Primera Division Apertura Final Stage stats & predictions
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The Thrill of the Primera Division Apertura Final Stage
The Costa Rican Primera Division Apertura Final Stage is set to captivate football fans with its high-stakes matches tomorrow. As teams vie for supremacy, expert betting predictions offer insights into potential outcomes, adding an extra layer of excitement for enthusiasts and bettors alike. This guide delves into the key matchups, team analyses, and strategic considerations that could influence the results.
Key Matchups and Team Analyses
Tomorrow's schedule features several pivotal clashes that could determine the fate of the league. Each team brings its unique strengths and challenges to the field, making these encounters particularly thrilling.
Saprissa vs. Herediano
- Saprissa: Known for their robust defense and tactical discipline, Saprissa enters this match with a strong track record in previous encounters against Herediano.
- Herediano: With a dynamic attacking lineup, Herediano aims to exploit any defensive gaps left by Saprissa. Their recent form suggests they are well-prepared for this challenge.
Limonense vs. Alajuelense
- Limonense: Limonense has shown remarkable resilience in recent games, often turning matches around with their tenacious midfield play.
- Alajuelense: Alajuelense's experience in high-pressure situations makes them formidable opponents. Their strategic gameplay could be decisive in this matchup.
Betting Predictions: Expert Insights
Betting experts have analyzed various factors influencing tomorrow's matches, including team form, head-to-head records, and player performances. Here are some expert predictions:
Predictions for Saprissa vs. Herediano
- Prediction: A closely contested match with a slight edge to Saprissa due to their home advantage and defensive prowess.
- Betting Tip: Consider backing Saprissa to win or a draw at competitive odds.
Predictions for Limonense vs. Alajuelense
- Prediction: Alajuelense is favored to secure a narrow victory, leveraging their tactical acumen and experienced squad.
- Betting Tip: A bet on Alajuelense winning by a one-goal margin could yield attractive returns.
Tactical Considerations and Player Performances
The outcome of these matches may hinge on key tactical decisions and standout player performances. Let's explore some critical aspects:
Saprissa's Defensive Strategy
- Saprissa's ability to maintain a solid defensive line will be crucial against Herediano's potent attack.
- The role of their central defenders in neutralizing Herediano's strikers could be decisive.
Herediano's Offensive Tactics
- Herediano will likely focus on quick counter-attacks to exploit any lapses in Saprissa's defense.
- The performance of their leading forwards will be pivotal in breaking through Saprissa's backline.
Limonense's Midfield Battle
- Limonense's midfielders will need to control the tempo of the game against Alajuelense's disciplined setup.
- A strong showing from their playmakers could disrupt Alajuelense's rhythm and create scoring opportunities.
Alajuelense's Experienced Playmakers
- Alajuelense relies heavily on their seasoned midfielders to orchestrate attacks and dictate play.
- Their ability to deliver precise passes under pressure will be key to unlocking Limonense's defense.
In-Depth Analysis: Factors Influencing Outcomes
Injury Concerns and Squad Depth
Injuries can significantly impact team dynamics. Both Saprissa and Herediano must manage any injury concerns effectively to maintain their competitive edge. Similarly, Limonese[0]: # Copyright (c) Facebook, Inc. and its affiliates.
[1]: #
[2]: # This source code is licensed under the MIT license found in the
[3]: # LICENSE file in the root directory of this source tree.
[4]: import torch
[5]: from fairseq import utils
[6]: from fairseq.criterions import FairseqCriterion
[7]: class MMDLoss(FairseqCriterion):
[8]: def __init__(self, task):
[9]: super().__init__(task)
[10]: self.eps = task.args.mmd_eps
[11]: @staticmethod
[12]: def add_args(parser):
[13]: """Add criterion-specific arguments to the parser."""
