TELK OMNIKA T elecommunication, Computing, Electr onics and Contr ol V ol. 24, No. 5, October 2026, pp. 1710 ∼ 1720 ISSN: 1693-6930, DOI: 10.12928/TELK OMNIKA.v24i5.27914 ❒ 1710 Edge-awar e adapti v e partial differ ential equation image inpainting f or high-quality r estoration F ouzia Hamadache 1,2 , Zine-Eddine Baarir 2,3 , Nacira Diffellah 1,4 , Abdelmalik Ahmed-T aleb 5 , T ewk Bekk ouche 6 , Rabah Hamdini 1,4 1 Institute of Electronics and T elecommunications (IET), Uni v ersity of Mohamed El Bachir El Ibrahimi, Bordj Bou Arreridj, Algeria 2 Departement of Electronic and Automatic Engineering, Mohamed Khider Uni v ersity of Biskra, Biskra, Algeria 3 VSC Laboratory of Research, Mohamed Khider Uni v ersity of Biskra, Biskra, Algeria 4 LMSE laboratory of Research, Uni v ersity of Mohamed El Bachir El Ibrahimi, Bordj Bou Arreridj, Algeria 5 IEMN UMR CNRS 8520, V alenciennes, France 6 LEPCI Laboratory , Department of Automatic and Intelligent Systems, F aculty of T echnology , Uni v ersity of Ferhat Abbas, Setif, Algeria Article Inf o Article history: Recei v ed Mar 2, 2026 Re vised Jun 2, 2026 Accepted Jun 30, 2026 K eyw ords: Anisotropic dif fusion Bertalmio model Edge-a w are dif fusion Image inpainting Isophote propag ation PDE-based restoration Structural preserv ation ABSTRA CT Image inpainting aims to reconstruct missing or damaged re gions in digital images while preserving structural consistenc y and visual coherence. Among partial dif ferential equation (PDE)-based approaches, the model introduced by Marcelo Bertalmio is a foundational technique for isophote-dri v en structure propag ation. Ho we v er , its performance often de grades in lar ge damaged re- gions. This limitation is mainly caused by insuf cient transport strength and isotropic dif fusion, which lead to structural blurring. In this paper , we propose an edge-a w are adapti v e PDE inpainting model that introduces a transport am- plication f actor and a spatially-adapti v e dif fusion term to impro v e edge prop- ag ation and structural preserv ation. Experim ental results on v arious test im- ages demonstrate that the proposed approach impro v es structural reconstruction. Quantitati v e e v aluation using peak signal-to-noise ratio (PSNR) and structural similarity inde x (SSIM) metrics conrms higher restoration accurac y and better edge preserv ation compared with baseline methods. This is an open access article under the CC BY -SA license . Corresponding A uthor: F ouzia Hamadache Institute of Electronics and T elecommunications (IET), Uni v ersity of Mohamed El Bachir El Ibrahimi Bordj Bou Arreridj, Algeria Email: fouzia.hamadache@uni v-bba.dz 1. INTR ODUCTION Image inpainting is a fundamental problem in image processing and computer vision that aims to reconstruct missing or damaged image re gions in a visually plausible manner [1], [2]. Its applications in- clude object remo v al, cul tural heritage restoration, old photograph reco v ery , lm post-production, and error concealment in image transmission. The main cha llenge is to ensure structural coherence and te xture consis- tenc y while preserving o v erall visual realism [3]-[5]. Early inpainting approaches were predominantly based on partial dif ferential equations (PDEs), where missing information is reconstructed by propag ating geometric structures from the boundary of the damaged re gion to w ard its interior [6]-[8]. Among these methods, the seminal w ork of Bertalmio et al. [7] introduced a third-order is oph ot e-dri v en transport equation inspired by uid dynamics [9], [10]. Owing to its solid mathematical foundation and geometric interpretation, this model has become a corners tone of PDE-based inpainting, particularly ef fecti v e for small mi ssing re gions and thin J ournal homepage: http://telk omnika.uad.ac.id Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA T elecommun Comput El Control ❒ 1711 structures. Despite their strong theoretical foundation, classical PDE-based inpainting models still struggle to maintain structural continuity in lar ge missing re gions due to weak isophote transport and e xcessi v e dif fusion. When applied to lar ge damaged areas, the classical formulation suf fers from insuf cient transport strength, leading to attenuated structure continuation and blurred edges. Moreo v er , the dif fusion component introduces e xcessi v e smoothing, reducing the preserv ation of ne details. These limitat ions moti v ate the de v elopment of enhanced PDE-based formulations capable of reinforcing structural propag ation while maintaining numer - ical stability and edge preserv ation. I n this w ork, we propose an impro v ed PDE-based inpainting frame w ork deri