1 00:00:05,542 --> 00:00:08,800 In this paper, we study the Light Transport Operator 2 00:00:08,800 --> 00:00:12,571 This operator transports radiance fields in a scene 3 00:00:12,570 --> 00:00:16,400 to the next surface, where it combines them with material properties 4 00:00:17,257 --> 00:00:20,571 Applying the operator to a light source as shown here 5 00:00:20,570 --> 00:00:22,285 gives direct illumination 6 00:00:22,742 --> 00:00:26,457 Applying it a second time gives the next bounce, etc 7 00:00:27,142 --> 00:00:30,560 It is known that the light transport equilibrium in a scene 8 00:00:30,560 --> 00:00:35,257 can be obtained by adding all bounces produced by this operator 9 00:00:36,754 --> 00:00:40,409 When the scene is Lambertian, this operator can be 10 00:00:40,409 --> 00:00:44,800 re-formulated in the space of radiant exitance distributions 11 00:00:44,800 --> 00:00:50,506 Both operators T and Tb act on the infinite dimensional 12 00:00:50,506 --> 00:00:54,994 spaces of light distributions over the scene 13 00:00:56,057 --> 00:01:00,262 Consequently, many light transport problems are solved numerically 14 00:01:00,514 --> 00:01:03,622 by projecting radiance, or radiant exitance, 15 00:01:03,622 --> 00:01:07,165 onto some finite dimensional spaces of functions 16 00:01:07,160 --> 00:01:12,811 and express T (or Tb) in this space, as a matrix Tn 17 00:01:13,908 --> 00:01:19,062 These functions can be piecewise constant functions, as in old radiosity methods 18 00:01:19,337 --> 00:01:22,354 wavelets, spherical harmonics, polynomials, 19 00:01:22,350 --> 00:01:24,365 or any combination of these. 20 00:01:25,348 --> 00:01:28,617 Many methods rely on such finite rank approximations 21 00:01:29,805 --> 00:01:31,862 Finite element global illumination 22 00:01:31,860 --> 00:01:35,405 in which the implicit assumption is that 23 00:01:35,702 --> 00:01:39,485 for the actual solution L, the matrix approaches the operator 24 00:01:41,040 --> 00:01:43,040 Precomputed radiance transfer methods 25 00:01:43,040 --> 00:01:45,028 where the assumption is stronger 26 00:01:45,020 --> 00:01:48,480 The matrix should indeed give a uniform error 27 00:01:48,480 --> 00:01:51,988 across an entire subspace of light distributions 28 00:01:53,257 --> 00:01:56,171 Pretty much the same is expected for neural rendering 29 00:01:56,170 --> 00:02:00,502 where the learning phase implicitly builds a finite rank approximation of T 30 00:02:01,725 --> 00:02:05,524 Inverse lighting. In this case a finite rank approximation of T 31 00:02:05,524 --> 00:02:07,885 is used to approximate the inverse of T 32 00:02:09,165 --> 00:02:13,365 ...and dimensional analysis where the eigenvalues of Tn 33 00:02:13,365 --> 00:02:16,971 are supposed to converge to the eigenvalues of T 34 00:02:18,560 --> 00:02:23,588 While most analyses in the literature have been done with discretisation, 35 00:02:24,171 --> 00:02:26,834 in this paper we study the original operators 36 00:02:26,914 --> 00:02:32,457 and show that some of the above assumptions are not at all straightforward 37 00:02:34,000 --> 00:02:38,685 Generally speaking, linear operators in infinite dimensions behave badly 38 00:02:38,800 --> 00:02:42,445 for instance they may not have a countable set of eigenvalues 39 00:02:42,754 --> 00:02:44,057 contrary to a matrix 40 00:02:44,730 --> 00:02:46,491 and even if they do 41 00:02:47,371 --> 00:02:50,045 these eigenvalues may have infinite multiplicity 42 00:02:51,085 --> 00:02:52,457 A consequence of that 43 00:02:52,891 --> 00:02:56,331 is that the spectrum of a matrix approximating the operator 44 00:02:56,605 --> 00:02:59,257 may not converge to the spectrum of the operator itself 45 00:03:00,411 --> 00:03:02,320 Another consequence is that 46 00:03:02,800 --> 00:03:06,331 finite rank approximations are not guaranteed to be uniform 47 00:03:07,074 --> 00:03:10,617 In other words, there is no matrix approximation 48 00:03:11,165 --> 00:03:15,474 that will guaranty a maximum error across all light distributions 49 00:03:17,470 --> 00:03:22,274 A