[14]: # fmt: off
[15]: parser.add_argument('--mmd-loss', action='store_true',
[16]: help='add mmd loss')
[17]: parser.add_argument('--mmd-weight', type=float, default=0,
[18]: help='weight for mmd loss')
[19]: parser.add_argument('--mmd-eps', type=float, default=1e-5,
[20]: help='epsilon value added inside sqrt')
[21]: # fmt: on
[22]: def forward(self, model, sample=None):
"""
mmd_loss = self.mmd_loss(model.module.model.encoder.embed_tokens.weight[model.padding_idx],
model.module.model.decoder.embed_tokens.weight[model.padding_idx])
loss = mmd_loss
sample_size = sample['target'].size(0) if self.args.sentence_avg else sample['ntokens']
logging_output = {
'loss': utils.item(loss.data) if reduce else loss.data,
'ntokens': sample['ntokens'],
'nsentences': sample['target'].size(0),
'sample_size': sample_size,
}
return loss, sample_size, logging_output
"""
def mmd_loss(self,genuine,synth):
r"""Calculate MMD loss
Args:
genuine (torch.Tensor): tensor containing genuine signatures (from dataset).
synth (torch.Tensor): tensor containing synthetic signatures
(from current batch).
Returns:
mmd_loss (torch.Tensor) : calculated loss
"""
autograd_flag=True
alpha = torch.rand(genuine.shape)
alpha = alpha.cuda() if genuine.is_cuda else alpha
batches = zip(genuine.split(100000,dim=1),synth.split(100000,dim=1))
pos_term,sig_term = [],[]
for xb,yb in batches:
def reduce_metrics(logging_outputs) -> None:
"""
num_sentences=sum(log.get('nsentences',0) for log in logging_outputs)
num_words=sum(log.get('nwords',0) for log in logging_outputs)
sample_size=sum(log.get('sample_size',0) for log in logging_outputs)
num_valid=(sample_size==num_sentences)
if num_valid:
metrics.log_scalar('mcd-loss-valid', sum(log.get('mcd-loss',0) for log in logging_outputs)/num_sentences , sample_size , round=3)
metrics.log_scalar('accuracy-valid', sum(log.get('accuracy-token-valid',0) for log in logging_outputs)/num_words , num_words , round=3)
metrics.log_scalar('wps-valid', sum(log.get('wps-token-valid',0) for log in logging_outputs)/num_words , num_words , round=3)
if len(logging_outputs)>0:
metrics.log_scalar('mcd-loss-train-gpu', sum(log.get('mcd-loss-gpu',0) for log in logging_outputs)/sample_size )
metrics.log_scalar('accuracy-token-train-gpu', sum(log.get('accuracy-token-train-gpu',0) for log in logging_outputs)/sample_size )
metrics.log_scalar('wps-token-train-gpu', sum(log.get('wps-token-train-gpu',0)for log in logging_outputs)/sample_size )
"""
***** Tag Data *****
ID: 1
description: Implementation of MMD Loss calculation which involves complex tensor
operations.
start line: 7
end line: 91
dependencies:
- type: Class
name: MMDLoss
start line: 7
end line: 91
context description: This snippet implements an advanced method `mmd_loss` which calculates
Maximum Mean Discrepancy (MMD). The method involves random sampling using PyTorch,
splitting tensors into batches based on certain conditions which are nontrivial.
algorithmic depth: 5
algorithmic depth external: N
obscurity: 5
advanced coding concepts: 5
interesting for students: 5
self contained: N
*************
## Suggestions for complexity
1. **Adaptive Batch Size**: Modify `mmd_loss` function so that it dynamically adjusts batch sizes based on available GPU memory.
2. **Custom Kernel Function**: Implement support within `mmd_loss` method to allow users to define custom kernel functions instead of relying on default ones.
3. **Mixed Precision Support**: Integrate mixed precision training support within `MMDLoss` class using PyTorch’s AMP (Automatic Mixed Precision).
4. **Multi-GPU Support**: Enhance `MMDLoss` class methods so that they can efficiently run across multiple GPUs.
5. **Real-Time Logging**: Add real-time logging capabilities within `forward` method which logs intermediate values during training without significant overhead.
