Wiki source code of Inverse problems, spring 2017

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1 = Inverse problems, spring 2017 =
2
3 [[image:attach:ShearletLotus.png||height="250"]]
4
5 (% style="color: rgb(153,51,0);" %)Question: What on earth is this picture about?
6
7 (% style="color: rgb(153,51,0);" %)Answer: It is an example of X-ray tomography!(%%) **[[See this video>>url:https://www.youtube.com/watch?v=eWwD_EZuzBI||shape="rect"]].(% style="color: rgb(51,153,102);" %) (%%)[[And this page>>url:http://www.siltanen-research.net/IPexamples/xray_tomography||shape="rect"]].**
8
9 ----
10
11 === Class of 2017 (poster session): ===
12
13 **[[image:attach:8U8A0825.jpg||height="400"]]
14 **
15
16 {{panel}}
17 **Teacher:** [[Samuli Siltanen>>doc:mathstatHenkilokunta.Siltanen, Samuli]]
18
19 **Scope:** 15 cr (lecture part and project work combined)
20
21 **Type:** Advanced studies
22
23 **Teaching:**
24
25 **Topics: **Inverse problems are about measuring something indirectly and trying to recover that something from the data. For example, a doctor may take several X-ray images of a patient from different directions and wish to understand the three-dimensional structure of the patient's inner organs. But each of the 2D images only shows a projection of the inner organs; one has to actually calculate the 3D structure using a reconstruction algorithm. This course teaches how to
26
27 * model a (linear) measurement process as a matrix equation m = Ax + noise,
28 * detect if the matrix A leads to an ill-posed inverse problem,
29 * design and implement a regularized reconstruction method for recovering x from m. We study truncated singular value decomposition, Tikhonov regularization, total variation regularization and wavelet-based sparsity,
30 * measure tomographic data in X-ray laboratory,
31 * report your findings in the form of a scientific poster.
32
33 See [[this page>>url:http://www.siltanen-research.net/IPexamples/inverse_problems||shape="rect"]] for more information about inverse problems.
34
35 Here are the old homepages of this course: [[2015>>url:http://wiki.helsinki.fi/display/mathstatKurssit/Inverse+problems%2C+spring+2015||shape="rect"]], [[2014>>url:https://wiki.helsinki.fi/display/mathstatKurssit/Inverse+problems%2C+spring+2014||shape="rect"]], [[2013>>url:https://wiki.helsinki.fi/pages/viewpage.action?pageId=96704134||shape="rect"]]. This year the course is roughly the same than in those past years.
36
37 **Prerequisites: **Linear algebra, basic Matlab programming skills, interest in practical applications, and a curious mind. The course is suitable (and very useful) for students of mathematics, statistics, physics or computer science.
38 {{/panel}}
39
40 === [[image:attach:hammerpic1_512.bmp||height="250"]][[image:attach:BlurredNoisy_orig.png||height="250"]] ===
41
42 Consider the above photographic situation. You would like to have taken the left image~: sharp and beautiful. However, due to misfocusing and high ISO setting in the camera, you ended up with the blurry and noisy picture on the right. Is there any chance to save the day? Well, one can apply deblurring, one of the topics of this course. Below you see three different deblurring methods with varying success. From left to right: basic regularization, total variation, and total generalized variation. (Thanks to Professor Kristian Bredies for sharing his amazing TGV algorithm!)
43
44 [[image:attach:recon0.bmp||height="250"]][[image:attach:BlurredNoisy_tv.png||height="250"]][[image:attach:BlurredNoisy_tgv.png||height="250"]]
45
46 {{toc maxLevel="4" minLevel="2" indent="20px"/}}
47
48 == News ==
49
50 [[image:attach:Screen Shot 2017-02-17 at 08.26.40.png||height="400"]]
51
52 Results of the mid-course questionnaire (Feb 15). It seems that all is going pretty well in general. Let's discuss in the lecture about how to make finding material easier.
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54
55
56 (% style="color: rgb(255, 0, 255); color: rgb(255, 0, 0)" %)**[[~[~[image:attach:YouTubeLogo_smaller.png~]~]>>url:https://www.youtube.com/playlist?list=PLyIjfdC_fHWYSVIcrNtV9Hr7zAGE3GF-6||style="text-decoration: underline;" shape="rect"]] **(% style="color: rgb(255,0,255);" %)[[(% style="color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)New lecture video playlist>>url:https://www.youtube.com/playlist?list=PL5vSJKR6SrD9124QEgO3_h1VPFgjG2lSQ||shape="rect"]]
57
58 (% style="color: rgb(255,0,0);" %)**
59 **
60
61 ----
62
63 (% style="color: rgb(255,0,0);" %)** **
64
65 (% style="color: rgb(255,0,0);" %)**Lecture 16 (March 24)**
66
67 (% style="color: rgb(0,0,0);" %)Project work info. The goals of this session are
68
69 * (% style="color: rgb(0,0,0);" %)Dividing students into project teams of two members each,
70 * (% style="color: rgb(0,0,0);" %)Choosing the topic (tomography or deconvolution data, what is the focus of the project, what methods to try),
71 * (% style="color: rgb(0,0,0);" %)Understanding the goals of Phase 1 and Phase 2 of the project,
72 * (% style="color: rgb(0,0,0);" %)Agreeing on meeting times for reviewing Phase 1 of the projects.
73
74
75 (% style="color: rgb(0,0,0);" %)
76
77
78 (% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0)" %)**Lecture 15 (March 22)**
79
80 (% style="color: rgb(0,0,0);" %)Real-data X-ray tomography. How to go from projection images to line integral data? What are the crucial properties of the variants of tomography, such as limited-angle, region-of-interest, sparse and exterior tomography?
81
82 (% style="color: rgb(0,0,0);" %)
83
84
85 (% style="color: rgb(0, 0, 0); color: rgb(255, 0, 0)" %)**[[~[~[image:attach:YouTubeLogo_smaller.png~]~]>>url:https://www.youtube.com/playlist?list=PLyIjfdC_fHWYSVIcrNtV9Hr7zAGE3GF-6||style="text-decoration: underline;" shape="rect"]]**
86
87 (% style="color: rgb(255,0,255);" %)[[(% style="color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 1/2 (screen capture)>>url:https://www.youtube.com/watch?v=ZuepayW9U2I&list=PL5vSJKR6SrD9124QEgO3_h1VPFgjG2lSQ&index=24||shape="rect"]](%%)
88 [[(% style="color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 2/2 (screen capture)>>url:https://www.youtube.com/watch?v=0fR4cLh84ng&list=PL5vSJKR6SrD9124QEgO3_h1VPFgjG2lSQ&index=25||shape="rect"]](%%)
89
90 ----
91
92 (% style="color: rgb(255,0,0);" %)**
93 **
94
95 (% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0)" %)**Lecture 14 (March 17)**
96
97 Choosing the regularization parameter. There are several methods, applicable to more or less of the different regularization approaches.
