ELEC Dr Reji Mathew Electrical Engineering UNSW

Transcription

1 ELEC 4622 Dr Reji Mathew Electrical Engineering UNSW

2 Filter Design Circularly symmetric 2-D low-pass filter Pass-band radial frequency: ω p Stop-band radial frequency: ω s 1 δ p Pass-band tolerances: δ p Stop-band tolerances: δ s How to design this low pass filter? Windowing? ω p ω s ω δ s

3 Windowing Design a 2-D FIR filter Start with a desired frequency response h d (ω) Where h d (ω) satisfies the requirements (stop & pass band) Perhaps an ideal filter h d (ω) Then take the inverse DSFT to obtain h d [n] h d [n] may have infinite support Apply a spatial window w[n] to limit support to a finite region n n

4 Windowing Windowing in the spatial domain: affects filter characteristics in the frequency domain Does h(ω) which is the DSFT of h n still satisfy the original constrains? Multiplication of spatial signals equates to convolution of the corresponding DSFT of the signals Ideally - window will have as narrow a bandwidth as possible That is w(ω) should decay rapidly with increasing ω However, narrow band windows have large spatial support Constrained by the desired support size

5 Windowing Remember: we want a filter with zero phase h ω = h(ω) w(ω) and h d (ω) will both need to be real-valued So h ω will also be real-valued (zero phase) Follows on from the convolution equation Convenient: express window as spatially continuous function w x And w n = w x x=n We assume that the window is sufficiently narrow band Essentially bandlimited to the Nyquist region, ω [ π, π] 2 So FT of w x is identical to the DSFT of w n Therefore, we are only concerned with the properties of the spatially continuous function w x

6 Windowing Simplest case: window is a separable product of 1D functions Determines the extent of the window This trivial window is rarely used in practice because its frequency response decays only slowly.

7 Windowing Smooth function: means high frequencies components are avoided minimising the effective bandwidth of w(ω) one period of a cosine raised with its negative peaks just touch zero ( raised cosine ) t Multiply a window: To Restrict an impulse response that is infinite t

8 Windowing

9 Windowing For some applications, superior results can be obtained by using non-separable windows. When designing a circularly symmetric filter, a circularly symmetric window function is generally preferable w(s) = w t t= s Where w t can be the 1D window functions we discussed earlier.

10 Filter Design Frequency Sampling Find a finite set of filter tap values Which minimizes a weighted combination of squared differences between the desired and actual frequency responses Can uniformly distribute ω k in the region ω [ π, π] 2 Alternatively, can place more samples in the neighbourhood of critical frequency regions such as (pass and stop) band edges

11 Frequency Sampling Want to ensure that the designed filter h n is zero phase Remember that zero phase implies a symmetric filter. h n = h n The sin terms are cancelled out. Note summation over

12 Frequency Sampling We wish to minimize the weighted squared error expression weight which reflects the relative importance of matching the desired response at frequency ω k Summation for k = 1 K

13 Frequency Sampling

14 Frequency Sampling Minimization of the weighted squared error expression is a classic linear algebra problem whose solution is well known: Can adjust the weights and number of samples (K) Reduce the density of samples in critical frequency regions by increasing weights associated with a smaller number of samples

15 Filter Design Transformation Method (McClellan Transformation) 1D filter design problem is much easier Methods exist for designing optimum 1D zero phase FIR filters subject to constraints on tolerances in the pass- and stop-bands Map the 2-D design problem into a 1-D design problem Without losing the finite impulse response property. The method is relevant only for zero phase designs. An approximate circularly symmetric function specifies the 2D to 1D mapping

16 Transformation Method Frequency response of a 1-D zero phase filter with 2N + 1 taps Taking the DTFT of zero phase filter h n, defined for -N to N Remember that zero phase implies a symmetric filter. h n = h n The sin terms are cancelled out. Note summation from 1 to N

17 Transformation Method Frequency response of a 1-D zero phase filter with 2N + 1 taps Taking the DTFT of zero phase filter h n, defined for -N to N By application of Chebychev polynomials