## Conversation
<|user|>I'm trying to implement adaptive batch sizes within my `mmd_loss` function based on available GPU memory instead of fixed sizes like what I have now [SNIPPET]. Can you guide me through it?<|end|>
<|assistant|>Sure! To implement adaptive batch sizes based on available GPU memory within your `mmd_loss` function:
1. First, you need a way to query available GPU memory.
2. Then adjust your batching logic accordingly.
Here’s how you can modify your code:
python
import torch.cuda as cuda
class MMDLoss(FairseqCriterion):
...
def get_available_memory(self):
total_memory = cuda.get_device_properties(0).total_memory / (1024 ** 2)
reserved_memory = cuda.memory_reserved(0) / (1024 **2)
allocated_memory = cuda.memory_allocated(0) / (1024 **2)
free_memory = total_memory - reserved_memory - allocated_memory
return free_memory
def adaptive_split(self,tensor):
free_mem_mb = self.get_available_memory()
element_size_mb = tensor.element_size() * tensor.nelement() / (1024 **2)
max_batch_elements = int(free_mem_mb // element_size_mb * len(tensor))
return max_batch_elements
def mmd_loss(self,genuine,synth):
...
max_batch_elements_genuine = self.adaptive_split(genuine[:, :, :])
max_batch_elements_synth = self.adaptive_split(synth[:, :, :])
min_batch_elements=max(min(max_batch_elements_genuine,max_batch_elements_synth),100000)
# Continue with your existing logic but use min_batch_elements instead of fixed size.
This should adaptively split your tensors based on available GPU memory.<|end|>
<|user|>Great! Now I want users be able define custom kernel functions within my `mmd_loss`. How do I go about integrating that?<|end|>
<|assistant|>To allow users define custom kernel functions within your `mmd_loss`, you can pass an optional kernel function parameter when initializing your class or directly calling `mmd_loss`. Here’s how you can integrate it:
1. Add an optional parameter during initialization or function call.
2. Use this parameter inside your computation.
Here’s how you might modify your code:
python
class MMDLoss(FairseqCriterion):
...
def __init__(self, task=None, kernel_fn=None):
super().__init__(task)
self.eps = task.args.mmd_eps if task else None
self.kernel_fn = kernel_fn if kernel_fn else self.default_kernel
@staticmethod
def default_kernel(x,y):
# Example Gaussian Kernel implementation
sigma_square=1
x_y_dist=torch.sum((x-y)**2,dim=-1).view(-1)
norm_factor=torch.exp(-x_y_dist/sigma_square)
return norm_factor
def mmd_loss(self,genuine,synth):
...
pos_term,sig_term=[],[]
# Using custom kernel function provided by user or default one
k_xx=k_gg=self.kernel_fn(xb_xb,yb_yb)+self.eps
k_xy=k_gu=self.kernel_fn(xb_xb,yb_yb)+self.eps
# Continue using k_xx,k_xy etc., as per original logic but now with user-defined kernels possible.
Now users can pass their own kernel functions while initializing or calling `mmd_loss`.
1] == True:
print("Detected vertical tile")
continue
f.write(f"{tile.x},{tile.y},")
f.write(f"{tile.width},{tile.height},")
f.write(f"{int(tile.is_horizontal())},{int(tile.is_vertical())}n")
def process_tile_file(filename="tiles.txt"):
global tiles_data_file_name_set
if filename not in tiles_data_file_name_set:
tiles_data_file_name_set.add(filename)
try:
with open(filename,'r') as f:
lines=f.readlines()
n=int(lines.pop(0))
all_tiles=[]
x_max=99999999999999; y_max=x_max;
x_min=-x_max; y_min=x_min;
new_tiles=[]
i=lines.index("n")
lines.pop(i); i-=1;
while i>=0:
x_start=int(lines[i].split(",")[0])
y_start=int(lines[i].split(",")[1])
width=int(lines[i].split(",")[2])
height=int(lines[i].split(",")[3])
is_horizontal=int(lines[i].split(",")[4])!=False;
is_vertical=int(lines[i].split(",")[5])!=False;
tile=tiles[x_start][y_start]
tile.width=width; tile.height=height;
tile.set_is_horizontal(is_horizontal);
tile.set_is_vertical(is_vertical);
all_tiles.append(tile);
if x_start+width>x_max:x_max=x_start+width;
if y_start+height>y_max:y_max=y_start+height;
new_tiles.append(tile);
i-=1;
new_tiles.reverse();
generate_all_rotations(new_tiles);
while True:
merge_adjacent(new_tiles,x_max,y_max);
merge_diagonal(new_tiles,x_max,y_max);
n=len(new_tiles);
if n==len(all_tiles):break;
all_tiles=new_tiles.copy();
new_tiles.clear();
save_new_tile_image(all_tiles,x_min,y_min,x_max-x_min+1,y_max-y_min+1,"output.png");
save_new_tile_info(all_tiles,x_min,y_min,x_max-x_min+1,y_max-y_min+1,filename);
except Exception as e:
print(f"Error processing file {filename}: {e}")
### Solution Explanation:
The solution introduces several enhancements over the initial code snippet:
- **Dynamic File Handling:** By maintaining a set (`tiles_data_file_name_set`) tracking processed files ensures no redundant processing occurs even when files are added dynamically during execution.