98
99 * L-curve method (mainly for Tikhonov)
100 * Morozov discrepancy principle (Tikhonov, has been extended to TV in recent research from Hong Kong)
101 * S-curve method (for sparsity-promoting regularization)
102 * Multi-resolution consistence method (for TV, see [[this file>>url:http://www.siltanen-research.net/publ/NiinimakiLassasHamalainenKallonenKolehmainenNiemiSiltanen2016.pdf||shape="rect"]])
103
104 We will take a look of these methods and test some of them.
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106
107
108 (% style="color: rgb(255,0,0);" %)**[[~[~[image:attach:YouTubeLogo_smaller.png~]~]>>url:https://www.youtube.com/playlist?list=PLyIjfdC_fHWYSVIcrNtV9Hr7zAGE3GF-6||style="text-decoration: underline;" shape="rect"]]**(%%)
109 (% style="color: rgb(255,0,255);" %)[[(% style="color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 1/2 (screen capture)>>url:https://www.youtube.com/watch?v=55cIKWHIS84&list=PL5vSJKR6SrD9124QEgO3_h1VPFgjG2lSQ&index=22||shape="rect"]](%%)
110 (% style="color: rgb(255,0,255);" %)[[(% style="color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 2/2 (screen capture)>>url:https://www.youtube.com/watch?v=RpAo-AkS5X0&index=23&list=PL5vSJKR6SrD9124QEgO3_h1VPFgjG2lSQ||shape="rect"]]
111
112 ----
113
114 (% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0)" %)**
115 **
116
117 (% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0); color: rgb(255, 0, 0)" %)**Lecture 13 (March 15)**
118
119 (% style="color: rgb(0,0,0);" %)Wavelet transform in dimensions 1 and 2. Large-scale sparsity-promoting reconstruction with Iterative Soft Thresholding Algorithm (ISTA).
120
121 (% style="color: rgb(255,0,0);" %)**[[image:attach:MatlabLogoSmaller.png]]**(% style="color: rgb(255, 0, 0); color: rgb(0, 51, 102)" %)** **Matlab resources:
122
123 (% style="color: rgb(255, 0, 0); color: rgb(0, 51, 102)" %)[[attach:recon_DB_comp.m]]
124 [[attach:recon_Haar_comp.m]](% style="color: rgb(0,51,102);" %)
125 [[attach:scale01.m]]
126 [[attach:Smu_wavelet_oper.m]]
127 [[attach:Smu.m]]
128 [[attach:thresholding_demo.m]]
129 [[attach:wavetrans2D_inv.m]]
130 [[attach:wavetrans2D.m]]
131 [[attach:wavetrans2Donce_inv.m]]
132 [[attach:wavetrans2Donce.m]]
133
134 (% style="color: rgb(0, 51, 102); color: rgb(255, 0, 0)" %)**
135 **
136
137 (% style="color: rgb(0, 51, 102); color: rgb(255, 0, 0)" %)**[[~[~[image:attach:YouTubeLogo_smaller.png~]~]>>url:https://www.youtube.com/playlist?list=PLyIjfdC_fHWYSVIcrNtV9Hr7zAGE3GF-6||style="text-decoration: underline;" shape="rect"]]**
138
139 (% style="color: rgb(255,0,255);" %)[[(% style="color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 1/2 (screen capture)>>url:https://www.youtube.com/watch?v=IBZMTEMtIQk&list=PL5vSJKR6SrD9124QEgO3_h1VPFgjG2lSQ&t=5s&index=19||shape="rect"]](%%)
140 [[(% style="color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 2/2 (screen capture) >>url:https://www.youtube.com/watch?v=B8nENjuv3uU&index=21&list=PL5vSJKR6SrD9124QEgO3_h1VPFgjG2lSQ||shape="rect"]](% style="color: rgb(255,0,0);" %)** **
141
142 ----
143
144 (% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0); color: rgb(255, 0, 0)" %)**
145 **
146
147 (% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0); color: rgb(255, 0, 0); color: rgb(255, 0, 0)" %)**Lecture 12 (Friday, March 3)**
148
149 (% style="color: rgb(0,0,0);" %)Planning the remaining lectures of the course. We used the "double team" brainstorming method and ended up with this set of suggestions:
150
151 (% style="color: rgb(0,0,0);" %)[[image:attach:tuplatiimi.jpg||height="250"]]
152
153 (% style="color: rgb(0,0,0);" %)The results of the planning are seen in the themes for lectures 13, 14 and 15 above.
