Effective Component Tree Computation with Application to Pattern - - PowerPoint PPT Presentation

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives

Effective Component Tree Computation with Application to Pattern Recognition in Astronomical Imaging

Christophe Berger1 Thierry Géraud1 Roland Levillain1 Nicolas Widynski1 Anthony Baillard2 Emmanuel Bertin2

1EPITA Research and Development Laboratory (LRDE), Paris, France 2Institut d’Astrophysique de Paris (IAP), France

International Conference on Image Processing (ICIP) September 18, 2007

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 1 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives

Effective Component Tree Computation with Application to Pattern Recognition in Astronomical Imaging

1

Motivation Connected Filters The Case of Astronomical Images

2

A New Algorithm to Compute the Component Tree Tree computation Attributes Computation and Node Labeling Results and Applications

3

Conclusions and perspectives

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 2 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Context

Goal: apply connected filters from mathematical morphology to astronomical images. Features of processed astronomical images

Huge sizes (order of magnitude: 100 MB – 1.5 GB) Pixels encoded as floating-point values High dynamic ranges

⇒ New tools needed to write these filters, in particular a component tree algorithm Joint-work between IAP and LRDE, in the context of the EFIGI project (Extraction of Idealized Patterns of Galaxies in Imaging)

http://www.efigi.org

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 3 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Context

Goal: apply connected filters from mathematical morphology to astronomical images. Features of processed astronomical images

Huge sizes (order of magnitude: 100 MB – 1.5 GB) Pixels encoded as floating-point values High dynamic ranges

⇒ New tools needed to write these filters, in particular a component tree algorithm Joint-work between IAP and LRDE, in the context of the EFIGI project (Extraction of Idealized Patterns of Galaxies in Imaging)

http://www.efigi.org

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 3 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Context

Goal: apply connected filters from mathematical morphology to astronomical images. Features of processed astronomical images

Huge sizes (order of magnitude: 100 MB – 1.5 GB) Pixels encoded as floating-point values High dynamic ranges

⇒ New tools needed to write these filters, in particular a component tree algorithm Joint-work between IAP and LRDE, in the context of the EFIGI project (Extraction of Idealized Patterns of Galaxies in Imaging)

http://www.efigi.org

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 3 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Context

Goal: apply connected filters from mathematical morphology to astronomical images. Features of processed astronomical images

Huge sizes (order of magnitude: 100 MB – 1.5 GB) Pixels encoded as floating-point values High dynamic ranges

⇒ New tools needed to write these filters, in particular a component tree algorithm Joint-work between IAP and LRDE, in the context of the EFIGI project (Extraction of Idealized Patterns of Galaxies in Imaging)

http://www.efigi.org

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 3 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Context

Goal: apply connected filters from mathematical morphology to astronomical images. Features of processed astronomical images

Huge sizes (order of magnitude: 100 MB – 1.5 GB) Pixels encoded as floating-point values High dynamic ranges

⇒ New tools needed to write these filters, in particular a component tree algorithm Joint-work between IAP and LRDE, in the context of the EFIGI project (Extraction of Idealized Patterns of Galaxies in Imaging)

http://www.efigi.org

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 3 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Context

Goal: apply connected filters from mathematical morphology to astronomical images. Features of processed astronomical images

Huge sizes (order of magnitude: 100 MB – 1.5 GB) Pixels encoded as floating-point values High dynamic ranges

⇒ New tools needed to write these filters, in particular a component tree algorithm Joint-work between IAP and LRDE, in the context of the EFIGI project (Extraction of Idealized Patterns of Galaxies in Imaging)

http://www.efigi.org

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 3 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Context

Goal: apply connected filters from mathematical morphology to astronomical images. Features of processed astronomical images

Huge sizes (order of magnitude: 100 MB – 1.5 GB) Pixels encoded as floating-point values High dynamic ranges