- **Enhanced Merge Logic:** The merging functions (`merge_adjacent` & `merge_diagonal`) now consider diagonal adjacency as well as horizontal/vertical adjacency while merging tiles.
- **Robust Error Handling:** Comprehensive error handling ensures graceful recovery from potential issues such as file read errors or invalid data formats.
### Follow-up exercise:
**Exercise Description**
Extend your implementation further by introducing parallel processing capabilities where each file is processed concurrently without interfering with others' states.
**Requirements**:
- Use Python’s concurrency libraries (`concurrent.futures`, threading/multiprocessing).
- Ensure thread safety when accessing shared resources such as global variables or file handles.
- Optimize performance by balancing between CPU-bound tasks like image generation & IO-bound tasks like reading/writing files.
### Solution Outline:
python
import concurrent.futures
def process_tile_file_concurrently(file_list):
global tiles_data_file_name_set
with concurrent.futures.ThreadPoolExecutor() as executor:
futures=[executor.submit(process_tile_file,filename=fname) for fname in file_list]
concurrent.futures.wait(futures)
# Assuming file_list contains list of filenames needing processing.
process_tile_file_concurrently(file_list=["file_01.txt", "file_02.txt", ...])
This solution leverages threading via ThreadPoolExecutor ensuring each file gets processed concurrently while managing shared state safely through thread-safe mechanisms where necessary.
By implementing these exercises progressively layered upon each other students gain deeper insights into concurrency handling alongside complex algorithmic implementations involving image processing tasks specific nuances highlighted initially.
***** Tag Data *****
ID: 6
description: The main loop that merges adjacent tiles until no further merges are possible,
start line:126,end line:-9 This loop continuously merges adjacent tiles until no more merges can occur,
start line:-10,end line:-9 It uses both horizontal/vertical merging strategies described earlier,
start line:-10,end line:-9 The loop terminates once no more changes are detected after an iteration,
start line:-11,end line:-9 demonstrating iterative optimization techniques.
algorithmic depth:5 algorithmic depth external:'N'
obscurity:'N'
advanced coding concepts:'N'
interesting for students:'N'
self contained:'N'
************
## Challenging aspects
### Challenging aspects in above code
The given snippet outlines an iterative process designed specifically around merging adjacent elements until no further merges can occur between iterations—a non-trivial operation involving multiple layers of complexity:
#### Algorithmic Depth & Logical Complexity
* **Convergence Detection:** The loop continues until no more changes are detected between iterations (`if n==len(new_tiles): break`). Detecting convergence accurately requires careful consideration because missing even one change means potentially infinite looping or premature termination.
* **State Management:** Properly managing state between iterations (`all_tiles`, `new_tiles`) requires attention since any inconsistency here could lead either to incorrect results or infinite loops.
* **Tile Transformation:** The transformation logic (`generate_all_rotations`) adds another layer where every possible rotation needs generating correctly before attempting merges again—this must handle geometric transformations precisely without losing data integrity or causing overlaps/errors.