154
155 (% style="color: rgb(255,0,0);" %)** **
156
157 ----
158
159 (% style="color: rgb(255,0,0);" %)**
160 **
161
162 (% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0)" %)**Wednesday, March 1. The lecturer is sick (damn this flu season!). **
163
164 (% style="color: rgb(255, 0, 0); color: rgb(0, 0, 0)" %)Instead of the lecture, please read Section 3.2 of this note on the Haar wavelet transform:[[attach:FourierSeries_Wavelets_v3.pdf]]
165
166 (% style="color: rgb(255, 0, 0); color: rgb(0, 0, 0)" %) Study these Matlab routines and clarify to yourself how the codes and the text are connected.(% style="color: rgb(255,0,0);" %)** **
167
168 (% style="color: rgb(255,0,0);" %)**[[image:attach:MatlabLogoSmaller.png]]**(% style="color: rgb(255, 0, 0); color: rgb(0, 51, 102)" %)** **Matlab resources:
169
170 (% style="color: rgb(255, 0, 0); color: rgb(0, 51, 102)" %)[[attach:Wavelet_tr_test.m]]
171
172 (% style="color: rgb(255, 0, 0); color: rgb(0, 51, 102)" %)[[attach:Wavelet_tr_onestep.m]]
173
174 (% style="color: rgb(255, 0, 0); color: rgb(0, 51, 102)" %)[[attach:Wavelet_tr_inv_onestep.m]]
175
176 (% style="color: rgb(255, 0, 0); color: rgb(0, 51, 102)" %)[[attach:Wavelet_tr.m]]
177
178 (% style="color: rgb(255, 0, 0); color: rgb(0, 51, 102)" %)[[attach:Wavelet_tr_inv.m]](% style="color: rgb(255,0,0);" %) (% style="color: rgb(255, 0, 0); color: rgb(0, 51, 102)" %)**
179 **
180
181 ----
182
183 (% style="color: rgb(255,0,0);" %)**
184 **
185
186 (% style="color: rgb(255,0,0);" %)**Lecture 11 (February 24)**
187
188 (% style="color: rgb(0,0,0);" %)Recap of the construction of the tomographic "rectangle" phantom. Simulating tomographic data with no inverse crime, using higher-resolution simulation and interpolation. Applying
189
190 * (% style="color: rgb(0,0,0);" %)Truncated SVD (TSVD),
191 * (% style="color: rgb(0,0,0);" %)Tikhonov regularization, and
192 * (% style="color: rgb(0,0,0);" %)Total Variation regularization (% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0)" %)
193
194
195 (% style="color: rgb(0,0,0);" %)
196 (% style="color: rgb(0,0,0);" %)for the sparse-angle tomographic problem. Note that the Tikhonov regularization code above is written in a matrix-free fashion using the Conjugate Gradient (CG) method. We will talk about CG and other iterative solvers later in the course. The Total Variation computation is a two-dimensional adaptation of the quadratic programming formulation including inequality and equality constraints.
197
198 (% style="color: rgb(0,0,0);" %)
199 (% style="color: rgb(0,0,0);" %)The files MyDS2.m and MyDScol.m are simple routines for downsampling an image by a factor of two.
200
201 (% style="color: rgb(0,0,0);" %)
202 (% style="color: rgb(0,0,0);" %)
203
204 (% style="color: rgb(0,0,0);" %)See Sections 2.3, 4.4, 5.5, 6.2 and 5.2 in the book Mueller-Siltanen (2012).
205
206 (% style="color: rgb(0,0,0);" %)
207 (% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0)" %)[[image:attach:MatlabLogoSmaller.png]](% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0); color: rgb(0, 51, 102)" %) Matlab resources:
208
209 (% style="color: rgb(0,0,0);" %)
210 (% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0); color: rgb(0, 51, 102)" %)[[attach:SqPhantom.m]]
211 [[attach:SqPhantom_plot.m]]
212 [[attach:tomo01_RadonMatrix_comp.m]]
213 [[attach:tomo02_firstTSVD_comp.m]]
214 [[attach:tomo02_firstTSVD_plot.m]]
215 [[attach:tomo03_NoCrimeData_comp.m]]
216 [[attach:tomo03_NoCrimeData_plot.m]]
217 [[attach:tomo04_Tikhonov_comp.m]]
218 [[attach:tomo04_Tikhonov_plot.m]]
219 [[attach:tomo07_TV_comp.m]]
220 [[attach:tomo07_TV_plot.m]]
221 [[attach:MyDS2.m]]
222 [[attach:MyDScol.m]]
223
224
225 (% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0)" %)**[[~[~[image:attach:YouTubeLogo_smaller.png~]~]
226 >>url:https://www.youtube.com/playlist?list=PLyIjfdC_fHWYSVIcrNtV9Hr7zAGE3GF-6||style="text-decoration: underline;" shape="rect"]]**(% style="color: rgb(255,0,255);" %)[[(% style="color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 1/2 (screen capture)>>url:https://www.youtube.com/watch?v=x0wNtc9ZsRM&index=18&list=PL5vSJKR6SrD9124QEgO3_h1VPFgjG2lSQ||shape="rect"]](%%)[[(% style="color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)
227 >>url:https://www.youtube.com/watch?v=x0wNtc9ZsRM&index=18&list=PL5vSJKR6SrD9124QEgO3_h1VPFgjG2lSQ||shape="rect"]](%%)[[(% style="color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 2/2 >>url:https://www.youtube.com/watch?v=E6SpXWgjeqA&index=19&list=PL5vSJKR6SrD9124QEgO3_h1VPFgjG2lSQ||shape="rect"]](%%)[[(% style="color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)
228 >>url:https://www.youtube.com/watch?v=E6SpXWgjeqA&index=19&list=PL5vSJKR6SrD9124QEgO3_h1VPFgjG2lSQ||shape="rect"]]
229
230 ----
231
232 (% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0)" %)** **
233
234 (% style="color: rgb(255,0,0);" %)**THERE IS NO LECTURE on Wednesday, February 22. The lecturer is traveling. **
235
236 (% style="color: rgb(255, 0, 0); color: rgb(0, 0, 0)" %)Instead of the lecture, please run all the Matlab files given below related to Lecture 10 using this year's target signal AND with the photographic data collected in Lecture 9.
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238 (% style="color: rgb(255, 0, 0); color: rgb(0, 0, 0)" %)
239
240 ----
241
242 (% style="color: rgb(255, 0, 0); color: rgb(0, 0, 0)" %)
243
244
245 (% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0)" %)**Lecture 10 (February 17)**
246
247 (% style="color: rgb(0,0,0);" %)Discussed were three forms of variational regularization: Tikhonov, generalized Tikhonov and Total Variation regularization. We focused on the one-dimensional convolution model and used a simple finite-difference matrix L in the regularization terms.
248
249 (% style="color: rgb(0,0,0);" %)Matlab demonstrations were based on the routines developed in the 2015 Inverse Problems course. They are given below. I recommend you to run them through with this year's target and with the photographic data collected in Lecture 9.
250
251 (% style="color: rgb(0,0,0);" %)Note that whenever you change the dimension n of the unknown, you need to run deconv2_discretedata_comp.m, deconv3_naive_comp.m, deconv4_SVD_comp.m and deconv5_truncSVD_comp.m before running any of the Tikhonov or TV routines.