⇒ New tools needed to write these filters, in particular a component tree algorithm Joint-work between IAP and LRDE, in the context of the EFIGI project (Extraction of Idealized Patterns of Galaxies in Imaging)

http://www.efigi.org

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 3 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Component Tree

Convenient and versatile representation of an image Parenthood relationship between nodes maps component (spatial) inclusion Applications

Classification Image Filtering Segmentation Registration Compression

f g h i j a b c d e {a, b, c, d, e, f, g, h, i, j} {a, b, f} {a, b, f} {f} {d, e, h, i} {d, i} {d}

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 4 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Component Tree

Convenient and versatile representation of an image Parenthood relationship between nodes maps component (spatial) inclusion Applications

Classification Image Filtering Segmentation Registration Compression

f g h i j a b c d e {a, b, c, d, e, f, g, h, i, j} {a, b, f} {a, b, f} {f} {d, e, h, i} {d, i} {d}

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 4 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Component Tree

Convenient and versatile representation of an image Parenthood relationship between nodes maps component (spatial) inclusion Applications

Classification Image Filtering Segmentation Registration Compression

f g h i j a b c d e {a, b, c, d, e, f, g, h, i, j} {a, b, f} {a, b, f} {f} {d, e, h, i} {d, i} {d}

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 4 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Component Tree

Convenient and versatile representation of an image Parenthood relationship between nodes maps component (spatial) inclusion Applications

Classification Image Filtering Segmentation Registration Compression

f g h i j a b c d e {a, b, c, d, e, f, g, h, i, j} {a, b, f} {a, b, f} {f} {d, e, h, i} {d, i} {d}

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 4 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Connected Filters

Properties

Rely on attributes of components (no structuring element) Simplify the images Do not create nor shift contours

Relationship with the component tree

A connected filter can be expressed as a transformation on the component tree that does not add any branch.

Recent filters (about ten years of existence)

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 5 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Connected Filters

Properties

Rely on attributes of components (no structuring element) Simplify the images Do not create nor shift contours

Relationship with the component tree

A connected filter can be expressed as a transformation on the component tree that does not add any branch.

Recent filters (about ten years of existence)

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 5 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Connected Filters

Properties

Rely on attributes of components (no structuring element) Simplify the images Do not create nor shift contours

Relationship with the component tree

A connected filter can be expressed as a transformation on the component tree that does not add any branch.

Recent filters (about ten years of existence)

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 5 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Connected Filters

Properties

Rely on attributes of components (no structuring element) Simplify the images Do not create nor shift contours

Relationship with the component tree

A connected filter can be expressed as a transformation on the component tree that does not add any branch.

Recent filters (about ten years of existence)

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 5 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Connected Filters

Properties

Rely on attributes of components (no structuring element) Simplify the images Do not create nor shift contours

Relationship with the component tree

A connected filter can be expressed as a transformation on the component tree that does not add any branch.

Recent filters (about ten years of existence)

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 5 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Connected Filters

Properties

Rely on attributes of components (no structuring element) Simplify the images Do not create nor shift contours

Relationship with the component tree

A connected filter can be expressed as a transformation on the component tree that does not add any branch.

Recent filters (about ten years of existence)

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 5 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Connected Filters

Properties

Rely on attributes of components (no structuring element) Simplify the images Do not create nor shift contours

Relationship with the component tree

A connected filter can be expressed as a transformation on the component tree that does not add any branch.

Recent filters (about ten years of existence)

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 5 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Example 1/4

Morphological Opening Using a Structural Element (disc, radius = 15 pixels)

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 6 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Example 2/4

Morphological Closing Using a Structural Element (disc, radius = 15 pixels)

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 7 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Example 3/4

Morphological Area (attribute) opening (area ≈ π152 pixels)

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 8 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Example 4/4

Morphological Area (attribute) closing (area ≈ π152 pixels)

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 9 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Acquisition and Nature of Data

The observed image is convolved by a Point-Spread Function 32-bit, floating-point values Most pixels correspond to the (noisy) sky background (dark areas) Brighter pixels: objects (stars, galaxies) and optical effects (halos, etc.).