* **Merge Logic:** The actual merge logic itself is abstracted away but must handle complex scenarios such as overlapping areas correctly without corrupting data structures (`merge_adjacent`, `merge_diagonal`). Ensuring these methods work seamlessly together increases complexity significantly.
#### Extension Possibilities
Given these complexities already present within the provided snippet framework—here are ways we might extend them specifically tailored rather than generic extensions like multi-thread safety:
* **Hierarchical Merging:** Introduce hierarchical levels where certain types of merges have precedence over others depending on additional properties/tiers assigned dynamically during runtime—requiring dynamic priority management.
* **Dynamic Tile Properties:** Allow tiles themselves properties that evolve over time depending on interactions—such properties might affect whether they should merge at all times leading up till convergence detection becomes more sophisticated than just length comparison.
* **Complex Merge Conditions:** Extend merge conditions beyond simple adjacency—perhaps introduce constraints based on color gradients along edges requiring gradient-based checking algorithms before deciding whether two tiles should merge.
## Exercise
### Problem Statement
You're tasked with enhancing a given algorithm designed around iterative merging processes until convergence is achieved among tile-like objects arranged spatially according specific ruleset defined below:
#### Initial Setup [SNIPPET]:
python
while True:
merge_adjacent(new_tiles,x_max,y_max);
merge_diagonal(new_tiles,x_max,y_max);
n=len(new_tiles);
if n==len(all_tiles):break;
all_tiles=new_tiles.copy();
new_tiles.clear();
generate_all_rotations(new_tiles);
#### Requirements:
Your goal is threefold:
##### Part A – Hierarchical Precedence System
Implement hierarchical precedence levels among different types of merges—adjacency takes precedence over diagonal unless specific conditions met dynamically determined at runtime.
##### Part B – Dynamic Tile Properties Evolution
Each tile possesses evolving properties affecting its eligibility/ability/preference towards merging operations at different stages throughout iterations.
##### Part C – Gradient-Based Merge Constraints
Introduce gradient-based constraints along edges determining whether two adjacent/diagonal tiles should merge.
#### Constraints & Assumptions:
* Assume initial setup includes predefined list/array/tile object structures holding spatial coordinates plus basic attributes necessary starting point.
* You must ensure robustness handling edge cases such as empty lists/nested hierarchies properly converging without entering infinite loops.
## Solution
### Part A – Hierarchical Precedence System Implementation
python
def hierarchical_merge(tiles,xmax,ymax):
hierarchy_levels={'adjacency':[],'diagonal':[]}
adjacency_merge(tiles,hierarchy_levels['adjacency'],xmax,ymax);
diagonal_merge(tiles,hierarchy_levels['diagonal'],xmax,ymax);
combined_merges=hierarchy_levels['adjacency']+ hierarchy_levels['diagonal']
return combined_merges
def adjacency_merge(tiles,hierarchy_level,xmax,ymax):
#implement adjacency-specific merging logic here updating hierarchy_level
def diagonal_merge(tiles,hierarchy_level,xmax,ymax):
#implement diagonal-specific merging logic here updating hierarchy_level
while True:
new_combined_merges=hierarchical_merge(new_tles,xmax ymax);
n=len(new_combined_merges);
if n==len(all_tles):break;
all_tles=new_combined_merges.copy();
new_tles.clear();
generate_all_rotations(new_tles);
### Part B – Dynamic Tile Properties Evolution Implementation
python
class Tile(object):
def __init__(self,id,pos_x,pos_y,size,color,**properties):
self.id=id;
self.pos_x=pos_x;
self.pos_y=pos_y;
self.size=size;
self.color=color;
self.properties={};
for prop,valueinproperties.items():
setattr(self,f'prop_{prop}',value)
def update_properties(tile,timestep):
#implement property evolution logic here considering timestep
while True :
for tle_innew_tles:update_properties(tle,time_step++)
new_combined_merges=hierarchical_merge(new_tles,xmax ymax);
n=len(new_combined_merges);
if n==len(all_tles):break;
all_tles=new_combined_merges.copy();
new_tles.clear();
generate_all_rotations(new_tles);
### Part C – Gradient-Based Merge Constraints Implementation
python
def check_gradient_constraint(tile_a,tile_b,axis='x'):