252
253 (% style="color: rgb(0,51,102);" %)[[image:attach:MatlabLogoSmaller.png]] Matlab resources:
254
255 (% style="color: rgb(0,51,102);" %)[[attach:DC_convmtx.m]]
256 [[attach:deconv1_cont_comp.m]]
257 [[attach:deconv1_cont_plot.m]]
258 [[attach:deconv2_discretedata_comp.m]]
259 [[attach:deconv2_discretedata_plot.m]]
260 [[attach:deconv3_naive_comp.m]]
261 [[attach:deconv3_naive_plot.m]]
262 [[attach:deconv4_SVD_comp.m]]
263 [[attach:deconv4_SVD_plot.m]]
264 [[attach:deconv5_truncSVD_comp.m]]
265 [[attach:deconv5_truncSVD_plot.m]]
266 [[attach:deconv6_Tikhonov_comp.m]]
267 [[attach:deconv6_Tikhonov_plot.m]]
268 [[attach:deconv7_genTikhonov_comp.m]]
269 [[attach:deconv7_genTikhonov_plot.m]]
270 [[attach:deconv8_L1reg_comp.m]]
271 [[attach:deconv8_L1reg_plot.m]]
272 [[attach:deconv9_TVreg_comp.m]]
273 [[attach:deconv9_TVreg_plot.m]]
274
275
276 (% style="color: rgb(0,51,102);" %)
277 (% style="color: rgb(0, 51, 102); color: rgb(255, 0, 0)" %)**[[~[~[image:attach:YouTubeLogo_smaller.png~]~]
278 >>url:https://www.youtube.com/playlist?list=PLyIjfdC_fHWYSVIcrNtV9Hr7zAGE3GF-6||style="text-decoration: underline;" shape="rect"]]**(% style="color: rgb(0, 51, 102); color: rgb(255, 0, 255)" %)[[(% style="color: rgb(0, 51, 102); color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 1/3>>url:https://www.youtube.com/watch?v=ZMYSDWyRuuY&list=PL5vSJKR6SrD9124QEgO3_h1VPFgjG2lSQ&index=16||shape="rect"]](% style="color: rgb(0,51,102);" %)
279 (% style="color: rgb(0, 51, 102); color: rgb(255, 0, 255)" %)[[(% style="color: rgb(0, 51, 102); color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 2/3>>url:https://www.youtube.com/watch?v=4mra02nNNx8&list=PL5vSJKR6SrD9124QEgO3_h1VPFgjG2lSQ&index=16||shape="rect"]](% style="color: rgb(0,51,102);" %)
280 (% style="color: rgb(0, 51, 102); color: rgb(255, 0, 255)" %)[[(% style="color: rgb(0, 51, 102); color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 3/3 (screen capture)>>url:https://www.youtube.com/watch?v=tbipOeP1gwE&list=PL5vSJKR6SrD9124QEgO3_h1VPFgjG2lSQ&index=17||shape="rect"]](% style="color: rgb(0,51,102);" %)
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282
283
284 ----
285
286 (% style="color: rgb(0,51,102);" %)
287
288 (% style="color: rgb(255,0,0);" %)** Lecture 9 (February 15) **
289
290 Photographic data with varying levels of blur (from defocusing of the lens) and noise (from increasing the ISO) as measurements.
291
292 We photographed printouts of these two pictures posted to the wall:
293
294 [[image:attach:EANcode.jpg||thumbnail="true" height="150"]]{{view-file att--filename="edge.tif" height="150"/}}
295
296 Further, we picked out rows from the edge image and analysed the pixels values in Matlab.
297
298 [[image:attach:signals.png||height="400"]]
299
300 Here are the two edge images we shot and processed: 
301 [[attach:edge_blur3.tif]]
302 [[attach:edge_blur3_noisy.tif]]
303
304 [[image:attach:MatlabLogoSmaller.png]](% style="color: rgb(0,0,0);" %) Matlab resources:
305
306 (% style="color: rgb(0,0,0);" %)[[attach:deco03_data_meas.m]]
307 [[attach:deco03_data_plot.m]]
308 [[attach:deco04_TSVD_comp.m]]
309 [[attach:deco04_TSVD_plot.m]]
310
311
312 (% style="color: rgb(255,0,0);" %)**[[~[~[image:attach:YouTubeLogo_smaller.png~]~]
313 >>url:https://www.youtube.com/playlist?list=PLyIjfdC_fHWYSVIcrNtV9Hr7zAGE3GF-6||style="text-decoration: underline;" shape="rect"]]**(% style="color: rgb(255, 0, 0); color: rgb(255, 0, 255)" %)[[(% style="color: rgb(255, 0, 0); color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 1/>>url:https://www.youtube.com/watch?v=q5BgnBk7RWk&list=PL5vSJKR6SrD9124QEgO3_h1VPFgjG2lSQ&index=5||shape="rect"]](%%)4(% style="color: rgb(255,0,0);" %)
314 (% style="color: rgb(255, 0, 0); color: rgb(255, 0, 255)" %)[[(% style="color: rgb(255, 0, 0); color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 2/>>url:https://www.youtube.com/watch?v=F5W6IPP6698&list=PL5vSJKR6SrD9124QEgO3_h1VPFgjG2lSQ&index=6||shape="rect"]](%%)4(% style="color: rgb(255,0,0);" %)
315 (% style="color: rgb(255,0,255);" %)[[(% style="color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 3/4 (screen capture)>>url:https://www.youtube.com/watch?v=lyqFp7zSLhw&index=8&list=PL5vSJKR6SrD9124QEgO3_h1VPFgjG2lSQ||shape="rect"]](%%) (%%)
316 (% style="color: rgb(255,0,255);" %)[[(% style="color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 4/4 (screen capture)>>url:https://www.youtube.com/watch?v=aRSUp169qoI&index=7&list=PL5vSJKR6SrD9124QEgO3_h1VPFgjG2lSQ||shape="rect"]](% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0)" %)
317 (% style="color: rgb(255,0,0);" %)** **
318
319 ----
320
321 (% style="color: rgb(255,0,0);" %)**Lecture 8 (February 10) **
322
323 (% style="color: rgb(0,0,0);" %)Naive inversion for non-square matrices. More precisely, we studied the case when A is a k x n matrix with k>n and when all the singular values of A are strictly positive. In that case there is a unique least-squares solution f0 to the equation m = Af. Further, we have f0 = inv(A^T A) A^T m.
324
325 (% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0)" %)See Sections 4.1 and 5.2 in the book Mueller-Siltanen (2012).