18 Transformation Method

19 Transformation Method x = cos ω C n = cos nω C 0 = cos 0ω = 1 C 1 = cos ω C 2 = cos 2ω = 2 cos ω cos ω 1 = 2cos 2 ω 1 C 3 = cos 3ω = 2 cos ω cos 2ω cos ω = 4cos 3 ω 3 cos ω cos nω can be expressed as an nth-degree polynomial in cos ω

20 Transformation Method Frequency response of a 1-D zero phase filter with 2N + 1 taps Taking the DTFT of zero phase filter h n, defined for -N to N By application of Chebychev polynomials Coefficients b n can be obtained from a n Next: Frequency mapping from ω 1, ω 2 cos (ω)

21 Transformation Method μ ω is the DSFT of a finite support sequence μ[n] Essentially μ[n] controls the mapping (kernel) h[n] has finite support Summation of n-fold convolution of finite support sequence μ[n]

22 Transformation Method Given μ[n], what can we say about the Region of Support for the final 2D filter h[n]? And what is DSFT of μ[n], which governs the mapping ω 1, ω 2 cos (ω)?

23 Transformation Method Example: for a 7 tap 1D filter N=3 (-3 0 3) So μ n convolved three times (due to μ ω 3 ) Since μ n has a region of support [ 1, 1] 2 The support of h[n] will be [ 3, 3] 2

24 Transformation Method DSFT of μ n The mapping represents quite a good approximation to a circularly symmetric function. (plot next slide)

25

26 Transformation Method Steps to follow for the Frequency Transform Method 1. Select a transformation kernel, μ n. Possible to design your own. The original McClellan kernel is a common choice. 2. Map the 2-D frequency domain specifications on h ω back to specification on the 1-D kernel, h ω. Stop band requirement for the 2D filter: ω ω s Find the largest ω s such that μ ω cos (ω s ) for all ω ω s ω 1 tolerances map directly from 2-D to the 1-D ω ω 2

27 Transformation Method Steps to follow for the Frequency Transform Method 1. Select a transformation kernel, μ n. Possible to design your own. The original McClellan kernel is a common choice. 2. Map the 2-D frequency domain specifications on h ω back to specification on the 1-D kernel, h ω. 3. Design a suitable 1-D kernel using well established methods. Thereby determine the b n coefficients. ω 1 ω ω 2

28 Transformation Method Input Image x n μ n μ n μ n b[0] b[1] b[2] b[3] Example: N = 3 Weighted summation of convolutions of the mapping kernel. Weights b n as determined from the 1D filter design

29 Filtering Examples Gaussian Filters Normalisation chosen so that Spatially continuous impulse response has unit DC gain

30 Gaussian Filters In most cases the off-diagonal term, σ 1,2, is zero So that h(s) is a separable Gaussian function whose contours have an elliptical cross-section (circular if σ 1 = σ 2 ). s 2 s 1

31 Gaussian Filters The σ 1,2 term may be used to design arbitrarily rotated elliptical cross-sections. s 2 s 1

32 Gaussian Filters Gaussian filters have the property that their Fourier transform is also a Gaussian function

33 Gaussian Filters Then: DSFT of the sampled impulse response, h n, will be almost identical to the spatially continuous Fourier transform of h(s). DSFT of the sampled impulse response is Gaussian and also that h n = 1 n Since the spatial impulse response decays rapidly as n becomes large, can simply truncate the impulse response (i.e. apply a rectangular window) to some suitably large region of support.

34 Gaussian Filters Important properties Gaussian filters - their Fourier transform is also a Gaussian function Both spatial & frequency domain representations are real & positive. For simplicity - Ignore the possibility that σ 1,2 might not be zero, Horizontal and vertical spread of the Gaussian function is given by σ 1 and σ 2 Horizontal and vertical spread in the frequency domain is given by 1 σ 1 and 1 σ 2 So the product of the spatial & frequency spreads is always exactly 1.

35 Moving Average Filters Low complexity Low pass, high pass and band pass implementations A2 B2 A1 B1 (B1 A1 + 1)(B2 A2 + 1) multiplications & additions for each pixel (B1 A1 + 1) + (B2 A2 + 1) multiplications & additions per output pixel

36 Moving Average Filters Low-pass filter, whose frequency response is a sinc function Filter is separable into the row and column filters Can be implemented independent of filter s region of support?

