THE INSTITUTE OF ELECTRONICS,INFORMATION AND COMMUNICATION ENGINEERS

TECHNICAL REPORT OF IEICE.

† ††

†2–12–1

††3–7–2

E-mail: †yamasita@ide.titech.ac.jp, ††wakahara@hosei.ac.jp

GAT/GPT (Global Affine Transform/Global Projection Transform)GAT/GPT whole-to-part

ASIFT RANSACGAT/GPT

An initilal search method for region-based image matchingYukihiko YAMASHITA† and Toru WAKAHARA††

† Tokyo Institute of Technology2-12-1, O-okayama, Mekuro-ku, Tokyo, 152-8552

†† Hosei University3–7–2 Kajino-cho, Koganei-shi, Tokyo 184-8584

E-mail: †yamasita@ide.titech.ac.jp, ††wakahara@hosei.ac.jp

Abstract GAT/GPT (Global Affine Transform/Global Projection Transform) correlation methods for imagematching that is one of the most important methods in the field of image processing can perform a precise im-age matching. When GAT/GPT correlation methods are applied to whole-to-part matching, it can be realized bysliding windows with various sizes but it needs much calculation time. In this paper we propose a fast initial searchmethod using the sliding Discrete Fourier Transformation for the region-based image matching. Furthermore, byexperiments we compare and evaluate the proposed method against the combination of ASIFT and RANSAC.Key words Image matching, region-based matching, GAT/GPT correlation, initial search

1.

2

Scale-Invariant Feature Transform (SIFT) [1], [2]SIFT Affine

ASIFT [3] RANSAC [4]

[5] Lucus-Kanade[6]

[7] GAT/GPT(Global Affine Transform/Global Projection Transform)

[8] [10]

— 1 —

Copyright ©201 by IEICE This article is a technical report without peer review, and its polished and/or extended version may be published elsewhere.

- 119 -

(a) (b) (c) (d) (e) LoG

1

GPTkNN MNIST

99.7% kNN

MNISTwhole-to-whole

Computer Vision

whole-to-part

whole-to-part

[11] [14]SIFT

ASIFT RANSAC

2.

2. 11

0 [−K, K]G[n] = e−(πn/K)2/(2σ2) σ

π −π

σK/π f [n] 12

K∑k=−K

G[k]f [n− k] (1)

K∑k=−K

kG[k]f [n− k] (2)

K∑k=−K

k2G[k]f [n− k] (3)

β = π/K

G[k] �P∑

p=0

ap cos(βpk)

kG[k] �P∑

p=1

bp sin(βpk)

k2G[k] �P∑

p=0

dp cos(βpk)

P DFTP 2 4

cp[n] =K∑

k=−K

f [n− k] cos(βpk)

sp[n] =K∑

k=−K

f [n− k] sin(βpk)

(1)-(3)

K∑k=−K

G[k]f [n− k] �P∑

p=0

apcp[n]

K∑k=−K

kG[k]f [n− k] �P∑

p=1

bpsp[n]

K∑k=−K

k2G[k]f [n− k] �P∑

p=0

dpcp[n]

1 Lenna

2. 2Kober

u[n] =n∑

k=1

f [k]eiβpk (4)

cp[n] sp[n]

— 2 —- 120 -

0

2

4

6

8

10

0.6 0.8 1 1.2 1.4 1.6 1.8 2

Err

or

(%)

Sigma

TotalInterval

Order

0

2

4

6

8

10

0.6 0.8 1 1.2 1.4 1.6 1.8 2

Err

or

(%)

Sigma

TotalInterval

Order

0

5

10

15

20

0.6 0.8 1 1.2 1.4 1.6 1.8 2

Err

or

(%)

Sigma

TotalInterval

Order

(a) G[n], K = 256, P = 2 (b) nG[n], K = 256, P = 2 (c) LoG, K = 256, P = 2

0

2

4

6

8

10

0.6 0.8 1 1.2 1.4 1.6 1.8 2

Err

or

(%)

Sigma

TotalInterval

Order

0

2

4

6

8

10

0.6 0.8 1 1.2 1.4 1.6 1.8 2

Err

or

(%)

Sigma

TotalInterval

Order

0

2

4

6

8

10

0.6 0.8 1 1.2 1.4 1.6 1.8 2

Err

or

(%)