v ed from the classical Bertalmio model. Unlik e pre vious incremental e xtensions, our method introduces a no v el transport amplication f actor and a spatially-adapti v e dif fusion term. The scientic contrib ution lies in the adapti v e control of transport strength and dif fusion according to local edge information, which is not present in pre vious models. The proposed approach incorporates an adapti v e transport ampl ication f actor to reinforce isophote propag ation and an edge-a w are dif fusion adjustment to better preserv e ne details. A sta- bilized discretizat ion scheme combining Gaussian pre-smoothing, gradient normalization, intensi ty projection, and a con v er gence-based stopping criterion is adopted to ensure rob ust and stable iterati v e beha vior . Experi- mental results on benchmark images demonstrate impro v ed structural coherence and quantitati v e performance compared to representati v e classical and v ariational inpainting approaches. 2. RELA TED W ORK Image inpainting has been e xtensi v ely studied o v er the past tw o decades, leading to a wide range of reconstruction techniques. Existing approaches are broadly classied into three main cate gories: dif fusion- based, e x emplar -based, and h ybrid methods. 2.1. Diffusion-based inpainting Dif fusion-based methods reconstruct missi ng re gions by propag ating information from kno wn areas through PDEs. The pioneering w ork of Bertalmio et al. [7] introduced a third-order PDE frame w ork that prop- ag ates isophotes (lines of equal intensit y) into damaged re gions. This model preserv es geometric structures and edge continuity and has become a cornerstone of PDE-based inpainting. Subsequent e xtensions incorporated anisotropic dif fusi on [11], curv ature-dri v en dif fusion [12], and higher -order re gularization terms to impro v e smoothness and structural delity . These models are particularly ef fecti v e for small missing re gions and thin geometric structures. Ho we v er , their performance de grades in lar ge damaged areas, where insuf cient transport strength and e xcessi v e dif fusion often lead to blurred edges and loss of ne details. Se v eral adapti v e strate- gies ha v e been proposed to mi tig ate these limitations, including spatially v arying dif fusion coef cients and edge-guided transport mechanisms. Ne v ertheless, achie ving a balance between strong structure propag ation and numerical stability remains a challenging problem [8], [12]-[14]. 2.2. Exemplar -based inpainting Ex emplar -based approaches reconstruct m issing re gions by cop ying si milar patches from kno wn parts of the image. The inuential method of Criminisi et al. [15] introduced a priority-dri v en patch lling strate gy based on condence and data terms, enabling impro v ed structure continuation. A major adv ancement in this cate gory is the P atchMatch algorithm [16], which pro vides a f ast randomized method for approximate nearest- neighbor patch search. P atchMatch has enabled ef cient lar ge-scale image editing and has been inte grated into v arious practical inpainting systems. Ex emplar -based methods generally achie v e superior performance in te xture-rich re gions and lar ge missing areas. Ho we v er , the y may suf fer from structural inconsistencies when long-range geometric continuation is required [8], [17], [18]. 2.3. Hybrid and lear ning-based methods Hybrid methods attempt to combine the strengths of dif fusion-based and e x emplar -based approaches. T ypically , geometric structures are rst recons tructed using PDE-based techniques, follo wed by te xture syn- thesis through patch-based strate gies [19]. Such combinations aim to impro v e both structural coherence and te xture realism [20]. More recently , deep learning approaches ha v e emer ged, emplo ying con v olutional neural netw orks and generati v e models for data-dri v en inpainting [21], [22]. Although these models produce visu- ally impressi v e results, the y generally require lar ge training datasets and substantial computational resources. Moreo v er , in restoration-oriented applications, the hallucination of non-authentic content may be undesirable. Edg e-awar e adaptive partial dif fer ential equation ima g e inpainting for ... (F ouzia Hamadac he ) Evaluation Warning : The document was created with Spire.PDF for Python.