326
327 (% style="color: rgb(0,0,0);" %)[[image:attach:MatlabLogoSmaller.png]](% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0)" %) Matlab resources:(% style="color: rgb(0,0,0);" %)
328 \\[[attach:QuadraticTest.m]]
329
330 (% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0)" %)**[[~[~[image:attach:YouTubeLogo_smaller.png~]~]>>url:https://www.youtube.com/playlist?list=PLyIjfdC_fHWYSVIcrNtV9Hr7zAGE3GF-6||style="text-decoration: underline;" shape="rect"]]
331 **(% style="color: rgb(0,0,0);" %) (% style="color: rgb(255,0,255);" %)[[(% style="color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Here's the "video proof" about the minimum claim>>url:https://www.youtube.com/watch?v=zoszwTcw79M||shape="rect"]](%%) (% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0)" %)**
332 **(% style="color: rgb(255,0,255);" %)[[(% style="color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 1/>>url:https://www.youtube.com/watch?v=_Ur4QmeBC4Q&index=1&list=PL5vSJKR6SrD9124QEgO3_h1VPFgjG2lSQ||shape="rect"]](%%)[[(% style="color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)4 >>url:https://www.youtube.com/watch?v=_Ur4QmeBC4Q&index=1&list=PL5vSJKR6SrD9124QEgO3_h1VPFgjG2lSQ||shape="rect"]](% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0)" %)
333 (% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0); color: rgb(255, 0, 255)" %)[[(% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0); color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 2/4 (screen capture) >>url:https://www.youtube.com/watch?v=phHhQxj6EDo&list=PL5vSJKR6SrD9124QEgO3_h1VPFgjG2lSQ&index=2||shape="rect"]](% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0)" %)
334 (% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0); color: rgb(255, 0, 255); color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)[[(% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0); color: rgb(255, 0, 255); color: rgb(255, 0, 255); color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 3/4>>url:https://www.youtube.com/watch?v=4dbRl__CABk&list=PL5vSJKR6SrD9124QEgO3_h1VPFgjG2lSQ&index=3||shape="rect"]](% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0)" %)
335 (% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0); color: rgb(255, 0, 255); color: rgb(255, 0, 255); color: rgb(255, 0, 255); color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)[[(% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0); color: rgb(255, 0, 255); color: rgb(255, 0, 255); color: rgb(255, 0, 255); color: rgb(255, 0, 255); color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 4/4>>url:https://www.youtube.com/watch?v=7emS_y2KYsw&index=4&list=PL5vSJKR6SrD9124QEgO3_h1VPFgjG2lSQ||shape="rect"]]
336
337 (% style="color: rgb(255,0,0);" %)** **
338
339 ----
340
341 (% style="color: rgb(255,0,0);" %)**Lecture 7 (February 8) **
342
343 (% style="color: rgb(0,0,0);" %)Building the computational model Af=m for the 2D tomography problem. That is,
344 -designing a phantom for test cases (we went for three rectangles in empty background in the unit square), called SqPhantom.m,
345 -constructing and saving the matrix A for a selection of sizes of f and collections of X-ray projection directions,
346 -calculating and saving the Singular Value Decomposition for the matrix A.
347
348 (% style="color: rgb(0,0,0);" %)Also, we tried out Truncated Singular Value Decomposition (TSVD) for robust reconstruction.
349
350 (% style="color: rgb(0,0,0);" %)[[image:attach:MatlabLogoSmaller.png]](% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0)" %) Matlab resources:
351 [[attach:SqPhantom.m]](% style="color: rgb(0,0,0);" %)
352 [[attach:SqPhantom_plot.m]]
353 [[attach:tomodata_test.m]]
354 [[attach:tomo01_RadonMatrix_comp.m]]
355 [[attach:tomo02_firstTSVD_comp.m]]
356 [[attach:tomo02_firstTSVD_plot.m]]
357
358 (% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0)" %)
359
360
361 (% style="color: rgb(0, 0, 0); color: rgb(255, 0, 0)" %)**[[~[~[image:attach:YouTubeLogo_smaller.png~]~]>>url:https://www.youtube.com/playlist?list=PLyIjfdC_fHWYSVIcrNtV9Hr7zAGE3GF-6||style="text-decoration: underline;" shape="rect"]]**
362
363 (% style="color: rgb(255,0,255);" %)[[(% style="color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 1/5>>url:https://www.youtube.com/watch?v=m1Xzb9H8fyY&list=PLyIjfdC_fHWYSVIcrNtV9Hr7zAGE3GF-6&index=7||shape="rect"]](%%)
364 [[(% style="color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 2/5>>url:https://www.youtube.com/watch?v=LZC1kS_CDrg&list=PLyIjfdC_fHWYSVIcrNtV9Hr7zAGE3GF-6&index=8||shape="rect"]](%%)
365 [[(% style="color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 3/5 (screen capture) >>url:https://www.youtube.com/watch?v=QkIUGupT3mA&list=PLyIjfdC_fHWYSVIcrNtV9Hr7zAGE3GF-6&index=9||shape="rect"]](%%)
366 [[(% style="color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 4/5 (screen capture) 
367 >>url:https://youtu.be/JQmsPr8wSP8||shape="rect"]](%%)[[(% style="color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Lecture, part 5/5 (screen capture)>>url:https://www.youtube.com/watch?v=QVqLBMoPWZw||shape="rect"]]
368
369 ----
370
371 (% style="color: rgb(255,0,0);" %)**Lecture 6 (February 3) **
372
373 (% style="color: rgb(0,0,0);" %)
374
375 (% style="color: rgb(0,0,0);" %)Recap of the trinity
376
377 (% style="color: rgb(0,0,0);" %)(1) measurement data vector given by a device,
378 (2) continuum model of the physical measurement process, and
379 (3) computational model used in practical inversion.
380 The aim is to build two different example models, namely 1D deconvolution and 2D tomography. These two cases will be used throughout the course.
381
382 (% style="color: rgb(0,0,0);" %)Construction of a matrix model m=Af using Matlab's radon.m routine. The matrix is constructed column by column by applying radon.m consecutively to unit vectors. While computationally inefficient, this approach provides an easy way for constructing system matrices for tomographic problems. The matrix A can then be studied by, for example, singular value decomposition.