Sigma

TotalInterval

Order

(a) G[n], K = 256, P = 4 (b) nG[n], K = 256, P = 4 (c) LoG, K = 256, P = 4

2 Interval Order

cp[n] + isp[n] = e−iβpn(u[n + K]− u[n−K − 1]) (5)

u[n]

u[n] = u[n− 1] + f [n]eiβpn (6)

1IIR

v[n] = e−iβpv[n− 1] + f [n] (7)

cp[n] sp[n]

cp[n]+ isp[n] = (−1)p(v[n+K]−v[n−K]+x[n−K]) (8)

K p

(2017) 2

v[n+1] = 2 cos(βp)v[n]+v[n−1]+f [n+1]−eiβpf [n] (9)

cp[n] sp[n]cp[n]

2

2. 3P

σK/π K σ

σ K

2 P = 2 P = 4 G[n] nG[n] LoG2 2

P = 2 σ = 1.1 1%5% LoG 10% P = 4

σ = 0.85 0.1%0.5% LoG 1%2. 4

GPU

Lenna Barbara 1524288 (= 512× 512× 2)

3 K = 256 c1[n] 10,0002

2 IIR 12%

3.

3. 1D(θ) θ

Q

— 3 —- 121 -

0

2

4

6

8

10

0 100000 200000 300000 400000 500000

Err

or

(%)

n

KernelKernelB

IIRIIRB

2ndIIR2ndIIRB

0

0.2

0.4

0.6

0.8

1

0 100000 200000 300000 400000 500000

Err

or

(%)

n

KernelKernelB

IIRIIRB

(a) c1[n], K = 256 (b) c1[n], K = 256 ( )

3 Kernel (6) IIR 1 IIR (7) 2ndIIR 2IIR (9) B

D(θ) =Q∑

q=−Q

dneiqθ (10)

d−q = dq dq

2Q + 1I(x, y)

IX(x, y) IY(x, y)

‖I1(x, y)‖ =√

(IX(x, y))2 + (IY(x, y))2 (11)

(x, y)δ(θ) ‖I1(x, y)‖δ(θ − θ(x, y))IX(x, y) = ‖I1(x, y)‖ cos θ(x, y) IY(x, y) =

‖I1(x, y)‖ sin θ(x, y)

dq = 12π

∫ π

−π

‖I1(x, y)‖δ(θ − θ(x, y))e−iqθdθ

= 12π‖I1(x, y)‖e−iqθ(x,y)

� (IX(x, y))− iIY(x, y))q

2π (‖I1(x, y)‖2 + ε)(q−1)/2 (12)

ε ε = 10−12

θ0

d′q

d′q = dqe−iqθ0 (13)

3. 2I (x, y)

S S

P = 4

I S/8IX IY

(12)

(a) (b)

4

2Q + 1 P = 2

S S/4 (x, y)S L

θ = 2πl/L (l = 0, 1, . . . , L− 1)

2

1 21

(x, y) 1 θ0

θ0 (x, y) M

S/4

4 M

(xm, ym) (m = 1, 2, . . . , M)

(x′m, y′m) (m = 1, 2, . . . , M)

x′m = S(xm cos θ0 − ym sin θ0) + x (14)

y′m = S(xm sin θ0 + ym cos θ0) + y (15)

S/4

(13)

— 4 —- 122 -

Algorithm 1Require: I S0 rS

SMax S TS

(xm, ym) (m = 1, 2, . . . , M)S ⇐ S0

while S ≤ SMax doI S/8IX IY

2Q + 1 S S/4for (x, y) ∈ TS do

S

for θ0 ∈ ( ) doθ0 (14) (15)

S/4(13) M(2Q + 1)

1 (x, y)1

end forend forS ⇐ rSS

end while

M(2Q + 1)1 (x, y)

(x, y) S

0.2Alogrithm 1

3. 3

(x, y) S

1

ASIFT Affine

2(rS = 2) S0

√2S0 2

√2

4.