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 10 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Acquisition and Nature of Data

The observed image is convolved by a Point-Spread Function 32-bit, floating-point values Most pixels correspond to the (noisy) sky background (dark areas) Brighter pixels: objects (stars, galaxies) and optical effects (halos, etc.).

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 10 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Acquisition and Nature of Data

The observed image is convolved by a Point-Spread Function 32-bit, floating-point values Most pixels correspond to the (noisy) sky background (dark areas) Brighter pixels: objects (stars, galaxies) and optical effects (halos, etc.).

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 10 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Acquisition and Nature of Data

The observed image is convolved by a Point-Spread Function 32-bit, floating-point values Most pixels correspond to the (noisy) sky background (dark areas) Brighter pixels: objects (stars, galaxies) and optical effects (halos, etc.).

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 10 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Quantization

Example

Histogram (log-log scale)

1 10 100 1000 10000 1 10 100 1000 10000 1e+05 1e+06 1e+07

16-bit, linear quantization Most pixels between 0 and 255 The slope that appears on this plot is due to the presence of blur. ⇒ Need for an optimal quantization;

• r

⇒ direct processing of floating-point values with no quantization

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 11 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Quantization

Example

Histogram (log-log scale)

1 10 100 1000 10000 1 10 100 1000 10000 1e+05 1e+06 1e+07

16-bit, linear quantization Most pixels between 0 and 255 The slope that appears on this plot is due to the presence of blur. ⇒ Need for an optimal quantization;

• r

⇒ direct processing of floating-point values with no quantization

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 11 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Quantization

Example

Histogram (log-log scale)

1 10 100 1000 10000 1 10 100 1000 10000 1e+05 1e+06 1e+07

16-bit, linear quantization Most pixels between 0 and 255 The slope that appears on this plot is due to the presence of blur. ⇒ Need for an optimal quantization;

• r

⇒ direct processing of floating-point values with no quantization

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 11 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Quantization

Example

Histogram (log-log scale)

1 10 100 1000 10000 1 10 100 1000 10000 1e+05 1e+06 1e+07

16-bit, linear quantization Most pixels between 0 and 255 The slope that appears on this plot is due to the presence of blur. ⇒ Need for an optimal quantization;

• r

⇒ direct processing of floating-point values with no quantization

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 11 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Connected Filters The Case of Astronomical Images

Quantization

Example

Histogram (log-log scale)

1 10 100 1000 10000 1 10 100 1000 10000 1e+05 1e+06 1e+07

16-bit, linear quantization Most pixels between 0 and 255 The slope that appears on this plot is due to the presence of blur. ⇒ Need for an optimal quantization;

• r

⇒ direct processing of floating-point values with no quantization

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 11 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Overview

Based upon a variant of the Union-Find algorithm [Tarjan, 1975] Three-step strategy

1

Ordering of the sites of the image f following a relationship R such as p R q ⇔          f(p) > f(q), or f(p) = f(q) and p is before q in the classical video scan order

2

Actual computation of the component tree

3

Canonization (compression)

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 12 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Overview

Based upon a variant of the Union-Find algorithm [Tarjan, 1975] Three-step strategy

1

Ordering of the sites of the image f following a relationship R such as p R q ⇔          f(p) > f(q), or f(p) = f(q) and p is before q in the classical video scan order

2

Actual computation of the component tree

3

Canonization (compression)

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 12 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Overview

Based upon a variant of the Union-Find algorithm [Tarjan, 1975] Three-step strategy

1

Ordering of the sites of the image f following a relationship R such as p R q ⇔          f(p) > f(q), or f(p) = f(q) and p is before q in the classical video scan order

2

Actual computation of the component tree

3

Canonization (compression)

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 12 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Overview

