using linear contrast transformation converting the intermediate image back into the original intensity range.

In order to find parameters of the bias model describing the undesirable illumination, we need to define a criterion, with respect to which the parameters would be optimized. In the works of Likar et al. [

Although the brightness

Illustration of Parzen windowing method for estimating density probability of image intensities.

Thanks to our formulation of the criterion (

From here on, the _{x1}_{x1}

By applying product rule, we obtain from (

Calculation of this integral is approximated using the rectangle rule as

Further from (

The image

The grid controlling the compensation field is chosen adequately sparse as required by the smooth character of the to be compensated illumination unevenness. An example illustrating behaviour of the criterion function

(a): Shannon entropy-based criterion and its partial derivative as functions of the parameter

The overall illumination correction algorithm is based on the reciprocal illumination model ^{-1}

Flow diagram of iterative optimization algorithm.

Various types of optimization techniques were studied in order to find the optimizer best suited for the particular properties of the used optimization criterion in the problem of non-homogenous illumination correction. The tested methods were namely, downhill simplex [

The comparison of results of the individual methods can be found in the next paragraph.

The proposed algorithm is supposed to be able to deal with non-homogenous illumination of retinal image data (384×384 pixel, 10

0—black background introduced by HRT II; 60—vessels; 120—vessel centers (brighter due to reflections); 90—retinal tissue; 30—optic disc; 250—optic disc cup). Further, according to the image acquisition model (

(a): Ideal model image created by manual segmentation of a real image, (b): Model image after noise addition, (c): Model image after noise addition and convolution with PSF of the imaging system. (d): Final model image corrupted by non-homogenous illumination. (e): the image after processing by the presented algorithm.

A set of 40 simulated images was created via this model. Normalized parameters of the illumination field were uniformly distributed on interval [

The seven optimization algorithms mentioned in the previous section were tested and the average results obtained when using the simulated image set are summarized in Table

Study of suitability of different optimizers for the task of optimization of Shannon’s entropy. The best values are highlighted.

Nongradient-based methods | Gradient-based methods | ||||||
---|---|---|---|---|---|---|---|

Amoeba | Powell | CRS | L-BFGS | L-BFGS-B | CG | ||

0.16 | 0.23 | 0.14 | 0.13 | 0.14 | 0.18 | ||

68.6 | 95.6 | 514 | 14.7 | 45.9 | 410 | ||

N | 163 | 226 | 1201 | 26.4 | 57.4 | 615 | |

44.5 | 18.6 | 52.9 | 55.9 | 51.3 | 36.9 |

(a) Artificially introduced illumination field. Error illumination field (illumination and restoration field multiplied, ideally black) using: (b) Amoeba optimizer, (c) Powel optimizer, (d) CRS, (e) LBFGS optimizer, (f) Gradient descent optimizer (g) Conjugate gradient optimizer.

In frame of experiments, we tested also the influence of different number of image samples used for probability approximation. As can be seen in Table

Influence of different number of samples used for entropy evaluation on the algorithm precision and speed.

GD, 50 bins | |||
---|---|---|---|

coverage | 0.15 | 0.5 | 1 |

45.8 | 46.1 | 212 | |

57.4 | 57.5 | 67.6 |

The next set of experiments concerned real retinal images with clearly visible illumination inhomogeneities that however were not known. The algorithm was tested on the set of 336 real (clinically obtained) retinal images. Obviously, the quality evaluation of non-homogenous illumination compensation could only be performed subjectively, due to lack of the golden standard (i.e., ideal illumination in each case). In majority of cases (about 95%), a substantial improvement was visible; in the remaining cases, the image remained practically unchanged, that is, no image was further distorted by the correction. On Figure

(a) Retinal image highly corrupted by non-homogenous illumination, (b) Image after multiplicative correction by recovered bias field, (c) Normalized bias field controlled by 3×3 parameters automatically obtained using proposed algorithm, (d) Intensity profiles along the indicated row.

A method for efficient illumination correction was proposed, implemented, and verified: quantitatively on simulated images with known deterioration and qualitatively on an extensive set of real retinal images lacking this knowledge. It is based on estimating a B-spline polynomial shading model, the inversion of which provides correction of the input image, optimal in sense of the used information-based criterion—Shannon’s entropy. Previously published methods using a similar principle were modified namely, as to the type of the used correction function concerns, and as a novelty, the computation of the criterion is based on probability distribution estimates by Parzen windowing method, which enables us to derive analytical expressions for derivatives of the criterion. This consequently substantially speeds up the computations. Along with the efficient B-spline distortion model, it results in the possibility of efficient usage of gradient-based optimizers. We compared the efficiency and precision of three non-gradient and four gradient-based optimizers and found the classical gradient descent optimizer with variable step as the best for the purpose of illumination correction, formulated in the suggested way.

The quantitative tests were done on an image set artificially created with respect to characteristics of the imaging device (the method is primarily aimed at improving quality of retinal images taken by means of the HRT II CSLO); these tests have shown that the designed algorithm is capable of removing up to about 70% of artificially introduced illumination variability. Finally, the method was successfully qualitatively tested on a set containing 336 real CSLO clinically obtained retinal images.

According to results of some further tests, involving subsequent registration of multiple images, preprocessing of the retinal images by the proposed algorithm had a clearly positive influence on reliability of the registration of the images using the registration method developed by our group [

This research has been supported by the National Research Centre DAR (sponsored by the Ministry of Education, Czech Republic) project no. 1M0572 and partly also by the research frame no. MSM 0021630513.