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Images: 2D signals

Module by: Robert Nowak. E-mail the author

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Summary: This module introduces image processing, 2D convolution, 2D sampling and 2D FTs.

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Image Processing

Figure 1: Images are 2D functions fxy f x y
Figure 1 (figure1.png)

Linear Shift Invariant Systems

Figure 2
Figure 2 (figure2.png)

HH is LSI if:

  1. H α 1 f1xy+α2f2xy=Hf1xy+Hf2xy H α 1 f1 x y α2 f2 x y H f1 x y H f2 x y for all images f1 f1, f2 f2 and scalar.
  2. Hfx−x0y−y0=gx−x0y−y0 H f x x0 y y0 g x x0 y y0
LSI systems are expressed mathematically as 2D convolutions: gxy=∫-∞∞∫-∞∞hx−αy−βfαβdαdβ g x y β α h x α y β f α β where hxy h x y is the 2D impulse response (also called the point spread function).

2D Fourier Analysis

ℱuv=∫-∞∞∫-∞∞fxyⅇ-ⅈuxⅇ-ⅈvydxdy ℱ u v y x f x y u x v y where ℱℱ is the 2D FT and uu and vv are frequency variables in xu x u and yv y v .

2D complex exponentials are eigenfunctions for 2D LSI systems:

∫-∞∞∫-∞∞hx−αy−βⅇⅈu0αⅇⅈv0βdαdβ=∫-∞∞∫-∞∞h α ′ β ′ ⅇⅈu0x− α ′ ⅇⅈv0y− β ′ dα′dβ′=ⅇⅈu0xⅇⅈv0y∫-∞∞∫-∞∞h α ′ β ′ ⅇ-ⅈ u 0 α ′ ⅇ-ⅈ v 0 β ′ d α ′ dβ′ β α h x α y β u0 α v0 β β′ α′ h α ′ β ′ u0 x α ′ v0 y β ′ u0 x v0 y β′ α ′ h α ′ β ′ u 0 α ′ v 0 β ′ (1)
where ∫-∞∞∫-∞∞hα′β′ⅇ-ⅈu0 α ′ ⅇ-ⅈv0 β ′ dα′dβ′≡Hu0v0 β′ α′ h α′ β′ u0 α ′ v0 β ′ H u0 v0 Hu0v0 H u0 v0 is the 2D Fourier transform of hxy h x y evaluated at frequencies u0 u0 and v0 v0 .

Figure 3
Figure 3 (figure3.png)

gxy=hxy*fxy=∫-∞∞∫-∞∞hx−αy−βfαβdαdβ g x y h x y f x y β α h x α y β f α β (2)
Guv=Huvℱuv G u v H u v ℱ u v

Inverse 2D FT

gxy=12π2∫-∞∞∫-∞∞Guvⅇⅈuxⅇⅈvydudv g x y 1 2 2 v u G u v u x v y (3)

2D Sampling Theory

Figure 4: Think of the image as a 2D surface.
Figure 4 (figure4.png)

We can sample the height of the surface using a 2D impulse array.

Figure 5: Impulses spaced Δx Δ x apart in the horizontal direction and Δy Δ y in the vertical
Figure 5 (figure5.png)

fsxy=Sxyfxy fs x y S x y f x y where fsxy fs x y is sampled image in frequency

2D FT of sxy s x y is a 2D impulse array in frequency Suv S u v

Figure 6
Figure 6 (figure6.png)

multiplication in time ⇔ convolution in frequency multiplication in time ⇔ convolution in frequency Fsuv=Suv*ℱuv Fs u v S u v ℱ u v

Figure 7: ℱuv ℱ u v is bandlimited in both the horizontal and vertical directions.
Figure 7 (figure7.png)
Figure 8: periodically replicated in ( u , v ) ( u , v ) frequency plane
Figure 8 (figure8.png)

Nyquist Theorem

Assume that fxy f x y is bandlimited to ±Bx ± Bx , ±By ± By :

Figure 9
Figure 9 (figure9.png)

If we sample fxy f x y at spacings of Δx<πBx Δ x Bx and Δy<πBy Δ y By , then fxy f x y can be perfectly recovered from the samples by lowpass filtering:

Figure 10: ideal lowpass filter, 1 inside rectangle, 0 outside
Figure 10 (figure10.png)
Figure 11
Aliasing in 2D
(a) (b)
Figure 11(a) (figure11-1.png)Figure 11(b) (figure11-2.png)

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