Based upon a variant of the Union-Find algorithm [Tarjan, 1975] Three-step strategy

1

Ordering of the sites of the image f following a relationship R such as p R q ⇔          f(p) > f(q), or f(p) = f(q) and p is before q in the classical video scan order

2

Actual computation of the component tree

3

Canonization (compression)

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 12 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Overview

Based upon a variant of the Union-Find algorithm [Tarjan, 1975] Three-step strategy

1

Ordering of the sites of the image f following a relationship R such as p R q ⇔          f(p) > f(q), or f(p) = f(q) and p is before q in the classical video scan order

2

Actual computation of the component tree

3

Canonization (compression)

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 12 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Component tree and max-tree

f R

4 B 1 I 2 G 3 E 1 J 3 C 3 D 1 H 4 A 2 F

{B, C, D} {B, C, D} {B} {A, E} {A} {A, E, F, G} {H, I, J} {E} {A} {F, G} {C, D} {B} λ=4 λ=3 λ=2 λ=1 {A, B, C, D, E, F, G, H, I, J} Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 13 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Image Level Sets

f 4 1 2 3 1 3 3 1 4 2

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 14 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Image Level Sets

f 4 1 2 3 1 3 3 1 4 2 R A B C D E F G H I J

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 14 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Image Level Sets

f 4 1 2 3 1 3 3 1 4 2 R A B C D E F G H I J Level 4 (λ = 4) Component tree Max-tree λ = 4

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 14 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Image Level Sets

f 4 1 2 3 1 3 3 1 4 2 R A B C D E F G H I J Level 4 (λ = 4) Component tree Max-tree λ = 4 A {A} {A}

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 14 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Image Level Sets

f 4 1 2 3 1 3 3 1 4 2 R A B C D E F G H I J Level 4 (λ = 4) Component tree Max-tree λ = 4 A B {A} {B} {A} {B}

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 14 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Image Level Sets

f 4 1 2 3 1 3 3 1 4 2 R A B C D E F G H I J Level 4 (λ = 4) Component tree Max-tree λ = 4 A B Level 3 (λ = 3) A B λ = 3 {B} {A} {A} {B} {B} {A}

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 14 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Image Level Sets

f 4 1 2 3 1 3 3 1 4 2 R A B C D E F G H I J Level 4 (λ = 4) Component tree Max-tree λ = 4 A B Level 3 (λ = 3) A B λ = 3 {B} {A} {A} {B} C {B, C} {A} {C}

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 14 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Image Level Sets

f 4 1 2 3 1 3 3 1 4 2 R A B C D E F G H I J Level 4 (λ = 4) Component tree Max-tree λ = 4 A B Level 3 (λ = 3) A B λ = 3 {B} {A} {A} {B} C D {B, C, D} {A} {C, D}

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 14 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Image Level Sets

f 4 1 2 3 1 3 3 1 4 2 R A B C D E F G H I J Level 4 (λ = 4) Component tree Max-tree λ = 4 A B Level 3 (λ = 3) A B λ = 3 {B} {A} {A} {B} C D E {B, C, D} {A, E} {C, D} {E}

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 14 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Image Level Sets

f 4 1 2 3 1 3 3 1 4 2 R A B C D E F G H I J Level 4 (λ = 4) Component tree Max-tree λ = 4 A B Level 3 (λ = 3) A B λ = 3 {B} {A} {A} {B} C D E Level 2 (λ = 2) A B C D E λ = 2 {B, C, D} {A, E} {C, D} {E} {B, C, D} {A, E}

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 14 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Image Level Sets

f 4 1 2 3 1 3 3 1 4 2 R A B C D E F G H I J Level 4 (λ = 4) Component tree Max-tree λ = 4 A B Level 3 (λ = 3) A B λ = 3 {B} {A} {A} {B} C D E Level 2 (λ = 2) A B C D E λ = 2 {B, C, D} {A, E} {C, D} {E} F {B, C, D} {A, E, F} {F}