37 Moving Average Filters x(0,0) x(0,1) x(0,2) x(0,3) x(0,4) x(0,5) x(0,6) x(0,7) x(1,0) x(1,1) x(1,2) x(1,3) x(1,4) x(1,5) x(1,6) x(1,7) x(2,0) x(2,1) x(2,2) x(2,3) x(2,4) x(2,5) x(2,6) x(2,7) x(3,0) x(3,1) x(3,2) x(3,3) x(3,4) x(3,5) x(3,6) x(3,7) x(4,0) x(4,1) x(4,2) x(4,3) x(4,4) x(4,5) x(4,6) x(4,7) Each subsequent scaled output value requires only two additions

38 Moving Average Filters

39 Unsharp Masking One of the oldest image processing operations Ad-hoc technique: artificially increasing sharpness of an image x n

40 Image Acquisition Models Scene illuminant reflects off scene surfaces; the resulting radiant flux passes through an optical system, is integrated over sensor surfaces, converted to an electrical signal and quantized

41 Image Acquisition Models Scene surface reflectivities as a function of wavelength λ, and position s = (s 1, s 2 ) Radiant power spectral density ( scene radiance ) at the surface of the sensor Spectral power density of the illuminant as a function of position and wavelength Note: Variation in s and λ Example of variations in λ : Outdoor and indoor images

42 Image Acquisition Models We ignore surface orientation and its impact upon scene radiance. Also, we ignore fluorescence.

43 Optics Add the impact of optics to our simple model Resulting scene radiance at the surface of the sensor Point spread function (PSF) of the optical system In general, the optical PSF depends on both wavelength λ and spatial position z (ie varies across the sensor). We ignore the dependence of h o z,λ(s) on z. optics are not ideal, but are at least linear

44 Optics This is the convolution integral. Linear systems which are modeled by convolution are known as Linear Shift Invariant (LSI) systems. LSI property is valid only when we ignore the dependence of h o z,λ(s) on z. LSI model is sufficiently accurate in the so-called centre-field

45 Optics Away from centre field a number of aberrations appear Example: Chromatic aberrations h o z,λ(s) introduces shifts which depend upon both position z and wavelength λ. Different wavelengths (colours) experience different geometric distortions do not properly line up blurring

46 Optics Simplifications ignore spatial variations of the optical system PSF restrict to centre field Even with these simplifications the PSF is not simply an impulse non-ideal causes of these non-idealities are refractive and diffractive limitations Refractive: lens imperfections Diffractive: finite lens size wavelength dependent Narrow aperture: comparable in size to incident light wavelength

47 Sensor Integration Integration across all wavelength Spectral sensitivity 2D sequence of pixel values Integration over the aperture of the sensor Ideal sensor

48 Sensor Integration

49 Sensing Devices CCD photons converted into electrons: accumulated in charge wells collection of 1-D analog shift registers entire array of pixels must be shifted difficult to retrieve a subset of the pixels clocking the out pixels : Consumes a large amount of power

50 Sensing Devices CMOS Most popular even at high quality end Sensing element CMOS transistor with a transparent gate Or a photodiode connected to a CMOS amplifier At each photoreceptor is a CMOS circuit, which amplify the signal & provide addressing capability Pixels in CMOS sensors addressed in a manner similar random access memory Low power, inexpensive manufacturing System-on-a-chip: A-D conversion, image processing on the same chip

51 Sensing Devices CCD and CMOS typically respond to infra-red light Infra-red is outside the visible spectrum Therefore require infra-red absorbtion filters in the optical path

52 Combining Optical and Aperture Effects Combine the effects of optical and sensor integration into a single PSF h v λ (s) Integral over wavelength Convolution with optical PSF Multiplication with the aperture

53 Combining Optical and Aperture Effects

54 Combining Optical and Aperture Effects

55 Combining Optical and Aperture Effects System PSF is obtained by convolving the optical PSF with a flipped version of the sensor integration aperture. Sensor samples are then obtained by convolving radiant scene intensity at the sensor surface by the system PSF impulsively sampling the resulting spatially and integrating over the spectral response characteristic of the sensor

56 Combining Optical and Aperture Effects What s wrong if the system PSF is an impulse? What s wrong if the system PSF is excessively smooth with extent larger than the separation between pixels (excessively smooth disc)?