Superior reconstruction by IFT Instead of that, for every point t , the IFT finds a path from a regional minimum in V 0 (component X ) whose maximum altitude to reach t along that path is minimum. X 1 0 1 1 I = ( D I , I ) V 0 = ( D I , V 0 ) V = ( D I , V ) Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction by IFT The IFT minimizes V ( t ) = ∀ π t ∈ Π( D I , A 1 , t ) { f srec ( π t ) } min where f srec is defined by f srec ( � t � ) = V 0 ( t ) f srec ( π s · � s , t � ) = max { f srec ( π s ) , I ( t ) } . Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction by IFT Indeed, the problem could also be easily solved without the closing operation, by marker imposition � I ( t ) if t ∈ S , V 0 ( t ) = + ∞ otherwise, where S represents seed spels (e.g., the border of I ). Original image of a carcinoma. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction by IFT Indeed, the problem could also be easily solved without the closing operation, by marker imposition � I ( t ) if t ∈ S , V 0 ( t ) = + ∞ otherwise, where S represents seed spels (e.g., the border of I ). Original image of a carcinoma. Its binarization. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction by IFT Indeed, the problem could also be easily solved without the closing operation, by marker imposition � I ( t ) if t ∈ S , V 0 ( t ) = + ∞ otherwise, where S represents seed spels (e.g., the border of I ). Original image of a carcinoma. Its binarization. A closing of basins (marker imposition). Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction by IFT Indeed, the problem could also be easily solved without the closing operation, by marker imposition � I ( t ) if t ∈ S , V 0 ( t ) = + ∞ otherwise, where S represents seed spels (e.g., the border of I ). Original image of a carcinoma. Its binarization. A closing of basins (marker imposition). Its residue. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction by IFT Indeed, the problem could also be easily solved without the closing operation, by marker imposition � I ( t ) if t ∈ S , V 0 ( t ) = + ∞ otherwise, where S represents seed spels (e.g., the border of I ). Original image of a carcinoma. Its binarization. A closing of basins (marker imposition). Its residue. An opening by reconstruction. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Inferior reconstruction by IFT Similarly, the inferior reconstruction of I from V 0 requires ≤ V 0 ( t ) I ( t ) for all t ∈ D I in order to eliminate domes rather than basins. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Inferior reconstruction by IFT Similarly, the inferior reconstruction of I from V 0 requires ≤ V 0 ( t ) I ( t ) for all t ∈ D I in order to eliminate domes rather than basins. In this case, for every point t , the IFT finds a path from a regional maxima in V 0 whose minimum altitude to reach t along that path is maximum. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Inferior reconstruction by IFT The IFT maximizes V ( t ) = ∀ π t ∈ Π( D I , A 1 , t ) { f irec ( π t ) } max for path function f irec defined by f irec ( � t � ) = V 0 ( t ) f irec ( π s · � s , t � ) min { f irec ( π s ) , I ( t ) } . = Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Inferior reconstruction by IFT The IFT maximizes V ( t ) = ∀ π t ∈ Π( D I , A 1 , t ) { f irec ( π t ) } max for path function f irec defined by f irec ( � t � ) = V 0 ( t ) f irec ( π s · � s , t � ) min { f irec ( π s ) , I ( t ) } . = Marker imposition using a set S of seed spels is also valid. � I ( t ) if t ∈ S , V 0 ( t ) = −∞ otherwise. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior and inferior reconstructions Therefore, we define the superior reconstruction by Ψ srec ( I , V 0 , A 1 ) , V 0 ≥ I , Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior and inferior reconstructions Therefore, we define the superior reconstruction by Ψ srec ( I , V 0 , A 1 ) , V 0 ≥ I , the inferior reconstruction by Ψ irec ( I , V 0 , A 1 ) , V 0 ≤ I . Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior and inferior reconstructions Therefore, we define the superior reconstruction by Ψ srec ( I , V 0 , A 1 ) , V 0 ≥ I , the inferior reconstruction by Ψ irec ( I , V 0 , A 1 ) , V 0 ≤ I . The way V 0 is created gives other specific names to them. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior and inferior reconstructions For instance, Closing by reconstruction: V 0 = Ψ C ( I , A r ). Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior and inferior reconstructions For instance, Closing by reconstruction: V 0 = Ψ C ( I , A r ). Opening by reconstruction: V 0 = Ψ O ( I , A r ). Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior and inferior reconstructions For instance, Closing by reconstruction: V 0 = Ψ C ( I , A r ). Opening by reconstruction: V 0 = Ψ O ( I , A r ). h -Basins: residue Ψ srec ( I , V 0 ) − I , V 0 = I + h , and h ≥ 1. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior and inferior reconstructions For instance, Closing by reconstruction: V 0 = Ψ C ( I , A r ). Opening by reconstruction: V 0 = Ψ O ( I , A r ). h -Basins: residue Ψ srec ( I , V 0 ) − I , V 0 = I + h , and h ≥ 1. h -domes: residue I − Ψ irec ( I , V 0 ), V 0 = I − h , and h ≥ 1. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior and inferior reconstructions For instance, Closing by reconstruction: V 0 = Ψ C ( I , A r ). Opening by reconstruction: V 0 = Ψ O ( I , A r ). h -Basins: residue Ψ srec ( I , V 0 ) − I , V 0 = I + h , and h ≥ 1. h -domes: residue I − Ψ irec ( I , V 0 ), V 0 = I − h , and h ≥ 1. Closing of basins or opening of domes: V 0 is created by marker imposition. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Levelings Superior and inferior reconstructions can also be combined into a leveling transformation to correct edge blurring created by linear smoothing [6]. Original image. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Levelings Superior and inferior reconstructions can also be combined into a leveling transformation to correct edge blurring created by linear smoothing [6]. Original image. Regular Gaussian filtering. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Levelings Superior and inferior reconstructions can also be combined into a leveling transformation to correct edge blurring created by linear smoothing [6]. Original image. Regular Gaussian filtering. Leveling transformation. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Levelings This leveling operator uses the following sequence of transformations from I and the impaired image V 0 . Algorithm – Leveling algorithm 1. X ← Ψ D ( V 0 , A 1 ) ∩ I . 2. I R ← Ψ iref ( I , X , A 1 ) . 3. Y ← Ψ E ( I , A 1 ) ∪ I R . 4. S R ← Ψ srec ( I R , Y , A 1 ) . Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction computation For superior reconstruction: First, all nodes t ∈ D I are trivial paths with initial connectivity values V 0 ( t ). Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction computation For superior reconstruction: First, all nodes t ∈ D I are trivial paths with initial connectivity values V 0 ( t ). The initial roots are identified at the global minima of V 0 ( t ). Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction computation For superior reconstruction: First, all nodes t ∈ D I are trivial paths with initial connectivity values V 0 ( t ). The initial roots are identified at the global minima of V 0 ( t ). They may conquer their adjacent nodes by offering them better paths. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction computation For superior reconstruction: First, all nodes t ∈ D I are trivial paths with initial connectivity values V 0 ( t ). The initial roots are identified at the global minima of V 0 ( t ). They may conquer their adjacent nodes by offering them better paths. The process continues from the adjacent nodes in a non-decreasing order of path values. if max { f srec ( π s ) , I ( t ) } < f srec ( π t ) then π t ← π s · � s , t � . Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction computation For superior reconstruction: First, all nodes t ∈ D I are trivial paths with initial connectivity values V 0 ( t ). The initial roots are identified at the global minima of V 0 ( t ). They may conquer their adjacent nodes by offering them better paths. The process continues from the adjacent nodes in a non-decreasing order of path values. if max { f srec ( π s ) , I ( t ) } < f srec ( π t ) then π t ← π s · � s , t � . Essentially the regional minima in V 0 ( t ) compete among themselves and some of them become roots (i.e., minima in V ( t )). Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction computation The optimum-path forest with filtered values V ( t ) (right) resulting from the superior reconstruction of I = ( D I , I ) (left) from marker V 0 = ( D I , V 0 ) (center) contains unconquered regions (black dots) and the winner regional minima (red dots) as roots. �� �� 8 30 8 25 10 10 � � 3 8 25 ��������� ��������� ��� ��� � � ��� ��� � � � � 20 ��� ��� � � 5 20 8 � � 15 15 ��� ��� 25 20 20 � � 30 ��� ��� 30 40 5 � � 5 0 ���� ���� 10 15 10 �� �� � � ����� ����� 20 20 20 �� �� 8 8 �� �� ���� ���� �� �� 4 10 5 10 12 10 ���� ���� 5 Images I (left), V 0 (center), and V (right). Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction algorithm Algorithm – Superior reconstruction algorithm 1. For each t ∈ D I , do 2. Set V ( t ) ← V 0 ( t ) . 3. If V ( t ) � = + ∞ , then insert t in Q. 4. While Q is not empty, do 5. Remove from Q a spel s such that V ( s ) is minimum. For each t ∈ A 1 ( s ) such that V ( t ) > V ( s ) , do 6. Compute tmp ← max { V ( s ) , I ( t ) } . 7. 8. If tmp < V ( t ) , then If V ( t ) � = + ∞ , remove t from Q. 9. 10. Set V ( t ) ← tmp. 11. Insert t in Q. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Organization of this lecture Basic definitions. Superior and inferior reconstructions. Their relation with watershed-based segmentation. Fast binary filtering. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction and watershed transform Suppose we make a hole in each minimum of an image I and submerge its surface in a lake, such that each hole starts a flooding with water of different color. A watershed segmentation is obtained by preventing the mix of water from different colors. Original image I . Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction and watershed transform Suppose we make a hole in each minimum of an image I and submerge its surface in a lake, such that each hole starts a flooding with water of different color. A watershed segmentation is obtained by preventing the mix of water from different colors. Original image I . IFT-watershed segmentation. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction and watershed transform Suppose we make a hole in each minimum of an image I and submerge its surface in a lake, such that each hole starts a flooding with water of different color. A watershed segmentation is obtained by preventing the mix of water from different colors. Original image I . IFT-watershed segmentation. Classical watershed segmentation requires to detect and label each minimum before the flooding process. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction and watershed transform During superior reconstruction, we may force each regional minimum in I to produce a single optimum-path tree in P with a distinct label in L . Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction and watershed transform During superior reconstruction, we may force each regional minimum in I to produce a single optimum-path tree in P with a distinct label in L . By definition, the resulting optimum-path forest is a watershed segmentation. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction and watershed transform During superior reconstruction, we may force each regional minimum in I to produce a single optimum-path tree in P with a distinct label in L . By definition, the resulting optimum-path forest is a watershed segmentation. Moreover, by choice of V 0 , we may also eliminate the influence zones of “irrelevant” minima and considerably reduce the over-segmentation problem. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction and watershed transform During superior reconstruction, we may force each regional minimum in I to produce a single optimum-path tree in P with a distinct label in L . By definition, the resulting optimum-path forest is a watershed segmentation. Moreover, by choice of V 0 , we may also eliminate the influence zones of “irrelevant” minima and considerably reduce the over-segmentation problem. A change of topology in Ψ srec ( I , V 0 , A r ) for r > 1 also helps on that. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction and watershed transform This requires a simple modification in f srec . � I ( t ) if t ∈ R , f srec ( � t � ) = V 0 ( t ) + 1 otherwise, f srec ( π s · � s , t � ) = max { f srec ( π s ) , I ( t ) } , where R is found on-the-fly with a single root for each regional minimum of the filtered image V . The condition V 0 ( t ) + 1 > I ( t ) guarantees that all spels in D I will be conquered. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction and watershed transform The choice of V 0 ( t ) = I ( t ) + h , h ≥ 0 will preserve all minima of I whose basins have depth greater than h . For h = 0, all minima will be preserved. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction and watershed transform The choice of V 0 ( t ) = I ( t ) + h , h ≥ 0 will preserve all minima of I whose basins have depth greater than h . For h = 0, all minima will be preserved. 20 20 23 20 18 21 5 8 5 10 13 10 (b) (c) (a) (a) Image I . (b) Image V 0 + 1 for h = 2. (c) Image V = Ψ srec ( I , V 0 , A 1 ) with indication of optimum paths in P . Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Superior reconstruction and watershed transform The choice of V 0 ( t ) = I ( t ) + h , h ≥ 0 will preserve all minima of I whose basins have depth greater than h . For h = 0, all minima will be preserved. 