The Dynamic Range Problem. High Dynamic Range (HDR) Multiple Exposure Photography. Multiple Exposure Photography. Dr. Yossi Rubner.
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1 The Dynamic Range Problem High Dynamic Range (HDR) starlight Domain of Human Vision: from ~10-6 to ~10 +8 cd/m moonlight office light daylight flashbulb Dr. Yossi Rubner Some slides stolen from: Jack Tumblin and from Bill Freeman who stole them from Alyosha Efros who stole them from Paul Debevec 1???? 0 55 Range of Typical Displays: from ~1 to ~100 cd/m Multiple Exposure Photography Multiple Exposure Photography Real world 10-6 High dynamic range 10 6 Real world 10-6 High dynamic range 10 6 Picture Low contrast Picture Low contrast 3 4 1
2 Multiple Exposure Photography Multiple Exposure Photography Real world 10-6 High dynamic range 10 6 Real world 10-6 High dynamic range 10 6 Picture Low contrast Picture Low contrast 5 6 Multiple Exposure Photography Multiple Exposure Photography Real world 10-6 High dynamic range 10 6 Real world 10-6 High dynamic range 10 6 Picture Low contrast Picture Low contrast 7 8
3 High Dynamic Range Image - Example High Dynamic Range Images LDR HDR high low From: Danny Luong 10 9 Raanan Fattal, Dani Lischinski, Michael Werman HDR Problems In fact, there are problems: How to create an HDR image using limited dynamic range camera? How to display an HDR image on a limited dynamic range screen? How?
4 HDR image from multiple images Ways to Vary exposure: Options: Shutter speed Aperture ISO Neutral density filter 13 From: Bill Freeman, Frédo Durand, MIT - EECS Tradeoffs Shutter speed Range: ~30 sec to 1/4000sec (6 orders of magnitude) Pros: reliable, linear Cons: sometimes noise for long exposure Aperture Range: ~f/1.4 to f/ (.5 orders of magnitude) Cons: changes depth of field Useful when desperate ISO Range: ~100 to 1600 (1.5 orders of magnitude) Cons: noise Useful when desperate Neutral density filter Range: up to 4 densities (4 orders of magnitude) & can be stacked Cons: not perfectly neutral (color shift), not very precise, need to touch camera (shake) Pros: works with strobe/flash, good complement when desperate From: Bill Freeman, Frédo Durand, MIT - EECS 14 MANY ways to make multiple exposure measurments Sequential Exposure Change: HDR Capture Ginosar et al 9, Burt & Kolczynski 93, Madden 93, Tsai 94, Saito 95, Mann 95, Debevec & Malik 97, Mitsunaga & Nayar 99, Robertson et al. 99, Kang et al. 03 Multiple Image Detectors: Doi et al. 86, Saito 95, Saito 96, Kimura 98, Ikeda 98, Aggarwal & Ahuja 01, time Multiple Sensor Elements in a Pixel: Handy 86, Wen 89, Murakoshi 94, Konishi et al. 95, Hamazaki 96, Street 98 Assorted Pixels: Generalized Bayer Grid: Trade resolution for multiple exposure,color Nayar and Mitsunaga oo, Nayar and Narasimhan 0 R R R R G G G G B B B B G G G G G G G G R R R R B B B B G G G G G G G G R R R R B B B B G G G G G G G G R R R R B B B B G G G G
5 Assorted-Pixel Camera ( Courtesy : Sony Kihara Research Lab ) Response Curve of DSLR Digital Still Camera Camera with Assorted Pixels Response Curve (log) Why Log? 55 Pixel value 0 We're sensitive to contrast (multiplicative) A ratio of 1: is perceived as the same contrast as a ratio of 100 to 00 Makes sense because illumination has a multiplicative effect Use the log domain as much as possible log Exposure = log (Radiance * t) (CCD photon count)
6 Reconstructing HDR Image Given N photos at different exposure Recover a HDR color for each pixel Reconstructing HDR Image Simple if we have the response curve: Z ij = value of i th pixel in j th image. e.g. Exposure = Radiance * t t increases 1 Reconstructing HDR Image Reconstructing HDR Image Image series Pixel Value Z loge i t t = 1/64 sec 1 3 t t = 1/1616 sec 1 3 t t = 1/4 sec 1 3 t t = 1 sec Pixel Value Z = f(exposure) Exposure = Radiance t log Exposure = log Radiance + log t 1 3 t t = 4 sec 4 6
7 Pixel value Assuming unit radiance for each pixel 3 Response curve Exposure is unknown, fit to find a smooth curve 1 Pixel value After adjusting radiances to obtain a smooth response curve Recovering Response Curve For each pixel site i in each image j, we have: Since f is monotonic, it is invertible: Let g(z) be the discrete inverse response function log Exposure log Exposure 5 6 Recovering Response Curve Find g that minimizes: Recovering Response Curve The solution can be only up to a scale, add a constraint Add a hat weighting function fitting term smoothness term 7 8 7
8 np 1 54 Sparse linear system 56 n g(0) : g(55) lne 1 : : lne n = Ax=b 9 function [g,le]=gsolve(z,b,l,w) Matlab code n = 56; A = zeros(size(z,1)*size(z,)+n+1,n+size(z,1)); b = zeros(size(a,1),1); k = 1; %% Include the data-fitting equations for i=1:size(z,1) for j=1:size(z,) wij = w(z(i,j)+1); A(k,Z(i,j)+1) = wij; A(k,n+i) = -wij; b(k,1) = wij * B(i,j); k=k+1; end end A(k,19) = 1; %% Fix the curve by setting its middle value to 0 k=k+1; for i=1:n- %% Include the smoothness equations A(k,i)=l*w(i+1); A(k,i+1)=-*l*w(i+1); A(k,i+)=l*w(i+1); k=k+1; end x = A\b; %% Solve the system using SVD g = x(1:n); le = x(n+1:size(x,1)); 30 Result: digital camera Reconstructed radiance map Kodak DCS460 1/30 to 30 sec Recovered response curve Pixel value log Exposure Slide stolen from Alyosha Efros who stole it from Paul Debevec 31 3 Slide stolen from Alyosha Efros who stole it from Paul Debevec 8
9 Varying shutter speeds Recovered response curve Red Green Blue RGB Reconstructed Radiance Map 35 HDR Cameras HDR sensors using CMOS Use a log response curve e.g. SMaL, Assorted pixels Fuji Nayar et al. Fuji SuperCCD Per-pixel exposure Filter Integration time Multiple cameras using beam splitters Other computational photography tricks 36 9
10 Problem : Contrast reduction Match limited contrast of the medium Preserve details Real world 10-6 High dynamic range 10 6 Basic Assumptions The eye responds more to local intensity differences than global illumination A HDR image must have some large magnitude gradients Fine details consist only of smaller magnitude gradients Picture Low contrast The second half: contrast reduction Input: high-dynamic-range image (floating point per pixel) Naïve technique Scene has 1:10,000 contrast, display has 1:100 Simplest contrast reduction?