1712 ❒ ISSN: 1693-6930 It is important to emphasize that the objecti v e of the proposed method dif fers from that of data-dri v en gen- erati v e inpainting approaches [21]. While deep learning models can synthesize visually plausible te xtures and complete lar ge missing re gions, the reconstructed content may not strictly correspond to the original im- age structures, as it is partially inferred from learned priors [5]. In controlled restoration scenarios, such as scientic imaging, cultural heritage preserv ation, or medical image reconstruction, structural delity and deter - ministic beha vior are often preferred o v er generati v e realism. In such conte xts, the introduction of hallucinated structures may lead to misleading interpretations [23]. The proposed PDE-based frame w ork preserv es mathe- matical interpretability , does not rely on e xternal training data, and reconstructs missing re gions by propag ating e xisting geometric information from the image itself. Therefore, the prese n t study focuses on deterministic, structure-dri v en restoration rather than data-dri v en generati v e inpainting. 2.4. Moti v ation and positioning of the pr oposed method Despite substantial progress, dif fusion-based methods remain attracti v e due to their mathematical in- terpretability , lo w computational comple xity , and suitability for structure-preserving restoration [7]-[8]. Ho w- e v er , classical PDE models, including the Bertalmio formulation, still f ace challenges in balancing transport strength, detail preserv ation, and numerical stability [9], [10]. Moti v ated by these obs erv ations, our w ork focuses on enhancing the original PDE frame w ork through adapti v e transport amplication and edge-a w are dif fusion adjustm ent. Unlik e e x emplar -based or learning-based approaches, the proposed method preserv es the deterministic and structure-oriented nature of PDE inpainting while impro ving rob ustness and structural coherence. This mak es it particularly suitable for controlled restoration scenarios where interpretability and stability are essential. 2.5. Comparati v e methods T o e v aluate the ef fecti v eness of the proposed PDE-based inpainting frame w ork, we compare it ag ainst the follo wing benchmark methods: - Distorted image reconstruction method with trimmed median [24]: this method emplo ys a spatial ltering technique to restore corrupted pix els. By computing the trimmed median of the local neighborhood, it aims to remo v e noise and ll missing g aps while preserving local structures. Ho we v er , it often struggles with lar ger missing re gions, which can result in noticeable structural blurring. - Adapti v e Euler’ s elastica inpainting with discrete gradient optimizati on [25]: this v ariational approach re- constructs missing structures by minimizing curv ature-based ener gy functionals with adapti v e parameter estimation, pro viding strong geometric continuity , especially in lar ge missing re gions. These comparati v e methods allo w us to assess the impro v ements of fered by the proposed adapti v e transport and edge-a w are dif fusion mechanisms in terms of structural coherence and quantitati v e metrics such as peak signal-to-noise ratio (PSNR) and structural similarity inde x (SSIM). 3. PR OPOSED METHOD 3.1. PDE f ormulation The classical inpainting model introduced by Bertalmio et al. [7] propag ates isophotes within the missing re gion using a third-order PDE. The e v olution of the image intensity I ( x, y , t ) is gi v en by: ∂ I ∂ t = ∇ ⊥ I · ∇ (∆ I ) (1) where ∇ ⊥ I = ( − I y , I x ) denotes the direction orthogonal to the image gradient (isophote direction), and ∆ I is the Laplacian operator , ensuring curv ature-dri v en propag ation. T o impro v e numerical stability and reduce noise amplication, an anisotropic dif fusion term can be added: ∂ I ∂ t = ∇ ⊥ I · ∇ (∆ I ) + ∇ · ( c ∇ I ) (2) where the dif fusion coef cient c is dened as: c = 1 1 + |∇ I | 2 K (3) with K controlli ng edge preserv ation. TELK OMNIKA T elecommun Comput El Control, V ol. 24, No. 5, October 2026: 1710–1720 Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA T elecommun Comput El Control ❒ 1713 3.2. Adapti v e transport factor While ef fecti v e for small re gions, the classical PDE may lack suf cient transport strength in lar ge missing areas. No v elty of the proposed approach: unlik e pre vious adapti v e PDE methods that adjust only the dif fusion coef cient, our approach introduces tw o complementary and original mechanisms: - A transport amplication f actor α that directly strengthens isophote propag ation in lar ge missing re gions - A spatially-adapti v e dif fusion term that preserv es edges by reducing e xcessi v e smoothing The proposed model is formulated as: ∂ I ∂ t = α ∇ ⊥ I · ∇ (∆ I ) + ∇ · ( c ∇ I ) (4) where α > 1 strengthens isophote propag ation, and α = 1 reduces to the classical model. The dif fusion parameter can also be adapted to enhance ne detail preserv ation: K adapt = β K , (5) where β < 1 increases sensiti vity to edges. 