383
384 (% style="color: rgb(0,0,0);" %)[[image:attach:MatlabLogoSmaller.png]](% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0)" %) Matlab resources:
385
386 (% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0)" %)[[attach:RadonMatrix.m]]
387
388
389 (% style="color: rgb(255,0,0);" %)**[[~[~[image:attach:YouTubeLogo_smaller.png~]~]>>url:https://www.youtube.com/playlist?list=PLyIjfdC_fHWYSVIcrNtV9Hr7zAGE3GF-6||shape="rect"]]**(% style="color: rgb(255,0,255);" %)[[(% style="color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Screen capture video>>url:https://youtu.be/GBT_Ub1jsqo||shape="rect"]]
390
391 (% style="color: rgb(255,0,0);" %)** **
392
393 ----
394
395 (% style="color: rgb(255,0,0);" %)**
396 **
397
398 (% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0)" %)**Lecture 5 (February 1)**
399
400 (((
401 (% style="color: rgb(0,0,0);" %)The lecturer is down with the flu and lost his voice! So the lecture is cancelled.
402 )))
403
404 (((
405 (% style="color: rgb(0,0,0);" %)
406
407 )))
408
409 (((
410 (% style="color: rgb(0,0,0);" %)Instead of the lecture, please open the following page:
411 )))
412
413 (((
414 (% style="color: rgb(0,0,0);" %)[[http:~~/~~/www.siltanen-research.net/IPexamples/xray_tomography>>url:http://www.siltanen-research.net/IPexamples/xray_tomography||shape="rect"]]
415 )))
416
417 (% style="color: rgb(0,0,0);" %)Read these items:
418
419 * (% style="color: rgb(0,0,0);" %)Background and Applications
420 * (% style="color: rgb(0,0,0);" %)Mathematical Model of X-ray Attenuation
421 * (% style="color: rgb(0,0,0);" %)Tomographic Imaging with Full Data
422 * (% style="color: rgb(0,0,0);" %)Tomographic Imaging with Sparse Data
423
424
425 (% style="color: rgb(0,0,0);" %)The most important is "(% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0)" %)Tomographic Imaging with Sparse Data."
426
427 (% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0)" %)Also, please read Section 2.3 of the textbook (Mueller&Siltanen 2012).
428
429 (((
430 (% style="color: rgb(255,0,0);" %)** **
431
432 ----
433
434 (% style="color: rgb(255,0,0);" %)**Lecture 4 (January 27)**
435 )))
436
437 (((
438 (% style="color: rgb(0,0,0);" %)Building a simulated continuum model for 1D convolution in Matlab. See Section 2.1.1 in the book Mueller-Siltanen (2012).
439 )))
440
441 (((
442 (% style="color: rgb(0,0,0);" %)[[image:attach:MatlabLogoSmaller.png]] Matlab resources:
443 )))
444
445 (((
446 (% style="color: rgb(0,0,0);" %)[[attach:PSF.m]],
447 )))
448
449 (((
450 (% style="color: rgb(0,0,0);" %)[[attach:PSF_plot.m]],
451 )))
452
453 (((
454 (% style="color: rgb(0,0,0);" %)[[attach:targetf.m]],
455 )))
456
457 (((
458 (% style="color: rgb(0,0,0);" %)[[attach:targetf_plot.m]],
459 )))
460
461 (((
462 (% style="color: rgb(0,0,0);" %)[[attach:contmeas.m]],
463 )))
464
465 (((
466 (% style="color: rgb(0,0,0);" %)[[attach:deco01_contmodel_comp.m]],
467 )))
468
469 (((
470 (% style="color: rgb(0,0,0);" %)[[attach:deco01_contmodel_plot.m]],
471 )))
472
473 (((
474 (% style="color: rgb(0,0,0);" %)[[attach:deco02_data_comp.m]],
475 )))
476
477 (((
478 (% style="color: rgb(0,0,0);" %)[[attach:deco02_data_plot.m]]
479
480 )))
481
482 (((
483 (% style="color: rgb(0,0,0);" %)
484
485 )))
486
487 (((
488 (% style="color: rgb(0,0,0);" %)[[~[~[image:attach:YouTubeLogo_smaller.png~]~]>>url:https://www.youtube.com/playlist?list=PLyIjfdC_fHWYSVIcrNtV9Hr7zAGE3GF-6||shape="rect"]] (% style="color: rgb(0, 0, 0); color: rgb(255, 0, 255)" %)[[(% style="color: rgb(0, 0, 0); color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Screen capture video 1>>url:https://www.youtube.com/watch?v=1t5dx0bfFck&list=PLyIjfdC_fHWYSVIcrNtV9Hr7zAGE3GF-6&index=2||shape="rect"]](%%), [[(% style="color: rgb(0, 0, 0); color: rgb(255, 0, 255); color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Screen capture video 2>>url:https://www.youtube.com/watch?v=7gKY79Wo8Oc&list=PLyIjfdC_fHWYSVIcrNtV9Hr7zAGE3GF-6&index=1&t=101s||shape="rect"]](% style="color: rgb(0,0,0);" %)
489
490 )))
491
492 (((
493 (% style="color: rgb(0,0,0);" %)
494
495 )))
496
497 (((
498 (% style="color: rgb(255,0,0);" %)** **
499
500 ----
501
502 (% style="color: rgb(255,0,0);" %)**Lecture 3 (January 25)**
503 )))
504
505 (% style="color: rgb(0,0,0);" %)Quick look at the first noise-robust reconstruction method of the course: truncated singular value decomposition (TSVD).
506
507 (% style="color: rgb(0,0,0);" %)We test TSVD with the simple convolution example from Lecture 2. (% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0)" %)Book chapters: 2.1, 4.2 and 4.1.1.
508
509 (% style="color: rgb(0,0,0);" %)//First observation about TSVD~:// the reconstruction is always a linear combination of singular vectors. Singular vectors are columns of the matrix V appearing in the singular value decomposition.
510
511 (% style="color: rgb(0,0,0);" %)//Second observation about TSVD~://  if we use only very few singular values, the reconstruction is not very accurate but extremely robust against noise. This means that the reconstruction does not change much form measurement to measurement even when there is large random noise added to the data.
512
513 (% style="color: rgb(0,0,0);" %)//Third observation about TSVD~:// it is not clear how to choose the truncation index r_\alpha in general. This is indeed a hard problem, and we will see some approaches to that later in the course.
514
515 (% style="color: rgb(0,0,0);" %)Also, we started to discuss the difference between the continuum model for convolution (b(% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0)" %)ook chapter 2.1.1(% style="color: rgb(0,0,0);" %)) and discrete model for convolution (% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0)" %)(book chapter 2.1.2).