11 Naïve: Gamma compression X > X γ (where γ=0.5 in our case) But colors are washed-out. Why? Input Gamma Gamma compression on intensity s are OK, but details (intensity high-frequency) are blurred Intensity Gamma on intensity 41 4 Oppenheim 1968, Chiu et al Reduce contrast of low-frequencies Keep high frequencies Low-freq. Reduce low frequency The halo nightmare For strong edges Because they contain high frequency Low-freq. Reduce low frequency High-freq. High-freq
12 Using Bilateral Filter Do not blur across edges Non-linear filtering Large-scale Output Start with Gaussian filtering Here, input is a step function + noise J = f I Detail 45 output input 46 Gaussian filter as weighted average Weight of depends on distance to x J(x)= f (x,) I() The problem of edges Here, Ι() pollutes our estimate J(x) It is too different J(x)= f (x,) I() x x x Ι() I(x) output input 47 output input 48 1
13 Principle of Bilateral filtering [Tomasi and Manduchi 1998] Penalty g on the intensity difference J(x)= 1 k(x) f (x,) g(i() I(x)) I() Bilateral filtering Spatial Gaussian f J(x)= 1 k(x) f (x,)g(i() I(x)) I() x Ι() I(x) x x output input 49 output input 50 Bilateral filtering Spatial Gaussian f Gaussian g on the intensity difference J(x)= 1 k(x) f (x,) g(i() I(x))I() k(x)= J(x)= 1 Normalization factor f (x,) k(x) g(i() I(x)) f (x,) g(i() I(x)) I() x Ι() I(x) output input 51 output input 5 13
14 Why do we say it is non-linear? It does not respect bila(f+g)=bila(f)+bila(g) k(x)= J(x)= 1 Normalization factor f (x,) k(x) g(i() I(x)) f (x,) g(i() I(x)) I() 53 output input 54 Other view The bilateral filter uses the 3D distance Input HDR image Contrast too high! Contrast reduction
15 Input HDR image Contrast reduction Input HDR image Contrast reduction Intensity Intensity Large scale Fast Bilateral Filter Input HDR image Contrast reduction Input HDR image Contrast reduction Intensity Large scale Intensity Large scale Reduce contrast Large scale Fast Bilateral Filter Detail Fast Bilateral Filter Detail
16 Input HDR image Contrast reduction Input HDR image Contrast reduction Output Intensity Large scale Reduce contrast Large scale Intensity Large scale Reduce contrast Large scale Fast Bilateral Filter Detail Preserve! Detail Fast Bilateral Filter Detail Preserve! Detail 61 6 Gradient Domain High Dynamic Range Compression The Method in 1D Raanan Fattal, Dani Lischinski, Michael Werman The Hebrew University of Jerusalem 500:1 7.5:1 log derivative attenuate exp integrate
17 Basic Method Take the log of the luminances Calculate the gradient at each point Scale the magnitudes of the gradients with a progressive scaling function (Large magnitudes are scaled down more than small magnitudes) Re-integrate the gradients and invert the log to get the final image The Method in D Given: a log-luminance luminance image H(x,y) Compute an attenuation map Compute an attenuated gradient field G: (, ) = (, ) Φ( ) G x y H x y H Problem: G is not integrable! Φ( H ) Solution Euler-Lagrange Equation Look for image I with gradient closest to G in the least squares sense. I must satisfy: F I d dx F I d dy F x I y =0 I minimizes the integral: F ( I G) F (, ) I G dx dy I I, = I G = Gx + Gy x y Substituting F we get: I G xi G I G x yy + = 0 + = + x xy xy y I = divg
18 Attenuation Details Multiscale Gradient Attenuation Images contain edges at multiple levels of detail How do we handle this? Compute gradients for many different resolutions of the image The set of different resolution images composes a Gaussian pyramid 69 log(luminance) Gradient magnitude Attenuation map 70 Multiscale Gradient Attenuation Final Gradient Attenuation Map Interpolate X = Interpolate X =
19 Performance Measured on 1.8 GHz Pentium 4: 51 x 384: 1.1 sec 104 x 768: 4.5 sec 16 1 Streetlight on a foggy night Dynamic range 100,000:1 Examples Can be accelerated using processor-optimized optimized libraries Examples Stanford Memorial Church DR 50, :
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