ASIFT+RANSACGraffiti (800 × 640 ) Boat (850 × 680) [15]Graffiti

6 (img1∼img6)Graffiti img1 Boat img1

(430, 310) 150√

2(425, 340) 200

√2

9 Affineimg2∼img6

2 20%

ASIFT+RANSAC Graffitiimg1 490 × 368

Boat img1ASIFT+RANSAC OpenCV SIFT

RANSAC python[16] OpenCV

version 3.4.15 Graffiti 6 Boat

img1

5 (b)–(f) 6 (b)–(f)

ASIFT+RANSAC

1 CPU In-tel(R) Core(TM) i5-6400 CPU @ 2.70GHz

ASIFT+RANSAC

— 5 —- 123 -

(a) (b) img2 (c) img3 (d) img4 (e) img5 (f) img6

5 Graffiti (b)–(f)

(a) (b) img2 (c) img3 (d) img4 (e) img5 (f) img6

6 Boat (b)–(f)

ASIFT+RANSAC

GPU

5.

GAT/GPT

ASIFT+RANSAC

GAT/GPT

[1] D.G. Lowe, “Distinctive image features from scale-invariantkeypoints,” Int. J. Comput. Vision, vol.60, no.2, pp.91–110,Nov. 2004.

[2] Y. Ke and R. Sukthankar, “PCA-SIFT: A more distinc-tive representation for local image descriptors,” Proceed-ings of the 2004 IEEE Computer Society Conference onComputer Vision and Pattern Recognition, pp.506–513,CVPR’04, IEEE Computer Society, Washington, DC, USA,2004. http://dl.acm.org/citation.cfm?id=1896300.1896374

[3] J.-M. Morel and G. Yu, “ASIFT: A new framework for fullyaffine invariant image comparison,” SIAM J. Img. Sci., vol.2,no.2, pp.438–469, April 2009.

[4] M.A. Fischler and R.C. Bolles, “Random sample consen-sus: A paradigm for model fitting with applications to im-

1 (second)

Image \ Method ASIFT+RAMSAC OursGraffiti 21.881 0.699Boat 168.366 1.132

age analysis and automated cartography,” Commun. ACM,vol.24, no.6, pp.381–395, June 1981.

[5] , 1989[6] S. Baker and I. Matthews, “Lucas-kanade 20 years on: A

unifying framework,” Int. J. Comput. Vision, vol.56, no.3,pp.221–255, Feb. 2004.

[7] S. Korman, D. Reichman, G. Tsur, and S. Avidan, “FAsT-Match: Fast affine template matching,” Int. J. Comput.Vision, vol.121, no.1, pp.111–125, Jan. 2017.

[8] T. Wakahara, Y. Kimura, and A. Tomono, “Affine-invariantrecognition of gray-scale characters using global affine trans-formation correlation,” IEEE Transactions on Pattern Anal-ysis and Machine Intelligence, vol.24, no.4, pp.384–395,April 2001.

[9] Y. Yamashita and T. Wakahara, “Affine-transformation and2D-projection invariant k-NN classification of handwrittencharacters via a new matching measure,” Pattern Recogni-tion, vol.52, pp.459–470, April 2016.

[10] S. Zhang, T. Wakahara, and Y. Yamashita, “Image match-ing using GPT correlation associated with simplified HOGpatterns,” Proceedings of 2017 Seventh International Con-ference on Image Processing Theory, Tools and Applica-tions, vol.1, pp.1–6, November–December 2017.

[11] E. Elboher and M. Werman, “Cosine integral images forfast spatial and range filtering,” Proceedings of 2011 18thIEEE International Conference on Image Processing, pp.89–92, Sept. 2011.

[12] K. Sugimoto and S.-I. Kamata, “Compressive bilateral fil-tering,” IEEE Transactions on Image Processing, vol.24,no.11, pp.3357–3369, Nov. 2015.

[13] “”

vol.117 no.48, IE2017-4 pp.19–24 May 2017[14] V. Kober, “Fast algorithms for the computation of sliding

discrete sinusoidal transforms,” IEEE Transactions on Sig-nal Processing, vol.52, no.6, pp.1704–1710, June 2004.

[15] Visual Geometry Group, University of Oxford, “Affinecovariant features”. http://www.robots.ox.ac.uk/~vgg/research/affine/.

[16] A. Alekhin, “Affine invariant feature-based image match-ing sample”. https://github.com/opencv/opencv/blob/master/samples/python/asift.py.

— 6 —- 124 -