1. Image Retrieval Based on Fractal Signatures John Y. Chiang Z. Z. Tsai
蔣依吾
國立中山大學 資訊工程學系
Web:

2. Block Diagram of Image retrieval System

3. Feature used for querying
Text, color, shape, texture , spatial relations
Image Retrieval Systems：
IBM QBIC： color , texture, shape
Berkeley BlobWorld： color, texture, location and shape of region (blobs)
Columbia VisualSEEK: text, color , texture, shape, spatial relations

4. Ideal image retrieval 相似之影像有相似之索引檔 不相似之影像有不相似之索引檔
(a) 高相關度影像資訊有高相關索引檔 (b) 索引檔相關度高，其影像資訊相關度高
不相似之影像有不相似之索引檔
(c) 索引檔相關度低，其影像資訊相關度低
(d) 影像資訊相關度低，其索引檔相關度低
Based on Fractal Orthogonal Basis,
image index satisfy the four properties.

5. Fractal Orthogonal Basis
Vines, Nonlinear address maps in a one-dimensional fractal model, IEEE Trans. Signal Processing,1993.
Training Orthonormal basis vectors.
Gram-Schmidt method
Compression: each image range block (R) decomposed into a linear combination,
Using linear combination coefficient as a feature vector。

6. Proof of Theorem theorem 1： Let f , g is distance space（K，d）functions，
Range block length k，f= , g= , i,j
xi‧xj=0 and | xi |=1 , S={s1s }
f has fixed point a，S: query image a’s index,
T={t1t2… }, g has fixed point b , T: database image
b’s index,
if a , b is close, then S , T is close；
if S , T is close, then a , b is close。

7. (a) Proof : a , b is close , S , T is close
Proof of Theorem 1
(a) Proof : a , b is close , S , T is close
a , b is close , then ||a－b||
thus a , b is close , S , T is close

8. (b) Proof: S , T is close , a , b is close
Proof of Theorem 1
(b) Proof: S , T is close , a , b is close
S , T is close , then
thus S , T is close , a , b is close

9. Proof of Theorem theorem 2： Let f , g is distance space（K，d）functions，
Range block length k，f= , g= , i,j
xi‧xj=0 and | xi |=1 , S={s1s }
f has fixed point a，S: query image a’s index,
T={t1t2… }, g has fixed point b , T: database image
b’s index,
if a , b is not close, then S , T is not close；
if S , T is not close, then a , b is not close。

10. (c) Proof : a , b is not close , S , T is not close
Proof of Theorem 2
(c) Proof : a , b is not close , S , T is not close
by theorem 1, a , b is close S , T is close
thus a , b is not close , S , T is not close
(d) Proof: S , T is not close , a , b is not close
by theorem 1, a , b is close S , T is close
thus S , T is not close , a , b is not close

11. Advantages based on Fractal Orthogonal Basis
Compressed data can be used directly as indices for query.
Image index satisfy the four properties.
Similarity measurement is easy.

12. Orthonormal Basis Vectors
(a) (b) (c)
Figure 3. The 64 fractal orthonormal basis vectors of (a) R, (b) G, and (c) B color components, respectively, derived from an ensemble of 100 butterfly database images. The size of each vector is enlarged by two for ease of observation..

13. Fourier transform vs Fractal Orthonormal basis

14. Original image
Compressed image

15. Experimental Results Image database The source of images:
Image size:320x240
Total of images:1013
Range block size:8x8
Domain block size:8x8
Domain block set: 64 domain blocks in each color plane

16. Figure 4. An image retrieval example
Figure 4. An image retrieval example. The features from R, G, B color components and brightness level of the query image are all selected. The rectangular area in the upper right-hand corner provides an enlarged viewing window for the image retrieved.

17. Figure 6. Retrieval results based on a sub-region of a query image with scaling factors 0.8 through 1.2 and rotation angles every 30 degrees.

18. Future work
1. With Multiple Instance Learning, finding “ideal” feature vector, as query feature.
2. Considering spatial relations of sub-regions.
3. Increasing performance.