gradient_threshold=10
a_gradient=get_gradient(tile_a,axis); b_gradient=get_gradient(tile_b,axis);
return abs(a_gradient-b_gradient)<=gradient_threshold
def get_gradient(tile,axis='x'):
#compute gradient along specified axis returning numerical value representing gradient magnitude
while True :
for tle_innew_tles:update_properties(tle,time_step++)
new_combined_merges=[] ;
for tle_innew_tles:hierarchical_merge([tle],xmax ymax)[check_gradient_constraint(prev_tile,tle,'x')and check_gradient_constraint(prev_tile,tle,'y')]append_to_new_combined_merges
n=len(new_combined_merges);
if n==len(all_tles):break;
all_tles=new_combined_merges.copy();
new_tles.clear();
generate_all_rotations(new_tles);
## Follow-up exercise
Consider extending functionality further:
##### Scenario A – Time-Varying Gradient Thresholds
Introduce time-varying gradient thresholds dependent upon external parameters varying throughout iterations requiring dynamic adjustment mid-process.
##### Scenario B – Multi-Dimensional Hierarchical Levels
Expand hierarchical system beyond binary levels allowing multi-tier prioritization influencing decision-making processes more granularly.
***** Tag Data *****
ID: '8'
description: Generate all rotations required by comparing rotations between consecutive/overlapping/unmerged/potential-to-be merged-slices/tiles,
start line:-6,end line:-6 This snippet generates all possible rotations needed when comparing two consecutive/unmerged slices/tiles/potential-to-be merged-slices/tiles.,
start line:-7,end line:-6 It includes detailed checks ensuring proper alignment before rotation.,
start line:-8,end line:-6 Making sure only necessary rotations are generated adds complexity.,
start is involved advanced geometry manipulation techniques including checking alignments before performing rotations.,
algorithmic depth :5 algorithmic depth external:'N'
obscurity :5 advanced coding concepts :5 interesting students :'N'
self contained :'Y'
************
## Challenging aspects
The provided code snippet deals with generating necessary rotations when comparing two unmerged slices/tiles/potential-to-be merged-slices/tiles under certain conditions involving geometric alignment checks before performing rotations. Here are some challenging aspects inherent to this problem:
### Challenging aspects present above include:
1. **Geometric Alignment Checks**: Ensuring proper alignment before performing rotations requires understanding geometric transformations deeply including translation vectors relative positions between slices/tiles/potential-to-be merged-slices/tiles etc., which involves vector mathematics precision especially when dealing with floating-point arithmetic issues.
2. **Conditional Rotation Generation**: Generating only necessary rotations implies conditional checks which involve understanding not only how individual slices align but also ensuring computational efficiency by avoiding redundant calculations - making sure only essential computations take place based on input conditions dynamically changing during execution time which adds logical complexity due branching paths involved contingent upon input variations making debugging harder especially when dealing with floating-point inaccuracies resulting from geometric transformations etc., .
3. **Handling Unmerged/Sliced States**: Managing unmerged/sliced states necessitates maintaining state information accurately across multiple iterations/slice comparisons adding another layer logical complexity due keeping track intermediate states ensuring consistency throughout entire process flow leading final desired output correct manner despite ongoing state transitions etc., .
### Extension ideas specific above logic include adding nuances such as :
- Implementing multi-step rotation sequences where intermediate steps depend recursively upon previous steps thereby increasing overall sequence length requiring additional recursive checks making problem solving even harder due nested dependencies introduced thereby increasing overall computational complexity .
- Extending functionality supporting arbitrary dimensional spaces beyond typical planar geometries thus requiring handling higher-dimensional rotational matrices/quaternions generalizing existing approach thereby adding further abstraction layers inherently increasing difficulty level comprehensively .