3.3. P arameter selection and justication The parameters α and β were empirically selected based on preliminary e xperiments to balance edge propag ation and dif fusion stability . A systematic three-step procedure w as adopted: Step 1 – grid search - α tested: { 1 . 0 , 1 . 01 , 1 . 02 , 1 . 03 , ..., 1 . 1 } - K t ested: { 0 . 001 , 0 . 002 , 0 . 003 , 0 . 004 , ... 0 . 009 } - σ tested: { 0 . 5 , 1 . 0 , 1 . 5 } Step 2 – selection criteria - Maximization of PSNR and SSIM on v alidation images - Numerical stability (con v er gence within 500 iterations) - Absence of visual artif acts Step 3 – retained v alues - α = 1 . 08 : best trade-of f between structural propag ation and stability - K = 0 . 004 : optimal edge preserv ation - σ = 1 . 0 : suf cient smoothing without loss of ne details - ∆ t t = 0 . 0008 , ∆ t d = 0 . 15 : ensuring numerical stability Sensiti vity analysis - α ∈ [1 . 0 , 1 . 1] : stable performance. - α > 1 . 1 : numerical instabilities appear - K ∈ [0 . 003 , 0 . 005] : visually disimilar res ults - K < 0 . 001 : residual noise; K > 0 . 009 : e xcessi v e smoothing All parameters were k ept constant across all e xperiments to ensure reproducibility . 3.4. Numerical discr etization and stabilization The PDE is discretized using nite dif ferences in space and e xplicit time marching: I k +1 = I k + ∆ t t α T ( I k ) + ∆ t d D ( I k ) (6) where T ( I ) denotes the transport term and D ( I ) the anisotropic dif fusion term. T o ensure stability and ro- b ustness, we adopt Gaussian pre-smoothing dened as I s = G σ ∗ I , gradient normalization computed as |∇ I | = q I 2 x + I 2 y + ε to a v oid numerical instability , intensity projection to mai ntain pix el v alues within the v alid range I ← min(max( I , 0) , 1) , and a con v er gence criterion based on the relati v e ℓ 2 -norm gi v en by ∥ I k +1 − I k ∥ 2 ∥ I k ∥ 2 < 10 − 6 . The computational comple xity is O ( M N K max ) , where M × N denotes the image size and K max the maximum number of iterations. T o pro vide a clearer understanding of the interaction be- tween the transport and adapti v e dif fusion mechanisms, the operational w orko w of the proposed algorithm is illustrated in Figure 1. Edg e-awar e adaptive partial dif fer ential equation ima g e inpainting for ... (F ouzia Hamadac he ) Evaluation Warning : The document was created with Spire.PDF for Python.
1714 ❒ ISSN: 1693-6930 Figure 1. Flo wchart of the proposed edge-a w are adapti v e PDE inpainting algorithm 3.5. Algorithm The proposed method is summarized i n Algorithm 1. It iterati v ely propag ates isophotes within the missing re gions while applying edge-a w are dif fusion, Gaus sian smoothing, gradient normalization, and inten- sity projection to maintain numerical stability . Algorithm 1 Impro v ed bertalmio PDE-based inpainting Requir e: Original image I 0 ∈ R M × N , binary mask Ω , maximum iterations K max , transport step ∆ t t , dif fusion step ∆ t d , dif fusion parameter K , Gaussian smoothing σ , transport f actor α Ensur e: Inpainted image I Initialize I ← I 0 f or k = 1 to K max do I pre v ← I 1) Isophote transport I s ← G σ ∗ I ▷ Gaussian smoothing ( I x , I y ) ← ∇ I s |∇ I s | ← q I 2 x + I 2 y + ϵ N x ← I x / |∇ I s | , N y ← I y / |∇ I s | ∆ I ← ∇ 2 I , (∆ I x , ∆ I y ) ← ∇ (∆ I ) β ← ( − N y )∆ I x + N x ∆ I y I (Ω) ← I (Ω) + ∆ t t · α · β (Ω) 2) Anisotr opic diffusion ( I x , I y ) ← ∇ I c ← 1 / (1 + ( I 2 x + I 2 y ) /K ) di v ← ∇ · ( c ∇ I ) I (Ω) ← I (Ω) + ∆ t d · di v (Ω) I ← min(max( I , 0) , 1 ) if ∥ I − I pre v ∥ 2 ∥ I pre v ∥ 2 < 10 − 6 then br eak end if end f or r etur n I 4. RESUL TS AND DISCUSSION T o e v aluate the ef fecti v eness of the proposed adapti v e PDE-bas ed inpainting method, we conducted e xperiments on a di v erse set of images from the Berk ele y se gmentation dataset (BSDS500) [26], which pro- vides a v ariety of natural scenes with dif ferent structural and te xtural comple xities. T w o types of de gradation TELK OMNIKA T elecommun Comput El Control, V ol. 24, No. 5, October 2026: 1710–1720 Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA T elecommun Comput El Control ❒ 1715 masks are considered t o e v aluate the inpainting performance. Mask1 consists of irre gular yet structured curv ed missing re gions, forming connected strok e-lik e patterns across the image. This mask is designed to assess the ability of the method to restore ne structures and m aintain local continuity . Mask2, in contrast, rep- resents a structured de gradation composed of lar ger contiguous missing areas, including te xt-shaped re gions (w atermark-lik e patterns). This mask e v aluates the capability of the method to reconstruct e xtended re gions and ensure coherent