516
517 (% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0); color: rgb(0, 0, 0)" %)[[image:attach:MatlabLogoSmaller.png]]Matlab resources:
518
519 (% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0); color: rgb(0, 0, 0)" %)[[attach:SimpleDeconvolutionTSVD.m]]
520
521 (% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0); color: rgb(0, 0, 0)" %) [[~[~[image:attach:YouTubeLogo_smaller.png~]~]>>url:https://www.youtube.com/playlist?list=PLyIjfdC_fHWYSVIcrNtV9Hr7zAGE3GF-6||shape="rect"]] (% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0); color: rgb(0, 0, 0); color: rgb(255, 0, 255)" %)[[(% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0); color: rgb(0, 0, 0); color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Screen capture video 1/2>>url:https://www.youtube.com/watch?v=yDfMc6-PXmE&t=3s||shape="rect"]](%%), [[(% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0); color: rgb(0, 0, 0); color: rgb(255, 0, 255); color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Screen capture video 2/2>>url:https://www.youtube.com/watch?v=GgmdFS8LjNU||shape="rect"]]
522
523 (% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0)" %)** **
524
525 ----
526
527 (% style="color: rgb(255, 0, 0); color: rgb(255, 0, 0)" %)**Lecture 2 (January 20)**
528
529 (% style="color: rgb(0,0,0);" %)Discrete convolution between two vectors. How noise is amplified in naive inversion attempts. Book chapters: 2.1.
530
531 (% style="color: rgb(0,0,0);" %)Lecture material: [[attach:Convolution.pdf]], [[attach:1D_convolution.pdf]]
532
533 (% style="color: rgb(0,0,0);" %)[[image:attach:MatlabLogoSmaller.png]]Matlab resources:
534
535 (% style="color: rgb(0,0,0);" %)[[attach:SimpleConvolution.m]]
536
537 (% style="color: rgb(0,0,0);" %)[[~[~[image:attach:YouTubeLogo_smaller.png~]~]>>url:https://www.youtube.com/playlist?list=PLyIjfdC_fHWYSVIcrNtV9Hr7zAGE3GF-6||shape="rect"]] (% style="color: rgb(255,0,255);" %)[[(% style="color: rgb(255, 0, 255); color: rgb(255, 0, 255)" %)Screen capture video>>url:https://www.youtube.com/watch?v=lCokUeI9aCE&list=PLyIjfdC_fHWYSVIcrNtV9Hr7zAGE3GF-6&index=5&t=128s||shape="rect"]](% style="color: rgb(0,0,0);" %)
538
539 (% style="color: rgb(255,0,0);" %)** **
540
541 ----
542
543 (% style="color: rgb(255,0,0);" %)**Lecture 1 (January 18)**
544
545 (% style="color: rgb(51,51,51);" %)Introduction to inverse problems. Practicalities about the course.
546
547 (% style="color: rgb(51,51,51);" %)Lecture material: [[PDF slides>>attach:Intro_v5_jakoon.pdf]]
548
549 (% style="color: rgb(51,51,51);" %)
550
551 ----
552
553 (% style="color: rgb(51,51,51);" %)
554
555
556 == (% style="color: rgb(255,0,0);" %)**[[image:attach:Uuzi.gif]] **(%%)Project work ==
557
558 == (% style="font-size: 14.0px;" %)Sign-up form for x-ray measurements: (%%)[[https:~~/~~/beta.doodle.com/poll/96rrp5ztg2psgmwm>>url:https://beta.doodle.com/poll/96rrp5ztg2psgmwm||style="font-size: 14.0px;" shape="rect"]](% style="font-size: 14.0px;" %) (%%) ==
559
560 **Kick-off session** on March 24 (usual lecture time and place): be there to get specific project work information.
561
562 **Phase 1** review interviews will take place in the time period April 5-7. Be sure to agree on a time slot with the lecturer!
563
564 **Phase 2** final goal is the Poster session on **Friday, May 5, at 10-12 in the Exactum 1st floor hallway.**
565
566 **
567 **
568
569 Project work assistants: Alexander Meaney, Zenith Purisha and Markus Juvonen.
570
571 The idea is to study an inverse problem both theoretically and computationally in **teams of two students**. The end product is a scientific poster that the team will present in a poster session on May 5 (details above).
572
573 The poster can be printed using the laboratory's large scale printer. Please send your poster via email as a pdf attachment to Markus Juvonen by Wednesday, 3rd May, 12 pm. Then your poster will be printed in time for the poster session.
574
575 The idea of the project work is to study an inverse problem both theoretically and computationally. The classical table of contents is recommended for structuring the first phase report and the poster:
576
577 1 Introduction
578 2 Materials and methods
579 3 Results
580 4 Discussion
581
582 Section 1 should briefly explain the topic in a way accessible to a non-expert.
583 Section 2 is for describing the data and the inversion methods used.
584 In section 3 the methods of Section 2 are applied to the data described in Section 2, and the results are reported with no interpretation; just facts and outcomes of computations are described.
585 Section 4 is the place for discussing the results and drawing conclusions.
586
587 The project is either about **X-ray tomography **or** digital image processing**. You can measure a dataset yourself in the Industrial Mathematics Laboratory.
588
589 **X-ray topic:** you can choose your own object to image in the X-ray lab. We can offer a range of tried-and-tested objects and tools to tailor them to your liking. Also, you can come up with your own idea of the measured object. The size of the object should not much exceed the size of an egg, and the chemical composition is important as X-ray attenuation contrast arises from electron densities in the materials. Please contact Alexander Meaney to find out if your object is good for imaging.
590
591 **Photographic topic:** take blurred or noisy photos of suitable targets.
592
593 In the project you are supposed to apply some of the regularization methods discussed in the course. Optimally, you should have an automatic method for choosing the regularization parameter and an automatic stopping criteria for the iteration. These are both difficult requirements, so have a simple approach as plan B if a more complicated approach does not work. Also, when choosing your objects of measurement, it's good to think about mathematical models of //a priori// information (piecewise constant, smooth, piecewise smooth) as it affects the choice of the regularizer.
594
595 (% style="color: rgb(255,0,0);" %)**First goal:**(%%) the two first sections (//Introduction// and //Materials and Methods)// should be preliminary written in LaTeX, but not necessarily in poster format. The most important things to explain are:
596
597 * what kind of data to measure,
598 * what inversion method to apply for the reconstruction, and
599 * how to implement the computation.
600
601 The grade of the first goal represents 30% of the final grade of the project work. Please agree on a meeting time (in the period April 5-7) with the lecturer for reviewing and grading the first goal.