## Exercise
Your task is two-fold involving both extending functionality described above followed implementing comprehensive solution encompassing full scope requirements outlined hereinbelow :
**Part A - Extend Functionality**
Based upon [SNIPPET] provided extend functionality incorporating following additional requirements :
a) Implement support handling arbitrary dimensional spaces beyond typical planar geometries thus requiring handling higher-dimensional rotational matrices/quaternions generalizing existing approach thereby adding further abstraction layers inherently increasing difficulty level comprehensively .
b) Incorporate multi-step rotation sequences where intermediate steps depend recursively upon previous steps thereby increasing overall sequence length requiring additional recursive checks making problem solving even harder due nested dependencies introduced thereby increasing overall computational complexity .
c ) Ensure efficient computational performance avoiding redundant calculations via conditional checks thus maintaining optimal efficiency especially under large datasets containing numerous slice comparisons necessitating thorough understanding dynamic programming principles alongside minimizing unnecessary computations wherever feasible .
**Part B - Full Solution Implementation**
Develop comprehensive solution implementing extended functionality encompassing full scope requirements specified above leveraging appropriate mathematical libraries/tools aiding matrix/quaternion manipulations geometric transformations conditional optimizations etc., . Provide detailed comments/docstrings explaining rationale behind design choices made optimizing clarity readability maintainability enabling seamless understanding ease comprehension facilitating future enhancements modifications extensions efficiently .
## Solution
Below provides fully detailed solution addressing both parts aforementioned :
python
import numpy as np
class GeometricTransformer:
def __init__(self):
pass
@staticmethod
def compute_rotation_matrix(angle_axis_pair):
angle_rad = np.radians(angle_axis_pair["angle"])
axis_normed=np.array(angle_axis_pair["axis"])/np.linalg.norm(angle_axis_pair["axis"])
cos_theta=np.cos(angle_rad)
sin_theta=np.sin(angle_rad)
ux_ux=np.outer(axis_normed,axis_normed)*(cos_theta - cos_theta*cos_theta + sin_theta*sin_theta )
ux_cross=np.array([[ 0,-axis_normed[-1],axis_normed[-2]],
[ axis_normed[-1], 0,-axis_normed[-3]],
[-axis_normed[-2], axis_normed[-3], 0]])
identity_matrix=np.eye(len(axis_normed))
rot_matrix=(identity_matrix * cos_theta)+(ux_ux + sin_theta * ux_cross )
return rot_matrix
@staticmethod
def rotate_point(point_vector,rmat):
return np.dot(rmat,np.array(point_vector))
def generate_necessary_rotations(slice_a,slice_b,dimensionality_check=True,multi_step=False):
assert dimensionality_check,"Dimensionality check failed"
angles_axes=[{"angle":90,"axis":[i,j]}
for i,j,k,l
in [(k,l,j,i),(l,k,i,j)]
for k,l,i,j
in [(dimensionality_check,dimensionality_check,dimensionality_check,dimensionality_check)]
]
necessary_rotations=[]
slice_diff=slice_b["center"] - slice_a["center"]
aligned=False
## Check initial alignment ##
try :
initial_alignment=slice_diff-np.round(slice_diff).astype(int)==np.zeros_like(slice_diff).astype(bool)
aligned=all(initial_alignment.tolist())
except ValueError :
aligned=False
## Compute Necessary Rotations ##
while not aligned :
best_angle_axis=None
best_error=float("inf")
## Try Each Angle-Axis Pair ##
iter_angles_axes=list(enumerate(angles_axes))
## Iterate Over All Angle-Axis Pairs ##
rotated_center=np.array(slice_a["center"])
## Initialize Minimum Error Tracking ##
min_error=float("inf")
## Iterate Over All Possible Rotations For Current Angle-Axis Pair ##
iter_angles_axes=list(range(len(rotated_center)))
## Update Minimum Error Tracking If Current Rotation Yields Lower Error ##
rotated_center=self.rotate_point(rotated_center,self.compute_rotation_matrix({"angle":angles_axes[idx]["angle"],"axis":angles_axes[idx]["axis"]}))
error=np.linalg.norm(rotated_center-slice_b["center"])
if error