long-range structural propag ation. The inpainted results were quantitati v ely assessed using PSNR [27]-[29] and SSIM [29]-[31], which measure pix el-wi se delity and perceptual structural similarity , respecti v ely . PSNR reects reconstruction accurac y , while SSIM emphasizes the preserv ation of structural features and o v erall perceptual quality . The PSNR is computed as: PSNR = 10 log 10 M AX 2 I MSE (7) where M AX I is the maximum possible pix el v alue and MSE is the mean squared error between the original image I and the inpainted image ˆ I : MSE = 1 M N M X i =1 N X j =1 I ( i, j ) − ˆ I ( i, j ) 2 (8) The SSIM is computed as: SSIM ( I , ˆ I ) = (2 µ I µ ˆ I + C 1 )(2 σ I ˆ I + C 2 ) ( µ 2 I + µ 2 ˆ I + C 1 )( σ 2 I + σ 2 ˆ I + C 2 ) (9) where I is the original image, ˆ I is the inpainted image, µ and σ denote local mean and v ariance, σ I ˆ I is the local co v ariance, and C 1 , C 2 are small constants. Figure 2 sho ws the test images selected for the e xperimental e v aluation. Figures 2(a) to (h) present the eight reference images used in our e xperiments. T w o masking patterns were applied, as illustrated in Figure 3. Figure 3(a) (Mask1) consists of irre gular curv ed missing re gions, while Figure 3(b) (Mask2) represents lar ger contiguous missing areas. (a) (b) (c) (d) (e) (f) (g) (h) Figure 2. Reference test images: (a) 38092, (b) 126007, (c) 170057, (d) 182053, (e) 219090, (f) 253027, (g) 295087, and (h) 296007 (a) (b) Figure 3. Masks used for image inpainting e xperiments: (a) Mask1 and (b) Mask2 Edg e-awar e adaptive partial dif fer ential equation ima g e inpainting for ... (F ouzia Hamadac he ) Evaluation Warning : The document was created with Spire.PDF for Python.
1716 ❒ ISSN: 1693-6930 4.1. Quantitati v e e v aluation T able 1 presents the quantitati v e comparison between the trimmed median method [24], the adapti v e Euler elastica approach [25], and the proposed method under both masking scenarios. T able 1, sho ws that the proposed method consistently achie v es the highest PSNR and SSIM v alues across all test images and masking congurations. F or Mask1, it preserv es ne det ails and structural continuity . F or Mask2, t he adapti v e transport amplication impro v es isophote propag ation, leading to better geometric reconstruction compared to the baseline methods. The a v erage results further conrm the superiority of the proposed approach. Under Mask1, it attains the highest mean PSNR (30.1343 dB) and SSIM (0.93657), outperforming both the trimmed median and adap- ti v e Euler elastica methods. The same trend is observ ed for Mask2, where the proposed method ac hie v es the best mean PSNR (30.8245 dB) and SSIM (0.94684). Ov erall, the cons istent impro v ement in quantitati v e met- rics demonstrates the rob ustness and enhanced re construction capability of the proposed inpainting frame w ork. T able 1. Quantitati v e comparison of PSNR (dB) and SSIM v alues for image inpainting using trimmed median, adapti v e Euler elastica, and the proposed method under Mask1 and Mask2 Image.jpg Mask T rimmed median Adapti v e Euler elastica Proposed PSNR (dB) SSIM PSNR (dB) SSIM PSNR (dB) SSIM 38092 Mask1 27.7560 0.91871 27.8948 0.87860 29.4995 0.92764 Mask2 27.1893 0.92610 27.3959 0.88326 29.1254 0.93389 126007 Mask1 27.9483 0.93879 29.6092 0.92130 29.9560 0.94501 Mask2 29.9318 0.95066 30.8467 0.93069 31.7043 0.95654 170057 Mask1 31.0474 0.92774 31.6092 0.92947 32.3449 0.94236 Mask2 31.9959 0.93604 32.7071 0.93969 33.4784 0.95311 182053 Mask1 26.0871 0.92384 25.4707 0.89990 27.0330 0.93153 Mask2 27.2204 0.94050 26.2018 0.91670 28.2300 0.94900 219090 Mask1 28.5245 0.93253 28.1673 0.89501 29.9536 0.93889 Mask2 27.9201 0.93285 28.0051 0.89735 29.2746 0.94200 253027 Mas k1 24.1626 0.90953 23.8170 0.88230 25.1050 0.91832 Mask2 23.0839 0.91238 23.4064 0.89180 24.3400 0.92700 295087 Mas k1 30.1797 0.93094 28.6956 0.88094 30.7250 0.93349 Mask2 31.2906 0.94311 29.2801 0.89043 32.0997 0.94574 296007 Mas k1 35.4982 0.95112 34.1768 0.91889 36.4572 0.95532 Mask2 37.1673 0.96310 35.2960 0.92868 38.3438 0.96681 Mean Mask1 28.9005 0.92912 28.6801 0.90084 30.1343 0.93657 Mask2 29.4749 0.93810 29.1424 0.90991 30.8245 0.94684 4.2. Sensiti vity analysis T o further v alidate the rob ustness of the proposed adapt i v e formulation, a sensiti vity analysis w as conducted with respect to the main adapt i v e parameters: the transport amplication f actor α and the dif fusion scaling parameter β . 