Autoorganised Structures for Extraction of Perceptual Primitives

Transcription

1 Autoorganised Structures for Extraction of Perceptual Primitives M. Penas, M.G.Penedo Dept. Computer Science Univ. of A Coruña, SPAIN M.J.Carreira Dept. Electronics and Computer Science Univ. of Santiago de Compostela, SPAIN

2 Contents Introduction to perceptual organisation. Efficient Spatial-Domain Implementation of Gabor Wavelets. Auto-organised Structures for Extraction of Perceptual Primitives. Self-Organised Maps (SOM). Growing-Cell Structures (GCS). Growing Neural Gas (GNG). Results. Discussion and Conclusion. Autoorganised Structures for Extraction of Perceptual Primitives 1

3 Introduction to Perceptual Organisation Perceptual Organisation in human enables detection of such relationships as symmetry, proximity, closure, collinearity, parallelism and repetition. Perceptual Organisation in Computer Vision is the study of how features are clustered prior to object recognition. Sarkar and Boyer considered organisation in layers of abstraction: signal level, primitive level, structural level and assembly level. Signal level. Extraction of directional primitives by means of Gabor filters. Data dimensionality reduction by means of auto-organised structures. Autoorganised Structures for Extraction of Perceptual Primitives 2

4 Efficient Spatial-Domain Implementation of Gabor Wavelets (I) Gabor Wavelets: Complex exponential signals modulated by gaussians. Applications to image processing. Maximisation of conjoint localisation in spatial and frequency domains. Mathematical expression: where g x0,y 0,f 0,θ 0 = exp {i [2πf 0 (x cos θ 0 + y sin θ 0 ) + φ]}gauss (x x 0, y y 0 ) (1) gauss(x, y) = a exp { π [ a 2 (x cos θ 0 + y sin θ 0 ) 2 + b 2 (x sin θ 0 y cos θ 0 ) 2] } (2) a, b shape of the wavelet; x 0, y 0 localisation in spatial domain; f 0, θ 0 localisation in frequency domain and φ is the phase. Gabor wavelets centred in origin (x 0 = 0, y 0 = 0), symmetric (a = b) and null phase (φ = 0). Autoorganised Structures for Extraction of Perceptual Primitives 3

5 Efficient Spatial-Domain Implementation of Gabor Wavelets (II) Implementations: Frequency domain. Multi-resolution frequency domain. Multi-resolution spatial domain: convolution of reduced versions of the image with unidimensional 11 component filters obtained from the decomposition of two dimensional Gabor filters. Complexity reduction. Autoorganised Structures for Extraction of Perceptual Primitives 4

6 Auto-organised Structures for Extraction of Perceptual Primitives (I) Reduction of input space dimensionality that leads to generalisation of results so that feature grouping is independent of the kind of images to process. Classified output space must maintain the topology of the input space. Auto-organised structures Self-Organised Maps (SOM). Growing-Cell Structures (GCS). Growing Neural Gas (GNG). Autoorganised Structures for Extraction of Perceptual Primitives 5

7 Auto-organised Structures for Extraction of Perceptual Primitives (II) Self-Organised Maps (SOM) Size and structure fixed prior to training. Processing elements represents clusters of patterns in the training set. Topological order in the output map. Performance: Initialisation: fix structure s size, weights, learning parameters, region of interest and maximum number of iterations. Learning: weights of the winning neuron and its neighbourhood are updated. Autoorganised Structures for Extraction of Perceptual Primitives 6

8 Auto-organised Structures for Extraction of Perceptual Primitives (III) Growing-Cell Structures (GCS) Based on SOM. Adds processing elements maintaining network topology. Performance: Initialisation process: create K-dimensional structure of K + 1 nodes and (K +1)K/2 connections, initialise weights, learning rate and resource value. Training process: repeat until stop criterion is satisfied. Learning phase: present P patterns to the map, for each of them determine winning neuron and update its weights and resource value, as those of its direct neurons. Repeat Q times. Growing phase: add new processing element, stablish and update weights and neighbourhood relations. Autoorganised Structures for Extraction of Perceptual Primitives 7

9 Auto-organised Structures for Extraction of Perceptual Primitives (IV) Growing Neural Gas (GNG) Insertion of new processing elements, modification and elimination of neighbourhood relations. Represent faithfully input space. Eliminate processing elements in areas of input space where probability is almost null. Performance: Initialisation process: two disconnected processing elements. Training process: repeat until stop criterion is satisfied. Learning phase: present P patterns in groups of Q (P >> Q). Determine two closest elements. Create or update the connection between them. Update connection between the winner and its direct neighbours. Remove connections and isolated elements. Growing phase: insertion of new processing elements as in GCS. Autoorganised Structures for Extraction of Perceptual Primitives 8

10 Results (I) Net input: IMAGE GABOR WAVELET 8 COMPONENT VECTOR ADD NINTH COMPONENT SCALE VECTOR INPUT TO THE NETWORK Response: Resp = arctan ( ( x x ) x π ) (3) where x is the modulus of the 8 component vector, x is the media of all the modulus and x is the typical deviation of all the modulus. Autoorganised Structures for Extraction of Perceptual Primitives 9

11 Results (II) Images Synthetic image Corridor Bridge Autoorganised Structures for Extraction of Perceptual Primitives 10

12 Results (III) SOM Training set: artificial images, lines in each main direction, same proportion, background not included. Topology: processing elements, rectangular vicinity. Training process: SOM-PAK. Ordering phase, learning rate 0.05 and neighbourhood size 12. Training phase, learning rate 0.02 and neighbourhood size 4. Autoorganised Structures for Extraction of Perceptual Primitives 11

13 Results (IV) GCS Training set: like SOM, including background. Objective: set of clusters, similar proportion of processing units. Topology and training: GCS map 200 processing units, P = input patterns, Q = 6000 adaption steps, 0.06 factor for winning element weight s updates, for neighbour weight s updates and 0.05 for resource values update. Autoorganised Structures for Extraction of Perceptual Primitives 12

14 Results (V) GNG Training set: Artificial images with or without background. Real images with the same proportion of all directional features. Topology and training: two dimensional GNG map with 100 processing units, Q = 3000 adaption steps, 0.2 winning neuron learning rate, direct neighbours learning rate, T = 50 age for deletion and for resource value reduction. Autoorganised Structures for Extraction of Perceptual Primitives 13

15 Discussion and conclusion (I) SOM High vicinity between clusters, background not enough separated from other directions. Noise presence, lost of some directional features. GCS Formation of clusters around winning elements of different main orientations or background, lower vicinity between clusters. GNG Excessive separation between clusters. primitives, bad generalisation capacity. Lost of intermediate directional D67.5 V D112.5 D67.5 V D112.5 D45 BG D135 D45 BG D135 D22.5 H D167.5 D22.5 H D167.5 SOM GCS GNG Autoorganised Structures for Extraction of Perceptual Primitives 14

16 Discussion and conclusion (II) Noise presence Directional primitives detection Example SOM Yes Quite good GCS No Good GNG No Quite good Autoorganised Structures for Extraction of Perceptual Primitives 15