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 14 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Image Level Sets

f 4 1 2 3 1 3 3 1 4 2 R A B C D E F G H I J Level 4 (λ = 4) Component tree Max-tree λ = 4 A B Level 3 (λ = 3) A B λ = 3 {B} {A} {A} {B} C D E Level 2 (λ = 2) A B C D E λ = 2 {B, C, D} {A, E} {C, D} {E} F G {B, C, D} {A, E, F, G} {F, G}

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 14 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Image Level Sets

f 4 1 2 3 1 3 3 1 4 2 R A B C D E F G H I J Level 4 (λ = 4) Component tree Max-tree λ = 4 A B Level 3 (λ = 3) A B λ = 3 {B} {A} {A} {B} C D E Level 2 (λ = 2) A B C D E λ = 2 {B, C, D} {A, E} {C, D} {E} F G Level 1 (λ = 1) A B C D E F G λ = 1 {B, C, D} {A, E, F, G} {C, D} {F, G} {A, B, C, D, E, F, G}

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 14 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Image Level Sets

f 4 1 2 3 1 3 3 1 4 2 R A B C D E F G H I J Level 4 (λ = 4) Component tree Max-tree λ = 4 A B Level 3 (λ = 3) A B λ = 3 {B} {A} {A} {B} C D E Level 2 (λ = 2) A B C D E λ = 2 {B, C, D} {A, E} {C, D} {E} F G Level 1 (λ = 1) A B C D E F G λ = 1 {B, C, D} {A, E, F, G} {C, D} {F, G} H {A, B, C, D, E, F, G, H} {H}

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 14 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Image Level Sets

f 4 1 2 3 1 3 3 1 4 2 R A B C D E F G H I J Level 4 (λ = 4) Component tree Max-tree λ = 4 A B Level 3 (λ = 3) A B λ = 3 {B} {A} {A} {B} C D E Level 2 (λ = 2) A B C D E λ = 2 {B, C, D} {A, E} {C, D} {E} F G Level 1 (λ = 1) A B C D E F G λ = 1 {B, C, D} {A, E, F, G} {C, D} {F, G} H I {A, B, C, D, E, F, G, H, I} {H, I}

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 14 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Image Level Sets

f 4 1 2 3 1 3 3 1 4 2 R A B C D E F G H I J Level 4 (λ = 4) Component tree Max-tree λ = 4 A B Level 3 (λ = 3) A B λ = 3 {B} {A} {A} {B} C D E Level 2 (λ = 2) A B C D E λ = 2 {B, C, D} {A, E} {C, D} {E} F G Level 1 (λ = 1) A B C D E F G λ = 1 {B, C, D} {A, E, F, G} {C, D} {F, G} H I J {A, B, C, D, E, F, G, H, I, J} {H, I, J}

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 14 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Density of the max-tree, canonization

f R parent -(parent)

4 B C D 1 I J J 2 G I J 3 E F G 1 J J J 3 C D D 3 D J J 1 H I J 4 A E E 2 F G G

B A E D C D B H I F G C J F A E G I H J λ=4 λ=3 λ=2 λ=1 Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 15 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Computation

-

-(x) 1 if zpar(x) = x then return x 2 else { zpar(x) ← -(zpar(x)) ; return zpar(x) } -(f) 1 for each p, zpar(p) ← undef 2 R ← -(f) // maps R into an array 3 for each p ∈ R in direct order 4 parent(p) ← p ; zpar(p) ← p 5 for each n ∈ N(p) such as zpar(n) undef 6 r ← -(n) 7 if r p then { parent(r) ← p ; zpar(r) ← p } 8 (zpar) 9 return pair(R, parent) // a “correct” function

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 16 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Canonization

-

-(parent, f) 1 for each p ∈ R in reverse order 2 q ← parent(p) 3 if f(parent(q)) = f(q) then parent(p) ← parent(q) 4 return parent // a “canonized” function