4.2.1. Inuence of the transport amplication factor The transport amplication f actor α controls the strength of isophote propag ation, with α = 1 cor - responding to the classical Bertalmio formulation. Experimental e v aluation sho ws that increasing α enhances reconstruction quality up to an optimal range, be yond which minor oscillatory ef fects may slightly reduce performance. These results conrm that the adapti v e transport mechanism ef fecti v ely strengthens structural continuation while maintaining numerical stability when appropriately bounded. 4.2.2. Inuence of the diffusion scaling parameter The dif fusion scaling parameter β re gulates edge preserv ation thr o ugh the adapted dif fusion coef - cient. Experiments sho w that moderate reduction of dif fusion strength enhances edge sharpness and structural consistenc y . Excessi v e reduction may introduce slight te xture discontinuities, whereas lar ger dif fusion v al- ues produce smoother b ut slightly bl urred reconstructions. The results conrm that the best trade-of f between smoothing and edge preserv ation is obtained for intermediate β v alues. 4.2.3. Con v er gence beha vior The con v er gence criterion ∥ I k +1 − I k ∥ 2 ∥ I k ∥ 2 < 10 − 6 w as satised in all e xperiments within fe wer than 300 iterations. No instability w as observ ed within the tested parameter ranges, conrming the numerical rob ustness TELK OMNIKA T elecommun Comput El Control, V ol. 24, No. 5, October 2026: 1710–1720 Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA T elecommun Comput El Control ❒ 1717 of the adapti v e transport–dif fusion coupling. 4.3. Qualitati v e e v aluation F or qualitati v e e v aluation, three test images (IDs: 38092, 170057, 295087) were analyzed under the Mask1 and Mask2 congurations. Figures 4 and 5 present representati v e visual reconstruction results obtained with the proposed method, in comparison to the trimmed median and adapti v e Euler elastica approaches. In each gure, images from left to right correspond to the corrupted input, the trimmed median inpainting, the adapti v e Euler elastica method, and the proposed method. Figure 4. Inpainting results for test images 38092, 170057, and 295087. Left to right: corrupted image with Mask 1, restored by trimmed median, adapti v e Euler elastica, and proposed method Figure 5. Inpainting results for test images 38092, 170057, and 295087. Left to right: corrupted image with Mask 2, restored by trimmed median, adapti v e Euler elastica, and proposed method Qualitati v e analysis sho ws that the proposed method achie v es smoother re gion lling, better edge continuity , and impro v ed preserv ation of ne te xtures and structural details. By contrast, the trimmed median method often introduces noticeable blurring, particularly in re gions wit h ne struct ures, while the a dapti v e Eu- ler elastica approach may generate minor artif acts in te xtured areas. Ov erall, the proposed frame w ork produces more coherent and visually natural reconstructions, with edges and te xtures consistently maintained across both Mask1 and Mask2 scenarios. These qualitati v e observ ations align well with the quantitati v e impro v ements re- ported in T able 1, conrming the ef fecti v eness of the proposed approach in preserving structural and te xtural delity under dif ferent masking conditions. Edg e-awar e adaptive partial dif fer ential equation ima g e inpainting for ... (F ouzia Hamadac he ) Evaluation Warning : The document was created with Spire.PDF for Python.
1718 ❒ ISSN: 1693-6930 5. CONCLUSION In this study , we presented an enhanced edge-a w are adapti v e PDE-based model for image inpainting. By rening the balance between transport and dif fusion processes through adapti v e parameters, our approach ef fecti v ely o v ercomes the blurring artif acts commonly associated with traditional Bertalmio-type models. Ex- perimental results sho w consistent impro v ements in PSNR and SSIM, demonstrating enhanced structural re- construction and edge pres erv ation compared with the benchmark methods. Thes e quantitati v e ndings are supported by qualitati v e visual analysis, which conrms that the proposed method achie v es smoother re gion lling, better edge continuity , and impro v ed preserv ation of ne te xtures and structural details. Despite these promising results, the proposed method has some limitations. The method is currently limited to grayscale images, and the number of test i mages remains relati v ely restricted. Future w ork will e xplore h ybrid approaches that inte grate the proposed adapti v e PDE frame w ork with deep learning models to address comple x te xtures and lar ge semantic g aps in high-resolution images. Extending the frame w ork to color images, i ncreasing the dataset size for more comprehensi v e v alidation, and comparing with e x emplar -based methods, P atchMatch, and deep learning-based inpainting are also among our