subclass of infinite dimensional linear operators however behave better: 50 00:03:23,245 --> 00:03:25,600 these are the so-called compact operators 51 00:03:26,422 --> 00:03:29,028 Compact operators are the operators with 52 00:03:29,154 --> 00:03:32,297 countable eigenvalues that converge to 0 53 00:03:33,017 --> 00:03:37,028 In this case, the spectrum of a matrix approaching the operator 54 00:03:37,520 --> 00:03:40,000 is going to converge to the spectrum of the operator 55 00:03:40,262 --> 00:03:42,857 and finite rank approximations of the operator 56 00:03:42,850 --> 00:03:48,251 come with a uniform error bound across all distributions 57 00:03:49,165 --> 00:03:51,988 So, now, the question is: 58 00:03:52,400 --> 00:03:55,245 Are the light transport operators compact? 59 00:03:57,240 --> 00:04:00,628 In this paper we show that T is never compact 60 00:04:01,565 --> 00:04:05,314 However, T still shares some key properties with compact operators 61 00:04:05,870 --> 00:04:08,182 In particular we show that in closed scenes 62 00:04:08,250 --> 00:04:09,828 T has a Schmidt expansion 63 00:04:10,491 --> 00:04:14,297 which stands for a SVD in infinite dimensions 64 00:04:15,565 --> 00:04:18,537 We also show that Tb is not compact either 65 00:04:19,348 --> 00:04:23,200 But still Tb keeps some interesting properties of compact operators 66 00:04:23,531 --> 00:04:25,531 First of all, it is not invertible 67 00:04:26,137 --> 00:04:29,805 and it acts as a low pass filter almost everywhere 68 00:04:30,994 --> 00:04:33,931 We also prove that the reflectance operator Kx 69 00:04:34,731 --> 00:04:39,085 which combines the incident illumination with the material at point x 70 00:04:39,565 --> 00:04:42,628 is compact, which makes it not invertible 71 00:04:43,828 --> 00:04:46,742 Finally in the paper we connect these findings 72 00:04:46,740 --> 00:04:51,257 to the various choices that characterize low rank approximations in the literature 73 00:04:53,211 --> 00:04:57,897 Now I'll quickly explain how we prove that Tb is not compact 74 00:04:58,731 --> 00:05:02,137 Compact linear operators map bounded sequences 75 00:05:02,130 --> 00:05:05,702 into sequences with converging subsequences 76 00:05:06,491 --> 00:05:09,131 So, to prove non-compactness 77 00:05:09,428 --> 00:05:11,931 we just need to find one sequence 78 00:05:11,930 --> 00:05:13,862 of radiosity functions 79 00:05:13,860 --> 00:05:17,348 that is mapped to a sequence which elements 80 00:05:17,360 --> 00:05:19,690 are always far away from each other 81 00:05:20,617 --> 00:05:22,537 In a finite dimensional space 82 00:05:22,530 --> 00:05:24,342 this would obviously not be possible 83 00:05:24,340 --> 00:05:27,074 for a bounded operator such as Tb 84 00:05:28,205 --> 00:05:31,234 But in infinite dimensions, we can 85 00:05:31,230 --> 00:05:33,211 create such a sequence 86 00:05:33,210 --> 00:05:37,051 by leveraging the increase of frequency in its elements 87 00:05:37,050 --> 00:05:40,685 so as to keep each element away from all other elements 88 00:05:42,468 --> 00:05:46,548 This is precisely what happens next to an abutting edge 89 00:05:46,868 --> 00:05:50,377 We choose each radiosity distribution Bn 90 00:05:50,370 --> 00:05:52,697 to be constant over a disk 91 00:05:52,690 --> 00:05:55,062 which radius decreases with n 92 00:05:56,194 --> 00:05:59,280 Keeping the Bn sequence bounded is easy 93 00:05:59,280 --> 00:06:02,548 we increase the radiant exitance over the disk 94 00:06:02,540 --> 00:06:06,377 so as to keep the norm of B_n constant with n 95 00:06:07,348 --> 00:06:10,525 The sequence of transported energy distributions 96 00:06:10,520 --> 00:06:13,782 Tb times Bn is of course bounded as well 97 00:06:13,794 --> 00:06:18,377 since the total energy received will always be less than the emitted power 98 00:06:19,428 --> 00:06:22,537 However, we show in our paper that 99 00:06:22,530 --> 00:06:27,337 the frequency of transported radiosity increases in this case 100 00:06:27,330 --> 00:06:30,937 keeping the transported distributions far away from each