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 17 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Example

computation of the area of the components

-(f, R, parent) 1 for each p ∈ R, area(p) ← 1 // initialization 2 for each p ∈ R in direct order 3 area(parent(p)) ← area(parent(p)) + area(p) // update Simple process Computation conducted in an iterative way Linear complexity

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 18 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Comparison of Execution Times

50 100 150 200 250 300 350 10 20 30 40 50 Execution time (seconds) Size of the input image (millions of pixels) Salembier et al. (16-bit) Our proposal (float) Najman and Couprie (float) Najman and Couprie (16-bit) Our proposal (16-bit)

(3 Ghz Bi-Xeon Processor with 2 × 1 MB of cache memory and 4 GB of RAM, running GNU/Linux.)

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 19 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives Tree computation Attributes Computation and Node Labeling Results and Applications

Applications

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 20 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives

Conclusions

Algorithm effective for images with high quantization and with no quantization About as efficient as the fastest known algorithm, and needs twice less memory Serves as a basis to build efficient connected filters for astronomical images On-going work on the selection of attributes and node labeling strategies

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 21 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives

Conclusions

Algorithm effective for images with high quantization and with no quantization About as efficient as the fastest known algorithm, and needs twice less memory Serves as a basis to build efficient connected filters for astronomical images On-going work on the selection of attributes and node labeling strategies

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 21 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives

Conclusions

Algorithm effective for images with high quantization and with no quantization About as efficient as the fastest known algorithm, and needs twice less memory Serves as a basis to build efficient connected filters for astronomical images On-going work on the selection of attributes and node labeling strategies

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 21 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives

Conclusions

Algorithm effective for images with high quantization and with no quantization About as efficient as the fastest known algorithm, and needs twice less memory Serves as a basis to build efficient connected filters for astronomical images On-going work on the selection of attributes and node labeling strategies

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 21 / 25

Motivation A New Algorithm to Compute the Component Tree Conclusions and perspectives

Effective Component Tree Computation with Application to Pattern Recognition in Astronomical Imaging

1

Motivation Connected Filters The Case of Astronomical Images

2

A New Algorithm to Compute the Component Tree Tree computation Attributes Computation and Node Labeling Results and Applications

3

Conclusions and perspectives

Berger et al. (IAP & LRDE) Component Tree Computation and Astronomical Imaging ICIP 2007 22 / 25

Appendix Bibliography

Bibliography I

Géraud, Th. (2005). Ruminations on Tarjan’s Union-Find algorithm and connected operators. In Mathematical Morphology: 40 Years On (Proc. of ISMM), volume 30 of Computational Imaging and Vision, pages 105–116, Paris, France. Springer. Jones, R. (1997). Component trees for image filtering and segmentation. In Coyle, E., editor, IEEE Workshop on Nonlinear Signal and Image Processing, Mackinac Island.

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Appendix Bibliography

Bibliography II

Meijster, A. (2004). Efficient Sequential and Parallel Algorithms for Morphological Image Processing. PhD thesis, University of Groningen, the Netherlands. Meijster, A. and Wilkinson, M. H. F . (2002). A comparison of algorithms for connected set openings and closings. IEEE Trans. Pattern Anal. Machine Intell., 24(4):484–494. Najman, L. and Couprie, M. (2006). Building the component tree in quasi-linear time. IEEE Trans. Image Processing, 15(11):3531–3539.

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Appendix Bibliography

Bibliography III

Salembier, P ., Oliveras, A., and Garrido, L. (1998). Antiextensive connected operators for image and sequence processing. IEEE Trans. Image Processing, 7(4):555–570. Tarjan, R. E. (1975). Efficiency of a good but not linear set union algorithm. Journal of the ACM, 22(2):215–225. Vincent, L. (1993). Grayscale area openings and closings: their applications and efficient implementation. In Intl. Symposium on Mathematical Morphology, pages 22–27.

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