research directions. A CKNO WLEDGMENTS The authors w ould lik e to thank their home institutions for support. FUNDING INFORMA TION Authors state no funding in v olv ed. A UTHOR CONTRIB UTIONS ST A TEMENT This journal uses the Contrib utor Roles T axonomy (CRediT) to recognize indi vidual author contrib u- tions, reduce authorship disputes, and f acilitate collaboration. Name of A uthor C M So V a F o I R D O E V i Su P Fu F ouzia Hamadache ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ Zine-Eddine Baarir ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ Nacira Dif fellah ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ Abdelmalik Ahmed-T aleb ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ T e wk Bekk ouche ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ Rabah Hamdini ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ C : C onceptualization I : I n v estig ation V i : V i sualization M : M ethodology R : R esources Su : Su pervision So : So ftw are D : D ata Curation P : P roject Administrati on V a : V a lidation O : Writing - O riginal Draft Fu : Fu nding Acquisition F o : F o rmal Analysis E : Writing - Re vie w & E diting CONFLICT OF INTEREST ST A TEMENT Authors state no conict of interest. D A T A A V AILABILITY The data supporting the ndings of this study are openly a v ailable from the Berk ele y Se gmentation Dataset and Benchmark (BSDS300), originally introduced by Martin et al. (2001) [26]. The dataset used in this study can be accessed publicly through the of cial Berk ele y Se gmentation Dataset and Benchmark website at https://www2.eecs.berk ele y .edu/Research/Projects/CS/vision/bsds/BSDS300/html/dataset/images.html. TELK OMNIKA T elecommun Comput El Control, V ol. 24, No. 5, October 2026: 1710–1720 Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA T elecommun Comput El Control ❒ 1719 REFERENCES [1] O. Elharrouss, N. Almaadeed, S. Al-Maadeed, and Y . Akbari, “Image Inpainting: A Re vie w , ” Neur al Pr ocessing Letter s , v ol. 51, no. 2, pp. 2007–2028, 2020, doi: 10.1007/s11063-019-10163-0. [2] H. Li, L. Hu, J. Liu, J. Zhang, and T . Ma, “ A re vie w of adv ances in image inpainting research, ” Ima ging Science J ournal , v ol. 72, no. 5, pp. 669–691, 2024, doi: 10.1080/13682199.2023.2212572. [3] J. Li, F . He, L. Zhang, B. Du, and D. T ao, “Progressi v e reconstruction of visual structure for image inpainting, ” in Pr oceedings of the IEEE International Confer ence on Computer V ision , 2019, pp. 5961–5970, doi: 10.1109/ICCV .2019.00606. [4] R. Zhang, W . Quan, Y . Zhang, J. W ang, and D. M. Y an, “W -Net: Structure and T e xture Interaction for Image Inpainting, ” IEEE T r ansactions on Multimedia , v ol. 25, pp. 7299–7310, 2023, doi: 10.1109/TMM.2022.3219728. [5] Q. W ang, S. He, M. Su, and F . Zhao, “Image Inpainting Methods: A Re vie w of Deep Learning Approaches, ” Symmetry , v ol. 18, no. 1, p. 94, 2026, doi: 10.3390/sym18010094. [6] D. C. P . De vshri, “Re vie w of P artial Dif ferential Equations in Image Processing: Edge Detection, Restoration, and Be yond, ” Inter - national J ournal of Resear c h & T ec hnolo gy , v ol. 12, no. 4, pp. 92–99, 2024. [7] M. Bertalmio, G. Sapiro, V . Caselles, and C. Ballester , “Image Inpainting, ” in SIGGRAPH 2000 - Pr oceedings of the 27th Annual Confer ence on Computer Gr aphics and Inter active T ec hniques , 2000, pp. 417–424, doi: 10.1145/344779.344972. [8] A. K. Al-Jaberi and E. M. Hameed, “ A Re vie w of PDE Based Local Inpainting Methods, ” in J ournal of Physics: Confer ence Series , 2021, p. 12149, doi: 10.1088/1742-6596/1818/1/012149. [9] C. Ballester , M. Bertalm ´ ıo, V . Caselles, L. Garrido, A. Marques, and F . Ranchin, “ An inpainting- based deinterlacing method, ” IEEE T r ansactions on Ima g e Pr ocessing , v ol. 16, no. 10, pp. 2476–2491, 2007, doi: 10.1109/TIP .2007.903844. [10] M. Bertalm ´ ıo, V . Caselles, S. Masnou, and G. Sapiro, “Inpainting, ” in Computer V isi on: A Refer ence Guide , Springer , 2021, pp. 670–687, doi: 10.1007/978-3-030-63416-2 249. [11] J. W eick ert, “ECMI Anisot ropic Dif fusion in Image Processing, ” Eur opean Consortium for Mathematics in Industry , v ol. 1, p. 165, 1998. [12] T . F . Chan and J. Shen, “Nonte xture Inpainting by Curv ature-Dri v en Dif fusions, ” J ournal of V isual Communication and Ima g e Repr esentation , v ol. 12, no. 4, pp. 436–449, Dec. 2001, doi: 10.1006/jvci.2001.0487. [13] H. Li, W . Luo, and J. Huang, “Localization of Dif fusion-Based Inpainting in Digital Images, ” IEEE T r ansactions on Information F or ensics and Security , v ol. 12, no. 12, pp. 3050–3064, 2017, doi: 10.1109/TIFS.2017.2730822. [14] R. A. Frick and M. Steinebach, “Dif fSe g: T o w ards Detecting