کاربرد فرآیندهای فضایی همبسته دوره‌ای و دوره نگار فضایی در پردازش تصویر

فرآیندهای فضایی همبسته دوره‌ای از جمله فرآیندهای پرکاربرد در تجزیه و تحلیل داده‌های فضایی می‌باشند، که در پردازش و تحلیل تصاویر دوره‌ای کاربرد دارند. در جهت تحلیل این نوع داده‌ها، در مرحله اول می‌بایست به تشخیص دوره‌ای بودن و تعیین مقدار دوره تناوب داده‌ها پرداخت. در این مقاله، ابتدا به معرفی فرآیندهای فضایی همبسته دوره‌ای و ویژگی‌های آنها پرداخته، سپس دوره‌نگار فضایی، به‌عنوان ابزاری برای تشخیص دوره‌ای بودن داده‌ها و تعیین مقدار دوره تناوب، معرفی می‌گردد و به بیان ویژگی‌های آن‌ها می‌پردازیم. در نهایت نحوه استفاده از دوره‌نگار فضایی در پردازش تصاویر دوره‌ای و تشخیص دوره‌ای بودن آن‌ها مورد بررسی قرار خواهد گرفت.

application of periodically correlated spatial processes and spatial periodogram in image processing

regarding spatial data analysis, detecting hidden dependencies in data observations is crucial. the second-order spatial process that exhibits periodic rhythm in the structure is called periodically correlated spatial process (pcsp), which necessitates explicitly the mean and covariance functions be periodic with the same periods. pcsp has many applications in variousfields since it can be considered an intermediary between stationary andnon-stationary spatial processes, including a large class of stationary andnon-stationary spatial processes. as one of the pcsp applications, imageprocessing can be used to analyze images with an alternating structure. tothis end, appropriate statistics should be used to estimate spectral density.first, this article introduces the periodic correlated spatial processes andtheir features. then, the spatial periodogram is presented as a tool for detectingthe periodicity of data and determining the value of the periodicperiod. in the following, the accuracy of spatial periodicity in detecting theperiodicity of data is investigated through simulation. in the end, its applicationis examined using authentic images.material and methodsa second order spatial process ξ = {ξt : t ∈ z2} is called periodicallycorrelated spatial process (pcsp) if there exists t = (t1, t2) ∈ n2, suchthat for any t, s ∈ z2 and n ∈ n2, its autocovariance function follow theequation rξ(t, s) = rξ(t + (n1t1, n2t2), s + (n1t1, n2t2)). if t1 and t2are the smallest natural numbers in this equation, then t is called periodof the spatial process ξ. if ξ is a pcsp with period t, then fξ, spectral density function of the spatial process ξ, is supported by ∪k∈dtdk,where dk = {(θ, η) : θ, η ∈ [0, 2π)2, θ − η = 2πkt = ( 2πk1t1, 2πk2t2)} anddt = {(i, j) : i = 0, . . . , t1 − 1, j = 0, . . . , t2 − 1}. for some arbitraryspatial process ξ and λ ∈ [0, 2π)2, the finite fourier transform (fft) of thefinite segment {ξt : t ∈ dn} is defined as dx(λ) = |n|−12σt∈dnxtei⟨t,λ⟩.also, spatial periodogram and two-dimensional spatial periodogram of thefinite segment {ξt : t ∈ dn} are defined as iξ(λ) = dξ(λ)dξ(λ) andjξ(λ, ˆλ) = dξ(λ)dξ(ˆλ). considering the above definition, jξ is an asymptoticallyunbiased estimator for fξ, also, can be said that jξ is near zeroeverywhere expect ∪k∈dtdk and iξ􀀀2πkn
= c|jξ􀀀0, 2πkn
|2 where c is positive.it is clear if ξ is a pcsp with period t and we have n observationfrom this process so iξ􀀀2πkn
, is positive for any kn multiple of 1t and nearzero for another.results and discussionconcerning the present study’s primary objective, recognizing the data’s periodicityand determining their periodicity, periodic photos are consideredprocesses, and their periodogram is depicted. if the figures are observedmeticulously, the presence of peaks indicates the process’s periodicity andthe peak’s location determines the period’s value.conclusionthere are too many periodic images, while the main objective of image processingis to recognize their periodicity and the type of their period. oneof the cases that should be considered is the moire effect. the moire effectmostly happens in images. their data periodically correlates with spatialprocesses, so if we are willing to erase that problem, we should notice theirperiod. one method to specify the period of these images is the spatial periodogram,which was elaborated in the present article.