18 20 21 20 18 19 20 5 6 5 10 11 10 (b) (c) (a) (a) Image I . (b) Image V 0 + 1 for h = 0. (c) Image V = Ψ srec ( I , V 0 , A 1 ) with indication of optimum paths in P . Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Watershed from grayscale marker For grayscale images V 0 , the simultaneous computation of a superior reconstruction in V and a watershed segmentation in L is called watershed from grayscale marker [4]. MR-image of a wrist. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Watershed from grayscale marker For grayscale images V 0 , the simultaneous computation of a superior reconstruction in V and a watershed segmentation in L is called watershed from grayscale marker [4]. MR-image of a wrist. A gradient image I . Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Watershed from grayscale marker For grayscale images V 0 , the simultaneous computation of a superior reconstruction in V and a watershed segmentation in L is called watershed from grayscale marker [4]. MR-image of a wrist. A gradient image I . The closing V 0 = Ψ C ( I , A 2 . 5 ). Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Watershed from grayscale marker For grayscale images V 0 , the simultaneous computation of a superior reconstruction in V and a watershed segmentation in L is called watershed from grayscale marker [4]. MR-image of a wrist. A gradient image I . The closing V 0 = Ψ C ( I , A 2 . 5 ). Segmentation in L for Ψ srec ( I , V 0 , A 3 . 5 ). Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Watershed from grayscale marker Algorithm – Watershed from Grayscale Marker 1. For each t ∈ D I , do 2. Set P ( t ) ← nil, λ ← 1 , and V ( t ) ← V 0 ( t ) + 1 . 3. Insert t in Q. 4. While Q is not empty, do 5. Remove from Q a spel s such that V ( s ) is minimum. If P ( s ) = nil then set V ( s ) ← I ( s ) , L ( s ) ← λ , and λ ← λ + 1 . 6. For each t ∈ A ( s ) such that V ( t ) > V ( s ) , do 7. Compute tmp ← max { V ( s ) , I ( t ) } . 8. 9. If tmp < V ( t ) , then 10. Set P ( t ) ← s, V ( t ) ← tmp, L ( t ) ← L ( s ) . 11. Update position of t in Q. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Organization of this lecture Basic definitions. Superior and inferior reconstructions. Their relation with watershed-based segmentation. Fast binary filtering. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Fast binary filtering via IFT For binary images I and Euclidean relations A r , it is also possible to exploit the IFT for fast computation of morphological operators, which can be decomposed into alternate sequences of erosions and dilations (or vice-versa). For instance, Ψ C ( I , A r ) = Ψ E (Ψ D ( I , A r ) , A r ) . Ψ CO ( I , A r ) = Ψ D (Ψ E (Ψ E (Ψ D ( I , A r ) , A r ) , A r ) , A r ) = Ψ D (Ψ E (Ψ D ( I , A r ) , A 2 r ) , A r ) . Ψ CO (Ψ CO ( I , A r ) , A 2 r ) Ψ D (Ψ E (Ψ D (Ψ E (Ψ D ( I , A r ) , A 2 r ) , = A 3 r ) , A 4 r ) , A 2 r ) . Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Fast binary filtering via IFT The basic idea is to extract the object’s (background’s) border S , Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Fast binary filtering via IFT The basic idea is to extract the object’s (background’s) border S , compute their propagation in sub-linear time outward (inward) the object for dilation (erosion), alternately. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Fast binary filtering via IFT The basic idea is to extract the object’s (background’s) border S , compute their propagation in sub-linear time outward (inward) the object for dilation (erosion), alternately. Each border propagation stops at the adjacency radius specified for dilation (erosion). Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Fast binary filtering via IFT This requires to constrain the computation of an Euclidean distance transform (EDT) either outside (dilation) or inside (erosion) the object up to a distance r from it. The EDT assigns to every spel in D I its distance to the closest spel in a given set S ⊂ D I (e.g., the object’s or background’s border). Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Fast binary filtering via IFT This requires to constrain the computation of an Euclidean distance transform (EDT) either outside (dilation) or inside (erosion) the object up to a distance r from it. r The EDT assigns to every spel in D I its distance to the closest spel in a given set S ⊂ D I (e.g., the object’s or background’s border). Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Fast binary filtering via IFT A spel s ∈ D I belongs to an object’s border S , when I ( s ) = 1 and ∃ t ∈ A 1 ( s ), such that I ( t ) = 0. Similar definition applies to backgroud’s border. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Fast binary filtering via IFT A spel s ∈ D I belongs to an object’s border S , when I ( s ) = 1 and ∃ t ∈ A 1 ( s ), such that I ( t ) = 0. Similar definition applies to backgroud’s border. For dilation, the value 1 is propagated to every spel t with value I ( t ) = 0 and distance � t − R ( π t ) � 2 ≤ r 2 , R ( π t ) ∈ S . Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Fast binary filtering via IFT A spel s ∈ D I belongs to an object’s border S , when I ( s ) = 1 and ∃ t ∈ A 1 ( s ), such that I ( t ) = 0. Similar definition applies to backgroud’s border. For dilation, the value 1 is propagated to every spel t with value I ( t ) = 0 and distance � t − R ( π t ) � 2 ≤ r 2 , R ( π t ) ∈ S . For erosion, the value 0 is propagated to every spel t with value I ( t ) = 1 and distance � t − R ( π t ) � 2 ≤ r 2 , R ( π t ) ∈ S . Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Fast binary filtering via IFT A spel s ∈ D I belongs to an object’s border S , when I ( s ) = 1 and ∃ t ∈ A 1 ( s ), such that I ( t ) = 0. Similar definition applies to backgroud’s border. For dilation, the value 1 is propagated to every spel t with value I ( t ) = 0 and distance � t − R ( π t ) � 2 ≤ r 2 , R ( π t ) ∈ S . For erosion, the value 0 is propagated to every spel t with value I ( t ) = 1 and distance � t − R ( π t ) � 2 ≤ r 2 , R ( π t ) ∈ S . During dilation (erosion), spels t whose distance � t − R ( π t ) � 2 > r 2 but � P ( t ) − R ( π t ) � 2 ≤ r 2 are stored in a new set S ′ for a subsequent erosion (dilation) operation. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Fast binary filtering via IFT The EDT is propagated in V from a set S ⊂ D I to every spel t ∈ D I in a non-decreasing order of squared distance using A √ 2 in 2D (8-neighbors) [7]. For fast dilation, it uses path function 0 if t ∈ S , f euc ( � t � ) + ∞ = if I ( t ) = 0, −∞ otherwise. � t − R ( π s ) � 2 . f euc ( π s · � s , t � ) = Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Fast binary filtering via IFT For fast erosion, it uses path function 0 if t ∈ S , f euc ( � t � ) = + ∞ if I ( t ) = 1, −∞ otherwise. � t − R ( π s ) � 2 . f euc ( π s · � s , t � ) = A dilated (eroded) binary image J = ( D I , J ) is created during the distance propagation process. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Fast dilation Algorithm – Fast Dilation in 2D up to distance r from S 1. For each t ∈ D I , set J ( t ) ← I ( t ) , R ( π t ) ← t and V ( t ) ← f euc ( � t � ) . 2. While S � = ∅ , remove t from S and insert t in Q. 3. While Q is not empty, do 4. Remove from Q a spel s such that V ( s ) is minimum. if V ( s ) ≤ r 2 , then 5. Set J ( t ) ← 1 . 6. For each t ∈ A √ 7. 2 ( s ) such that V ( t ) > V ( s ) , do Compute tmp ← � t − R ( π s ) � 2 . 8. 9. If tmp < V ( t ) , then 10. If V ( t ) � = + ∞ , remove t from Q. Set V ( t ) ← tmp and R ( π t ) ← R ( π s ) . 11. 12. Insert t in Q. Else insert s in S . 13. Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Fast binary filtering via IFT Sets S and S ′ may contain spels from multiple borders. Multiple borders, Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Fast binary filtering via IFT Sets S and S ′ may contain spels from multiple borders. Multiple borders, distances outside up to r = 10, Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Fast binary filtering via IFT Sets S and S ′ may contain spels from multiple borders. Multiple borders, distances outside up to r = 10, their dilation, Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Fast binary filtering via IFT Sets S and S ′ may contain spels from multiple borders. Multiple borders, distances outside up to r = 10, their dilation, erosion, Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Fast binary filtering via IFT Sets S and S ′ may contain spels from multiple borders. Multiple borders, distances outside up to r = 10, their dilation, erosion, closing, Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Fast binary filtering via IFT Sets S and S ′ may contain spels from multiple borders. Multiple borders, distances outside up to r = 10, their dilation, erosion, closing, closing by reconstruction, Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
Fast binary filtering via IFT Sets S and S ′ may contain spels from multiple borders. Multiple borders, distances outside up to r = 10, their dilation, erosion, closing, closing by reconstruction, opening, and Alexandre Xavier Falc˜ ao Image Processing using Graphs at ASC-SP 2010
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