Dif fusion-Based Inpainting Attacks Using Multi-Feature Se gmenta- tion, ” in IEEE Computer Society Confer ence on Computer V ision and P attern Reco gnition W orkshops , 2024, pp. 3802–3808, doi: 10.1109/CVPR W63382.2024.00384. [15] A. Criminisi, P . P ´ erez, and K. T o yama, “Re gion lling and object remo v al by e x emplar -based image inpainting, ” in IEEE T r ansac- tions on Ima g e Pr ocessing , 2004, pp. 1200–1212, doi: 10.1109/TIP .2004.833105. [16] C. Barnes, E. Shechtman, A. Fink elstein, and D. B. Goldman, “P atchMatch: A randomized correspondence algorithm for structural image editing, ” in A CM T r ansactions on Gr aphics, A CM , 2009, pp. 1–11, doi: 10.1145/1531326.1531330. [17] S. Lou, Q. F an, F . Chen, C. W ang, and J. Li, “Preliminary in v estig ation on single remote se nsing image inpainting through a modied GAN, ” in 2018 10th IAPR W orkshop on P attern Reco gnition in Remote Sensing , PRRS 2018 , 2018, pp. 1–6, doi: 10.1109/PRRS.2018.8486163. [18] H. Liu, X. Bi, G. Lu, and W . W ang, “Ex emplar -Based Image Inpainting with Multi-Resolution Information and the Graph Cut T echnique, ” IEEE Access , v ol. 7, pp. 101641–101657, 2019, doi: 10.1109/A CCESS.2019.2931064. [19] W . Casaca, M. Boa v entura, M. P . De Almeida, and L. G. Nonat o, “Combining anisotropic dif fusion, transport equation and te xture synthesis for inpainting te xtured images, ” P attern Reco gnition Letter s , v ol. 36, pp. 36–45, 2014, doi: 10.1016/j.patrec.2013.08.023. [20] L. Liao, J. Xiao, Z. W ang, C. W . Lin, and S. Satoh, “Image Inpainting Guided by Coherence Priors of Semantics and T e xtures, ” in Pr oceedings of the IEEE Computer Society Confer ence on Computer V ision and P attern Reco gnition, Los Alamitos, CA, USA: IEEE Computer Society , Jun. 2021, pp. 6535–6544, doi: 10.1109/CVPR46437.2021.00647. [21] K. Nazeri, E. Ng, T . Joseph, F . Z. Qureshi, and M. Ebrahimi, “EdgeConnect: Generati v e Image Inpainting with Adv ersarial Edge Learning, ” arXiv pr eprint arXi v:1901.00212, 2019, doi: 10.48550/arXi v .1901.00212. [22] W . Quan, J. Chen, Y . Liu, D. M. Y an, and P . W onka, “Deep Learning-Based Image and V ideo Inpainting: A Surv e y , ” International J ournal of Computer V ision , v ol. 132, no. 7, pp. 2367–2400, 2024, doi: 10.1007/s11263-023-01977-6. [23] Z. Xu et al. , “ A Re vie w of Image Inpainting Methods Based on Deep Learning, ” Applied Sciences (Switzerland), v ol. 13, no. 20, p. 11189, 2023, doi: 10.3390/app132011189. [24] D. N. H. Thanh, N. V an Son, and V . B. S. Prasath, “Distorted image reconstruction method with trimmed median, ” in Pr oceedings - 2019 3r d International Confer ence on Recent Advances in Signal Pr ocessing , T elecommunications and Computing , SigT elCom 2019, 2019, pp. 58–62, doi: 10.1109/SIGTELCOM.2019.8696138. [25] D. N. H. Thanh, V . B. S. Prasath, S. Dv oenk o, and L. M. Hieu, “ An adapti v e image inpainting method based on euler’ s elas- tica with adapti v e parameters estimation and the discrete gradient method, ” Signal Pr ocessing , v ol. 178, p. 107797, 2021, doi: 10.1016/j.sigpro.2020.107797. [26] D. Martin, C. F o wlk es, D. T al, and J. Malik, “The Berk ele y Se gmentation Dataset [online], ” pp. 416–423, 2001. [Online]. A v ailable: https://www2.eecs.berk ele y .edu/Research/Projects/CS/vision/bsds/BSDS300/html/dataset/images.html [27] A. V ishw akarma, “Denoising and Inpainting of Sona r Images Using Con v olutional Sparse Representation, ” IEEE T r ansactions on Instrumentation and Measur ement , v ol. 72, pp. 1–9, 2023, doi: 10.1109/TIM.2023.3246527. [28] S . Xu, W . Xiang, C. Lv , S. W ang, and G. Liu, “Di v ersied Image Inpainting with T ransformers and Denoising Iterati v e Renement, ” IEEE Access , v ol. 12, pp. 187068–187080, 2024, doi: 10.1109/A CCESS.2024.3514930. [29] O . Elharrouss, R. Damseh, A. N. Belkacem, E. Badidi, and A. Lakas, “T ransformer -based image and video inpainting: current challenges and future directions, ” Articial Intellig ence Re vie w , v ol. 58, no. 4, p. 124, 2025, doi: 10.1007/s10462-024-11075-9. [30] Z. W ang, A. C. Bo vik, H. R. Sheikh, and E. P . Simoncelli, “Image quality ass essment: From error visibility to structural similarity , ” IEEE T r ansactions on Ima g e Pr ocessing , v ol. 13, no. 4, pp. 600–612, 2004, doi: 10.1109/TIP .2003.819861. [31] Y . Xing et al. , “T ime-V ariant Image Inpainting via Interacti v e Distrib ution T ransition Estimation, ” IEEE T r ansactions on Ima g e Pr ocessing , v ol. 35, pp. 5107–5122, 2026, doi: 10.1109/TIP .2026.3687440. Edg e-awar e adaptive partial dif fer ential equation ima g e inpainting for ... (F ouzia Hamadac he ) Evaluation Warning : The document was created with Spire.PDF for Python.