At stochastic gradient estimation of image parameters the estimates convergence character and computational expenses essentially depend on image samples local sample size used for obtaining the stochastic gradient. In the paper the possibility of a priori optimization of the volume of a local sample to minimize computational costs at geometrical images deformations estimation is considered. The minimum of the given computational costs for the conventional unit of expectation of the improvement of the evaluation is chosen as an optimization criterion. The block diagram of one of the algorithms and the examples of calculation results are presented.
it is possible to neglect the brightness distortions, or the sampling coefficient of interframe correlation at interframe brightness distortions close to linear [8].
The key problem is to increase the speed of the SGA. Various approaches are being explored to
solve this problem. In particular, in [9] a procedure for stochastic gradient optimization of the second
order in a linear time was proposed, in [10] to accelerate the optimization process using algorithms
based on stochastic gradient descent the Nesterov moment method is applied, in [11] a convolutional
neural network is used to estimate the geometric mismatch parameters between two images in
accordance with a given geometric model (the affine model and the thin-plate spline transformation
are studied), in [12] the acceleration of the stochastic gradient descent is achieved by taking into
account the probability of smoothness of separate areas of the image. Also one of the approach is to
reduce the volume of a two-dimensional local sample Zt z(
Suppose that, in accordance with the given error in the estimation of the parameter , the mismatch ev ˆ of the parameter estimate ˆ and its exact value ev should change from max to min . Consider the possibility of minimizing the computing costs of the SGA by optimizing the volume of the local sample for each iteration of the estimation for the given conditions. We use the following optimality criterion.
At each t -th iteration of the stochastic gradient estimation, we will search for such a volume t of
the local sample that provides the minimum of the computational cost per the conditional unit of the
mathematical expectation of an parameter estimate t improvement
t k min gkt , k 1, 2, ... ,
(
Because at the estimation iterations the parameter mismatch consistently changes from max to min , then for the T iterations the proposed criterion will provide the minimum total computational costs
T
G g t , (
t1 where T – the number of iterations required to perform the condition T min ; T – the mismatch of the parameter estimate ˆ and its exact value at the T -th iteration.
A detailed analysis of computational costs requires consideration not only the features and structure
of the calculated ratio, but also many other influencing factors. These include the sampling time and
conditions of images, the class of computing device, the time spent on the operations of addition,
multiplication, division, access to memory, move data, and other auxiliary operations. Many of these
factors depend on the specific image recording devices and the computers used. Therefore, in this
paper, the computational cost components will not be concretized. However, we will assume that the
computational costs g t of performing the SGA t -th iteration contain two components:
g t = gZt + go , (
In this case, the cost of forming a local sample will be considered proportional to the volume of
the local sample: gZt g , where g – computational costs for the formation of a local sample of a
unit volume. Then
g t = g c1 t ,
(
For relay SGA, the mathematical expectation of the improvement in the estimation t of the parameter under study by the t -th iteration can be found [13] by using the drift probability of the estimates [14]
t M t1 t t t t o t t t t , (
start
A block diagram of one of the possible algorithms for finding the optimal volume of a local sample are presented on figure 1. Here, for simplicity, it is assumed that o 0 , then 1 and t Λt 2 1 . To sequentially calculate the volume t of the local sample at the t -th iteration, t 1,T , in the range of the deviation of the estimate from max to min , the initial conditions are given t 0 and o max . Next, the volume 1 of the local sample is calculated, at which the minimum of the reduced computational costs (the minimum of the ratio g k 1 ) is reached at the first iteration. Then, the local sample volume is estimated at the next iteration, i.e. is computed 2 and so on, until condition t min is fulfilled.
An example of computation results of optimal value of the local sample volume as a function of the mismatch is shown in figure 2. An interframe parallel image shift was evaluated. The value of the parameter c is chosen equal to 5%. Curve 1 corresponds to the noiseless images, and curve 2 corresponds to the signal-to-noise ratio x2 2 10 . It is assumed that the noise model of the researched images Z1 и Z2 is additive: z j x j j , where x j – image with dispersion x2 , j – independent Gaussian noise with zero mathematical expectation and variance 2 .
For the conditions corresponding to curves 1 and 2, the table 1 shows the results of the experiment. They show the loss in computing costs when using а constant volume of local sample ( const ) in comparison with the case of using the optimal volume of local sample. When specifying in SGA const , the value of corresponded to the average value avg of the optimal sample size, and avg 2 , avg 1 , avg 1 и avg 2 .
The approach to increasing the rate of algorithms for stochastic gradient estimation of image parameters is considered on the example of inter-frame deformations estimation. The purpose is achieved by optimizing the size of the two-dimensional local sample used at each iteration of the estimation to determine the stochastic gradient. A priori optimization based on the criterion of minimum computational costs for the case of estimating one parameter is used. At the same time at each iteration of the estimation, the local sample size providing a minimum of computational costs for the conventional unit of the mathematical expectation of the parameter estimate improvement is determined. It is shown that, as the number of iterations increases, the mismatch modulus of the estimate and the exact value of the parameter decreases, the proposed approach provides a minimum of total computational costs. To determine the mathematical expectation of an improvement of the studied parameter estimate, the probabilities of drift estimates (the probability of changing the estimates towards the exact value of the parameter and from it) are used.
The carried out modeling for one of the possible algorithms realizing the proposed approach confirmed the set purpose. So, for the given example of results, the gain in computational costs in comparison with the situation of using the constant volume of the local sample amounted to no less than 3.9% in the absence of noise, and not less than 1.8% for a signal-to-noise ratio equal to 10 (in variance). Thus, the proposed approach for algorithms of stochastic gradient estimation of image parameters makes it possible to determine the optimal size of the local sample for each estimation iteration, which ensures the minimization of computational costs.
The reported study was funded by RFBR and Government of Ulyanovsk Region according to the research projects 16-01-00276 and 18-41-730006.