other 101 00:06:32,040 --> 00:06:36,091 Consequently, no subsequence of Tb times Bn 102 00:06:36,491 --> 00:06:38,114 can ever converge 103 00:06:38,490 --> 00:06:42,125 Of course a much more detailed proof 104 00:06:42,148 --> 00:06:43,920 is provided in our paper 105 00:06:45,702 --> 00:06:49,542 The first consequence of light transport operators not being compact 106 00:06:49,737 --> 00:06:53,394 is that there is no uniform finite rank approximations of these operators 107 00:06:54,480 --> 00:06:55,428 In other words, 108 00:06:55,860 --> 00:06:57,817 For any finite rank approximation 109 00:06:58,000 --> 00:07:00,400 There is no guarantee on the error 110 00:07:00,400 --> 00:07:03,428 across all light distributions of unit norm 111 00:07:04,400 --> 00:07:07,188 This is why adaptive methods are needed 112 00:07:07,180 --> 00:07:09,817 when solving for global illumination, 113 00:07:10,525 --> 00:07:12,365 this is also why they require guidance 114 00:07:14,651 --> 00:07:16,130 Another consequence is that 115 00:07:16,210 --> 00:07:19,840 neither precomputed radiance transfer methods 116 00:07:19,874 --> 00:07:21,874 nor deep learning methods 117 00:07:22,274 --> 00:07:26,354 can give error guarantees beyond their learning space 118 00:07:28,000 --> 00:07:32,000 Similarly, spectral analysis based on finite rank approximations 119 00:07:32,022 --> 00:07:34,308 need some additional justification 120 00:07:35,428 --> 00:07:37,234 Another consequence is that 121 00:07:37,230 --> 00:07:41,462 approaching light transport by operators with bounded kernels 122 00:07:41,460 --> 00:07:45,348 is not going to converge uniformly to the real light transport operator 123 00:07:46,400 --> 00:07:48,994 That explains somehow the bias 124 00:07:49,382 --> 00:07:52,194 that shows up when bounding the weight 125 00:07:53,142 --> 00:07:56,754 of connecting close points in path tracing 126 00:07:58,628 --> 00:08:01,622 Since reflectance is compact, it is not invertible 127 00:08:01,890 --> 00:08:05,810 Consequently, solving for inverse reflectance 128 00:08:05,810 --> 00:08:05,828 Using a finite rank approximation Consequently, solving for inverse reflectance 129 00:08:05,828 --> 00:08:08,342 Using a finite rank approximation 130 00:08:08,377 --> 00:08:10,674 is probably not a good idea 131 00:08:11,690 --> 00:08:13,222 The same happens to Tb 132 00:08:13,382 --> 00:08:15,828 that is not invertible either 133 00:08:16,662 --> 00:08:19,108 If we try to for instance find 134 00:08:19,100 --> 00:08:22,537 the radiosity distribution in the top square that 135 00:08:22,530 --> 00:08:26,034 produces a step function on the bottom square 136 00:08:26,285 --> 00:08:29,394 the more we mesh, the higher the values 137 00:08:29,390 --> 00:08:31,474 and the higher the frequencies 138 00:08:32,914 --> 00:08:36,102 it looks like the solution somehow tries to escape 139 00:08:36,100 --> 00:08:39,097 the space in which we look for it 140 00:08:40,330 --> 00:08:42,274 It should be noted however that 141 00:08:42,270 --> 00:08:45,668 inverting multi-bounce transport is trivial. 142 00:08:46,377 --> 00:08:50,365 Our conclusion is that the light transport operators 143 00:08:50,360 --> 00:08:52,937 are generally not well behaved 144 00:08:52,930 --> 00:08:56,217 and one needs to be very cautious when using 145 00:08:56,210 --> 00:08:57,988 finite rank approximations 146 00:08:57,980 --> 00:09:02,811 We also prove that T_b in the Lambertian case 147 00:09:03,588 --> 00:09:07,840 is not invertible, which was not obvious to us at first 148 00:09:09,245 --> 00:09:12,971 Interestingly the cause of these deficiencies 149 00:09:12,970 --> 00:09:14,400 is not visibility 150 00:09:14,400 --> 00:09:17,988 Usually visibility is...what causes problems 151 00:09:18,628 --> 00:09:24,354 Here T is not compact because T is a partial integral operator 152 00:09:24,868 --> 00:09:28,422 and Tb is not compact because of abutting edges 153 00:09:29,668 --> 00:09:32,297 So, that concludes my presentation 154 00:09:32,290 --> 00:09:33,805 Thanks you for your attention