602
603 (% style="color: rgb(255,0,0);" %)**Second and final goal:**(%%) poster session. The poster will be printed in size A1. You may create your own poster (from scratch), or you can use e.g. [[this template>>url:https://wiki.helsinki.fi/download/attachments/113254781/posterA1_templ_IP2014.zip?version=1&modificationDate=1397043033121&api=v2||rel="nofollow" shape="rect"]] as a starting point and edit its layout, colors, fonts, etc. as much as you like.
604
605 You can take a look at old posters [[on this page.>>url:http://wiki.helsinki.fi/display/mathstatKurssit/Inverse+problems%2C+spring+2015||shape="rect"]]
606
607 **
608 **
609
610 ----
611
612
613
614 == Teaching schedule ==
615
616 Weeks 3-9 and 11-18, Wednesday 10-12 and Friday 10-12 in hall C123.
617
618 Easter holiday 13.-19.4.
619
620 ----
621
622 == (% style="color: rgb(255,0,0);" %)**[[image:attach:Uuzi.gif]] Exam**(%%) ==
623
624 The exam period is March 17-27. The deadline for returning it by email is at 12 o'clock noon on Monday, March 27.
625
626 (% style="color: rgb(255,0,0);" %)**Note:**(%%) Each student is expected to solve the problems and write the answers individually with **no collaboration**. A few students will be randomly picked for interviews to discuss the details of their answers.
627
628 [[attach:2017_home_exam.pdf]]
629
630 [[attach:2017_home_exam.tex]]
631
632
633
634 (% style="color: rgb(255,0,0);" %)**Note:**(%%) If you have questions about the exam, send email to the lecturer. The questions and answers will be shown here.
635
636 **Question:** Is it possible to answer the exam in Finnish? (Saako tenttiin vastata suomeksi?)
637 **Answer:** Yes. (Kyllä)
638
639 **Question:** Is the function F(s) considered for real or complex values of s in Problem 2b? (Käsitelläänkö F(s) -funktiota tehtävän 2 b-kohdassa kompleksi- vai reaalimuuttujan funktiona?)
640 **Answer:** It is considered for real values of s. (Reaalimuuttujan.)
641
642 **Question:** Can one use smaller resolution in Problem 3 if computational power is not sufficient for 128x128? (Voiko tehtävässä 3 käyttää pienempää resoluutiota, jos laskentateho ei riitä tuohon 128x128?)
643 **Answer:** Yes. (Voi.)
644
645 **Question:** In 1(a) and 1(b), should one take literally the requirement of taking as small k and n as possible? In other words, is it an optimization problem for finding the mathematically smallest possible values that yield the desired examples?
646 **Answer:** No. Use your common sense and pick smallish values for k and n in a way that gives a nice explanation.
647
648
649
650 ----
651
652
653
654 == Course material ==
655
656 The course follows this textbook:
657
658 Jennifer L Mueller and Samuli Siltanen: //Linear and Nonlinear Inverse Problems With Practical Applications// (SIAM 2012)
659
660 [[image:attach:BookCover2.png||height="250"]]
661
662 == [[Registration>>url:https://weboodi.helsinki.fi/hy/opintjakstied.jsp?html=1&Tunniste=57720||shape="rect"]] ==
663
664
665 (% style="color: rgb(96,96,96);" %)Did you forget to register? (%%)[[What to do?>>url:https://wiki.helsinki.fi/display/mathstatOpiskelu/Kysymys4||style="text-decoration: underline;" shape="rect"]]
666
667 == Exercises ==
668
669 Weekly exercises will appear here.
670
671 [[attach:Exercise6.pdf]]
672
673 [[attach:Exercise5.pdf]] (link in M3 fixed)
674
675 (% class="confluence-link" %)[[attach:Exercise4.pdf]]
676
677 (% class="confluence-link" %)[[attach:Exercise3.pdf]]
678
679 (% class="confluence-link" %) [[**Second exercise (PDF, Exercise T1(b) corrected TWICE on Feb 2, 2017)**>>attach:Exercise2_v3.pdf]]
680
681 (% class="confluence-link" %)[[attach:deco02_data_comp.m]](%%)
682 [[attach:deco02_data_plot.m]]
683
684 **Ex2 M2 example solution:** [[attach:Ex2_M2.m]] [[attach:PSF.m]] [[attach:targetf.m]]
685
686 (% style="color: rgb(128,0,0);" %)**[[First exercise (PDF)>>attach:Exercise1.pdf]]**
687
688 === Exercise classes ===
689
690 Teaching assistants:
691
692 [[Minh Nguyet Mach>>url:https://www.helsinki.fi/en/researchgroups/inverse-problems/people/postdoctoral-researchers#section-17661||shape="rect"]]
693
694 Santeri Kaupinmäki (firstname.lastname@helsinki.fi)
695
696
697
698 (% class="wrapped" %)
699 |=(((
700 Group
701 )))|=(((
702 Day
703 )))|=(((
704 Time
705 )))|=(((
706 Room
707 )))|=(% colspan="1" %)(((
708 Instructor
709 )))
710 |(((
711 1.
712 )))|(((
713 Wednesday
714 )))|(((
715 12:15 - 14:00
716 )))|(((
717 C128
718 )))|(% colspan="1" %)(((
719
720 )))
721 |(% colspan="1" %)(((
722 2.
723 )))|(% colspan="1" %)(((
724 Thursday
725 )))|(% colspan="1" %)(((
726 8:15 - 10:00
727 )))|(% colspan="1" %)(((
728 C128
729 )))|(% colspan="1" %)(((
730
731 )))
732
733 The weekly problem sets will be graded at the exercise sessions, thus attendance at one of the weekly exercise sessions is required in order for your work to be graded. In case of scheduling conflicts please contact one of the teaching assistants; we can make special arrangements for those who are unable to attend either of the exercise sessions.
734
735 The theoretical exercises are to be presented by the students. In each session the students mark the exercises they have completed and according to this list one student will be chosen per exercise to present their work.
736
737 The Matlab exercises will be checked individually at the exercise class.
738
739 Points for all completed exercises will be accumulated and part of the final grade.
740
741 == Course feedback ==
742
743 Course feedback can be given at any point during the course. Click [[here>>url:https://elomake.helsinki.fi/lomakkeet/11954/lomake.html||style="line-height: 1.4285;" shape="rect"]].